EFFECTS OF CORROSION AND VEHICULAR IMPACT
DAMAGE ON ULTIMATE CAPACITY
OF STEEL BRIDGE BEAMS
By
AARON MICHAEL FINLEY
Bachelor of Science
Oklahoma State University
Stillwater, Oklahoma
2004
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the degree of
MASTER OF SCIENCE
July, 2006
ii
EFFECTS OF CORROSION AND VEHICULAR IMPACT
DAMAGE ON ULTIMATE CAPACITY
OF STEEL BRIDGE BEAMS
Thesis Approved:
Dr. Charles M. Bowen, Ph.D.
Thesis Adviser
Dr. Robert N. Emerson, Ph.D.
Dr. G. Steven Gipson, Ph.D.
Dr. A. Gordon Emslie, Ph.D.
Dean of the Graduate College
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TABLE OF CONTENTS
CHAPTER1: INTRODUCTION AND BACKGROUND............................................ 1
1.1 INTRODUCTION.......................................................................................................... 1
1.2 BACKGROUND........................................................................................................... 1
1.3 OBJECTIVES AND SCOPE............................................................................................ 4
1.4 COMPUTER SIMULATION AND ANALYSIS .................................................................. 5
1.5 HAND CALCULATIONS............................................................................................... 8
CHAPTER 2: LITERATURE REVIEW....................................................................... 9
CHAPTER 3: LATERAL TORSIONAL BUCKLING OF CORROSION
DAMAGED MEMBERS................................................................................................ 14
3.1 SETUP AND TEST CASES .......................................................................................... 14
3.2 HOLES IN WEB ........................................................................................................ 19
3.2.1: Hole Depth..................................................................................................... 21
3.2.2: Hole Length ................................................................................................... 22
3.3 FLANGE THINNING .................................................................................................. 24
3.3.1: Full Length, Partial Width Flange Thinning.................................................. 24
3.3.2: Full Width, Partial Length Flange Thinning.................................................. 27
3.3.3: Web Holes and Flange Damage..................................................................... 29
3.3.4: Two Continuous Spans: Full Length, Partial Width Flange Thinning ......... 30
CHAPTER 4: FLEXURAL STRESS DISTRIBUTION IN CORROSION-DAMAGED
MEMBERS................................................................................................ 32
4.1 SETUP AND TEST CASES .......................................................................................... 33
4.1.1: Case 1, Concentrated Load ............................................................................ 34
4.1.2: Case 2, Uniformly Distributed Load.............................................................. 37
4.2 BASIC VIERENDEEL ANALYSIS................................................................................ 38
4.2.1: Example of finite element results vs. Vierendeel Calculations ..................... 43
4.2.2: Example Vierendeel Stress Calculations ....................................................... 45
4.3 SIMPLE SPANS WITH HOLES IN WEBS ...................................................................... 46
4.3.1: Set 1—12”x 12” Vertically Centered Hole at Quarterspan, Concentrated Load
at Midspan................................................................................................................. 46
4.3.2: Set 2—Preliminary Investigation of Vierendeel Applicability vs. Hole Size,
Holes at Quarterspan, Concentrated Load at Midspan ............................................. 51
4.3.3: Set 3—Four-Point Loading, Hole in No-Shear Region................................. 64
4.3.4: Set 4— Partial Length Uniformly Distributed Load, Hole at Midspan......... 68
4.3.5: Investigation of Stress Shift........................................................................... 70
4.3.6: Vierendeel Method Applicability .................................................................. 80
4.3.7: Stress Increase at Hole Corner, Inside Hole Edge ......................................... 87
4.3.8: Vertically Shifted Holes................................................................................. 89
CHAPTER 5: PLASTIC MOMENT CAPACITY ..................................................... 94
5.1 VERTICALLY CENTERED HOLES.............................................................................. 94
5.2 VERTICALLY ECCENTRIC HOLES............................................................................. 96
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CHAPTER 6: IMPACT DAMAGE ........................................................................... 101
CHAPTER 7: STRESS ANALYSIS WITH MISSING BEAM............................... 106
7.1 BRIDGE MODEL..................................................................................................... 106
7.2 LOADING CONDITIONS .......................................................................................... 107
7.3 FLEXURAL STRESSES IN BEAMS ............................................................................ 108
7.3.1: All Beams Present........................................................................................ 109
7.3.2: Beam 1 Removed......................................................................................... 110
7.3.3: Beam 4 Removed......................................................................................... 111
CHAPTER 8: CONCLUSIONS ................................................................................. 114
8.1 INTRODUCTION...................................................................................................... 114
8.2: FLANGE THINNING ............................................................................................... 115
8.3: WEB HOLES ......................................................................................................... 115
8.3.1: Buckling Capacity........................................................................................ 115
8.3.2: Web Holes and Flexural Stress Distribution................................................ 116
8.3.3: Web Holes and Plastic Moment Capacity (Mp) .......................................... 117
8.4: IMPACT DAMAGE ................................................................................................. 117
8.5: BRIDGE DECK CAPACITY LOSS DUE TO AN INCAPACITATED MEMBER ................ 118
8.6 FUTURE RESEARCH ............................................................................................... 118
BIBLIOGRAPHY......................................................................................................... 121
APPENDIX A: FINITE ELEMENT VS. VIERENDEEL PREDICTIONS FOR
STRESS SHIFT, BEAM SETS B AND C (SECTION 4.3.5).................................... 123
APPENDIX B: DIFFERENCE BETWEEN VIERENDEEL AND FINITE
ELEMENT, BEAM THEORY AND FINITE ELEMENT MAXIMUM STRESSES
AT INSIDE HOLE EDGE (SECTION 3.3.6)............................................................. 125
APPENDIX C: DIFFERENCE BETWEEN ABAQUS AND BEAM THEORY
FLEXURAL STRESS PREDICTIONS AT INSIDE HOLE EDGE, NEXT TO
HOLE............................................................................................................................ 129
APPENDIX D: MP LOSS VS. HOLE SIZE, BEAM SETS B AND C .................... 131
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LIST OF TABLES
TABLE 3.1: TEST CASES WITH STANDARD BOUNDARY CONDITIONS, LOAD AT NEUTRAL AXIS....................16
TABLE 3.2: TEST CASES WITH COMMONLY USED BOUNDARY CONDITIONS, LOAD AT TOP FLANGE ............17
TABLE 3.2: TEST CASES WITH MODIFIED BOUNDARY CONDITIONS, LOAD AT TOP FLANGE.........................19
TABLE 3.3: LTB CAPACITY WITH INCREASING HOLE DEPTH........................................................................21
TABLE 3.4: LTB CAPACITY WITH INCREASING HOLE LENGTH......................................................................22
TABLE 3.5: LTB CAPACITY UNDER UNIFORMLY DISTRIBUTED LOAD..........................................................23
TABLE 3.6: LTB CAPACITY WITH FULL LENGTH, PARTIAL WIDTH FLANGE THINNING ................................25
TABLE 3.8: LTB CAPACITY WITH FULL WIDTH, PARTIAL LENGTH FLANGE THINNING ................................27
TABLE 3.9: LTB CAPACITY WHEN WEB HOLES AND FLANGE THINNING BOTH PRESENT ............................29
TABLE 3.10: LTB CAPACITY FOR TWO CONTINUOUS SPANS, FULL LENGTH PARTIAL WIDTH FLANGE
THINNING ............................................................................................................................................30
TABLE 4.1: TOPICS OF CHAPTER 4.................................................................................................................33
TABLE 4.2: “Y” VALUES FOR SHEAR-INDUCED MOMENT IN SAMPLE BEAM .................................................72
TABLE 4.3: FIRST SERIES OF BEAMS TESTED FOR STRESS SHIFT (SET A)......................................................73
TABLE 4.4: SECOND SERIES OF BEAMS TESTED FOR STRESS SHIFT (SET B)..................................................76
TABLE 4.5: THIRD SERIES OF BEAMS TESTED FOR STRESS SHIFT (SET C).....................................................77
TABLE 5.1: MP LOSS FOR COMMON ROLLED SECTIONS, 4” HOLE BOTTOM OF WEB..................................100
TABLE 6.1: LOAD AND DISPLACEMENT VALUES FOR IMPACTED BEAMS.....................................................104
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LIST OF FIGURES
FIGURE 1.1: DEFORMATION OF FLANGE DUE TO IMPACT,
<HTTP://WWW.STEELSTRAIGHTENING.COM/ARIZONA.HTM>...................................................................3
FIGURE 1.2: TYPICAL 1" SHELL ELEMENT MESH.............................................................................................7
FIGURE 3.1: STANDARD BOUNDARY CONDITIONS.........................................................................................16
FIGURE 3.2: MODIFIED BOUNDARY CONDITIONS ..........................................................................................18
FIGURE 3.3: PHOTO OF CORRODED WEB .......................................................................................................19
FIGURE 3.4: FULL LENGTH, PARTIAL WIDTH FLANGE THINNING..................................................................25
FIGURE 3.5: LTB CAPACITY FOR THREE BEAM LENGTHS, FULL LENGTH PARTIAL WIDTH FLANGE DAMAGE
............................................................................................................................................................26
FIGURE 3.6: REMAINING LTB CAPACITY VS. LENGTH OF FULL-WIDTH DAMAGE ........................................28
FIGURE 3.7: REMAINING LTB CAPACITY VS. WIDTH OF FULL LENGTH FLANGE THINNING..........................31
FIGURE 4.1: TEST CASE 1, SIMPLY SUPPORTED CONDITIONS AT BOTTOM FLANGE.......................................34
FIGURE 4.2: FLEXURAL STRESS DISTRIBUTION, TEST CASE 1, MIDSPAN ......................................................35
FIGURE 4.3: FLEXURAL STRESS DISTRIBUTION, TEST CASE 1, QUARTERSPAN..............................................36
FIGURE 4.4: FLEXURAL STRESS DISTRIBUTION, TEST CASE 2, MIDSPAN ......................................................37
FIGURE 4.5: BASIC SETUP FOR VIERENDEEL METHOD ..................................................................................39
FIGURE 4.6: STRESS COMPONENTS OF VIERENDEEL METHOD.......................................................................40
FIGURE 4.7: CROSS SECTION OF TOP TEE SECTION .......................................................................................41
FIGURE 4.8: INSIDE HOLE EDGE, OUTSIDE HOLE EDGE, AND HOLE CENTER NOTATION...............................43
FIGURE 4.9: EXAMPLE FLEXURAL STRESS DISTRIBUTION, INSIDE HOLE EDGE.............................................44
FIGURE 4.10: EXAMPLE FLEXURAL STRESS DISTRIBUTION, OUTSIDE HOLE EDGE .......................................44
FIGURE 4.11: FLEXURAL STRESS DISTRIBUTION AT INSIDE HOLE EDGE .......................................................47
FIGURE 4.12: FLEXURAL STRESS DISTRIBUTION AT HOLE CENTER...............................................................48
FIGURE 4.13: FLEXURAL STRESS DISTRIBUTION AT OUTSIDE HOLE EDGE....................................................50
FIGURE 4.14: FLEXURAL STRESS DISTRIBUTION AT INSIDE HOLE EDGE, 10” X 10”......................................52
FIGURE 4.15: FLEXURAL STRESS DISTRIBUTION AT HOLE CENTER, 10” X 10”..............................................53
FIGURE 4.16: FLEXURAL STRESS DISTRIBUTION AT OUTSIDE HOLE EDGE, 10” X 10”...................................54
FIGURE 4.17: FLEXURAL STRESS DISTRIBUTION AT INSIDE HOLE EDGE, 20” X 20”......................................55
FIGURE 4.18: FLEXURAL STRESS DISTRIBUTION AT HOLE CENTER, 20” X 20”..............................................56
FIGURE 4.19: FLEXURAL STRESS DISTRIBUTION AT OUTSIDE HOLE EDGE, 20” X 20”...................................57
FIGURE 4.20: FLEXURAL STRESS DISTRIBUTION AT INSIDE HOLE EDGE, 30” X 30”......................................58
FIGURE 4.21: FLEXURAL STRESS DISTRIBUTION AT HOLE CENTER, 30” X 30”..............................................59
FIGURE 4.22: FLEXURAL STRESS DISTRIBUTION AT OUTSIDE HOLE EDGE, 30” X 30”...................................60
FIGURE 4.23: FLEXURAL STRESS DISTRIBUTION AT INSIDE HOLE EDGE, 40” X 40”......................................61
FIGURE 4.24: FLEXURAL STRESS DISTRIBUTION AT HOLE CENTER, 40” X 40”..............................................62
FIGURE 4.25: FLEXURAL STRESS DISTRIBUTION AT OUTSIDE HOLE EDGE, 40” X 40”...................................63
FIGURE 4.26: FOUR-POINT LOADING.............................................................................................................64
FIGURE 4.27: FLEXURAL STRESS DISTRIBUTION AT HOLE EDGE, 4 PT. LOADING..........................................65
FIGURE 4.28: FLEXURAL STRESS DISTRIBUTION AT MIDSPAN (HOLE CENTER), 4 PT. LOADING ...................66
FIGURE 4.29: FLEXURAL STRESS DISTRIBUTION 2’ FROM HOLE EDGE, 4 PT. LOADING ................................67
FIGURE 4.30: FLEXURAL STRESS DISTRIBUTION AT HOLE EDGE, PARTIAL UNIFORMLY DISTRIBUTED
LOADING .............................................................................................................................................68
FIGURE 4.31: STRESS SHIFT..........................................................................................................................70
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FIGURE 4. 32: GLOBAL AND SHEAR-INDUCED MOMENTS..............................................................................71
FIGURE 4.33: “Y” TERM FOR SHEAR-INDUCED MOMENT ..............................................................................71
FIGURE 4.34: HOLE SIZE REQUIRED TO CAUSE STRESS SHIFT VS. H/TW RATIO..............................................75
FIGURE 4.35: HOLE SIZE REQUIRED TO CAUSE STRESS SHIFT IN THREE SETS OF BEAMS .............................78
FIGURE 4.36: DIFFERENCES BETWEEN FINITE ELEMENT AND VIERENDEEL HOLE SIZES TO CAUSE STRESS
SHIFT...................................................................................................................................................79
FIGURE 4.37: MAXIMUM STRESS LOCATIONS BEFORE AND AFTER STRESS SHIFT ........................................81
FIGURE 4.38: DISCREPANCY BETWEEN FINITE ELEMENT AND VIERENDEEL MAX STRESSES (BEFORE STRESS
SHIFT)..................................................................................................................................................82
FIGURE 4.39: DIFFERENCE BETWEEN BEAM THEORY AND FINITE ELEMENT MAXIMUM STRESSES BEFORE
STRESS SHIFT ......................................................................................................................................83
FIGURE 4.40: DISCREPANCY BETWEEN FINITE ELEMENT AND VIERENDEEL MAX STRESS (AFTER STRESS
SHIFT)..................................................................................................................................................84
FIGURE 4.41: UPPER AND LOWER BOUNDS ON VIERENDEEL APPLICABILITY.................................................85
FIGURE 4.42: INSIDE HOLE EDGE, NEXT TO HOLE ........................................................................................87
FIGURE 4.43: DIFFERENCE BETWEEN BEAM THEORY AND FINITE ELEMENT STRESSES, ...............................88
INSIDE EDGE NEXT TO HOLE..........................................................................................................................88
FIGURE 4.44: MAXIMUM STRESS AT INSIDE HOLE EDGE WITH INCREASING VERTICAL ECCENTRICITY ........90
FIGURE 4.45: "A" AND "B" IN ECCENTRICITY DEFINITION .............................................................................90
FIGURE 4.46: MAXIMUM FLEXURAL STRESS AT OUTSIDE HOLE EDGE WITH INCREASING VERTICAL
ECCENTRICITY.....................................................................................................................................92
FIGURE 5.1: MP LOSS VS. HOLE SIZE ............................................................................................................95
FIGURE 5.2: MP LOSS VS. HOLE ECCENTRICITY ............................................................................................97
FIGURE 5.3: MP LOSS WITH ENTIRE WEB REMOVED.....................................................................................99
FIGURE 6.1: BLOCK AND BEAM BEFORE IMPACT ........................................................................................101
FIGURE 6.2: IMPACT DAMAGE AT 100 IN/S ..................................................................................................102
FIGURE 6.3: IMPACT DAMAGE AT 200 IN/S ..................................................................................................102
FIGURE 6.4: IMPACT DAMAGE 400 IN/S .......................................................................................................103
FIGURE 6.5: LOAD VS. DISPLACEMENT PLOTS FOR IMPACT DAMAGED BEAMS...........................................104
FIGURE 6.6: POST-BUCKLING CAPACITY LOSS FOR DIFFERENT IMPACT VELOCITIES .................................105
FIGURE 7.1: CROSS SECTION OF BRIDGE MODEL ........................................................................................106
FIGURE 7.2: LOAD POSITIONING ON BRIDGE ...............................................................................................108
FIGURE 7.3: FINITE ELEMENT MODEL, ALL BEAMS PRESENT .....................................................................109
FIGURE 7.4: FLEXURAL STRESS DISTRIBUTIONS, ALL BEAMS PRESENT......................................................110
FIGURE 7.5: FLEXURAL STRESS DISTRIBUTIONS, BEAM 1 REMOVED ..........................................................111
FIGURE 7.6: FLEXURAL STRESS DISTRIBUTIONS, BEAM 4 REMOVED ..........................................................112
1
CHAPTER1: INTRODUCTION AND BACKGROUND
1.1 Introduction
According to the Federal Highway Administration, approximately one third of the
nation’s bridges are either structurally deficient or functionally obsolete. A factor
contributing to the structural deficiency of steel bridge superstructures is damaged beams.
Damage may take the form of section loss due to corrosion, or geometric distortion due to
vehicular impact. These damage forms may cause reduced buckling capacity, elevated
flexural stresses, and reduced ultimate moment capacity.
1.2 Background
The National Bridge Inventory 2003 report lists over 615,000 bridges. Approximately
one third of these bridges are steel. Oklahoma contains over 23,000 bridges, 34% of
which are made of steel (2000). Steel bridges are susceptible to damage that may result
from corrosion and vehicular impact. Given the number of steel bridges throughout
Oklahoma and the nation, the number of structurally deficient bridges in the nation, and
the vulnerability of steel bridges to corrosion and impact damage, engineers could benefit
from a greater understanding of the effects corrosion and impact damage have on the
capacity of bridge members.
2
Corrosion is a commonly known problem with steel bridge members. Uniform corrosion
may reduce the cross sectional dimensions of a girder evenly, in which case the load-carrying
capacity of the girder is easily recalculated using the new dimensions of the
reduced section. However, localized corrosion is also common, and may be severe
enough to completely penetrate the girder web. Determination of girder capacity is more
difficult with local corrosion as compared to uniform corrosion. Localized thinning of a
girder and/or holes in the web will leave less area to sustain flexural and shear stresses.
Also, the geometric properties (such as moment of inertia and radius of gyration) are
altered, so the beam may have a greater propensity to buckle.
Although less common than corrosion damage, another issue affecting steel bridges is
structural damage due to accidental impacts. This circumstance may occur when an over-height
truck or equipment travels under an overpass. An example of such impact damage
is shown in Figure 1.1, in which the bottom flange of a girder has been deformed by
vehicular impact (2006).
3
Figure 1.1: Deformation of Flange Due to Impact, < http://www.steelstraightening.com/arizona.htm>
In the case of impact damage, the cross-sectional area of the girder may essentially
remain unchanged, which means the ultimate moment capacity should be changed only
slightly. However, the presence of a significant local deformation in the beam could
adversely affect the girder’s ability to resist buckling.
The Oklahoma Department of Transportation averages between 5 and 10 projects each
year involving the repair of impacted bridges, while they average 10 to 20 projects a year
involving the repair of corrosion damage (Allen 2004). Damaged bridge members are
encountered on a regular basis. It is costly to close bridges, but if the damaged members
retain enough capacity, closure may be unnecessary. The structural capacity of damaged
members must be evaluated to determine if a bridge must be closed. Therefore, it would
be useful to obtain a simple method for engineers to achieve a safe and acceptably
accurate assessment of a damaged girder’s remaining capacity. Bridge ratings may then
4
be adjusted or repairs conducted as necessary, so the public safety can be maintained
without over-expenditure of time and money in the analysis and repair processes.
1.3 Objectives and Scope
The objectives of this research were to:
1. Determine the effects of various damage configurations on steel bridge members.
2. Possibly develop a simplified method (performed easily by hand or spreadsheet)
to determine capacity of damaged members. The intent was to develop methods
that would provide a quick yet accurate assessment of remaining capacity without
requiring advanced computer analysis.
Damage examined includes scenarios likely caused by corrosion, such as holes of various
sizes and locations in beam webs, and thinned sections of flanges. Web holes were
studied to determine how they would affect flexural and shear stress distributions, as well
as lateral torsional buckling capacity. Flange damage from corrosion was primarily
examined to determine how lateral torsional buckling would be affected. Bearing
capacity of beams with holes in webs and/or corrosion damaged flanges was not
examined in this research, as it has been recently addressed in other work (Lindt and
Ahlborn 2004). Lower flange and web deformation, such as might be caused be over-height
vehicle impact, was also examined to determine its effect on flexural stress
distribution and ultimate moment capacity.
5
1.4 Computer Simulation and Analysis
The analysis performed utilized ABAQUS 6.4.2, a non-linear finite element program.
The research required more complex analysis (non-linear geometric and constitutive
analysis, as well as elastic and inelastic buckling analysis) than other standard finite
element packages (SAP, STAAD, etc.) are not capable of performing. ABAQUS also
provides the user with more flexibility in modeling damaged members (such as a
standard W-section with a section of web removed).
Several types of elements were examined and simple test cases were performed to
determine the most appropriate type. The first elements utilized were linear, four-node
constant stress tetrahedral elements, referred to by ABAQUS as C3D4 elements. These
types of elements are generally acceptable for standard cases if the mesh is refined
adequately. Test cases were run which modeled a W27x94 (discussed further in Chapter
3). The model was comprised of C3D4 elements approximately 2” on each side. When a
point load was applied at midspan, a flexural stress distribution at quarterspan had an
average error of approximately 1% throughout when compared to the theoretical
distribution predicted by elementary beam theory (Timoshenko beam theory). (The
flexural stress distribution was examined at quarterspan instead of midspan because the
distribution may be slightly distorted immediately beneath a point load. This is a local
phenomenon, and would not provide an adequate gauge of the model’s overall accuracy).
However, the elastic buckling capacity predicted by finite element analysis was
approximately 264% greater (192,224 lb vs. 52,750 lb) than the capacity predicted by
6
methods commonly used in the American Institute for Steel Construction design manual.
The AISC methods are based on commonly derived elastic buckling expressions, found
in Salmon and Johnson, 1996, and other texts. They calculate the critical moment,
assuming a beam loaded with a uniformly distributed moment. To calculate the critical
moment for other load configurations, the critical moment is multiplied by a scalar factor
known as Cb. For a simply supported beam with a point load at midspan, Cb =1.32. For
a uniformly distributed load, Cb=1.14. AISC methods also assume the load to be applied
at the neutral axis of the beam. If the load is applied at the top flange, a destabilizing
effect occurs which reduces the buckling capacity by approximately 1.4 (Galambos
1998). AISC methods are discussed further in Chapter 3. When a uniformly distributed
load was applied to the top flange of the model, the buckling capacity predicted by finite
element analysis was approximately 82% higher (462.1 lb/in vs. 253.1 lb/in.) than the
capacity predicted by AISC methods.
Since the tetrahedral elements produced inaccurate buckling results, another element type
was investigated. ABAQUS type B31 elements, which are first-order three-dimensional
Timoshenko beams in space, were used for the next test model. When the W27x94 with
a uniformly distributed load modeled with the beam element, the buckling capacity
predicted by finite element analysis was only about 6% less (236.5 lb/in vs. 253.1 lb/in)
than the capacity predicted by accepted theoretical results. Although the beam elements
proved more accurate for simple buckling analyses, they were not useful for complex
stress analyses because they do not allow localized modification of the beam geometry
(such as the inclusion of impact damage or web holes).
7
The next element type tested was a four-node shell element referred to by ABAQUS as
S4R. These are standard stress/displacement shell elements with reduced integration.
They account for finite membrane strains and arbitrarily large rotations, and are typically
suitable for large-strain analysis (2006). The W27x94 was modeled with square shell
elements approximately 1.5” on each side and a point load was applied at midspan. A
flexural stress distribution at quarterspan had approximately 1% average error throughout
when compared to the theoretical distribution predicted by beam theory. For this same
configuration, the finite element model predicted a buckling capacity about 6% less
(49,769 lb vs. 52,750 lb) than the AISC results. Based on these results, shell elements
were adopted for further tests because it was felt that they would yield results with
acceptable accuracy. They also allowed beam models to be geometrically modified to
simulate various forms of corrosion and impact damage. Figure 1.2 shows an example of
a typical mesh used, which is 1” on a side. The beam shown is again a W27x94, 30’ in
length.
Figure 1.2: Typical 1" Shell Element Mesh
8
1.5 Hand Calculations
The research also examined methods which do not require advanced software. These
calculations were primarily flexural stress calculations based on the Vierendeel
procedure, which is more fully described in Section 3.2. The Vierendeel procedure is a
means of predicting the flexural stress distributions on either side of a hole in a girder
web. It can be done completely by hand, or programmed into a spreadsheet.
Comparisons are made in Chapter 3 between Vierendeel and ABAQUS results to
determine if simple hand procedures can accurately predict flexural stress distributions in
the presence of web holes. In some cases, the Vierendeel procedure provides very good
predictions for the magnitude and location of the maximum flexural stress. However, the
accuracy of the method seems to be affected by specific beam geometry, making it hard
to clearly state when the method should be used.
Ultimate moment capacity was also computed by hand (and in Excel spreadsheets).
These calculations were performed for beams with web holes or thinned flanges. No new
techniques were used; section properties were recalculated based on modified geometries.
These calculations demonstrated clear trends which are more fully discussed in Chapter
5.
9
CHAPTER 2: LITERATURE REVIEW
The first step in this research was a review of relevant literature. Literature which
provided means to simply calculate capacity of damaged members would be especially
useful. These simply calculated results could then be compared to results acquired from
more advanced analysis using finite elements. Although there were several papers dealing
with corroded or impact damaged beams, few of them directly related with assessing the
remaining flexural capacity of the beams. For example, Frangopol and Nakib’s article
titled “Effects of Damage and Redundancy on the Safety of Existing Bridges” (Frangopol
and Nakib 1991) initially appeared to be closely related to the problem under
consideration. However, the article opens with discussion of the fact that there is
currently no method for quantifying structural redundancy levels in bridge systems, then
an example bridge is analyzed using finite elements to demonstrate how accidental
damage and corrosion damage would affect the redundancy of the structure.
Unfortunately, the focus on redundancy does not translate into remaining flexural
capacity, especially for individual bridge members.
Kayser and Nowak (Kayser and Nowak 1989) present analytical information on capacity
loss as a result of corrosion in steel bridges. The effects of corrosion loss on bending,
shearing, and bearing behaviors are all considered. For example, effects of corrosion on
bending performance are demonstrated in a graph of percent remaining ultimate moment
capacity vs. flange loss. However, the capacity loss is calculated based on the reduced
10
section properties which would result from uniform corrosion. A formula is provided for
predicting the depth of corrosion penetration over time, but no other calculations or
predictive formulae are provided. Conclusions focus on the fact that corrosion can lead
to web buckling in bearing, and bearing stiffeners can create a more corrosion-tolerant
structure. The material discussed clearly parallels the current project. However, uniform
corrosion is the main focus, and localized corrosion is mentioned only briefly.
Shanafelt and Horn (Shanafelt and Horn 1984) provide a subsection titled “Strength of
Damaged Member.” This subsection merely states that during damage assessment “a
complete evaluation of strength should be made.” However, no further discussion is
offered on how to best evaluate the strength of the damaged members. Informative
material is also presented about when impacted members should or should not be
straightened, yet this determination is not made on the basis of remaining capacity. A
main point is that by measuring the curvature of a deformed member, it can be
determined if the member has deformed plastically. If not, the member should not be
straightened. When adjacent members which have deformed plastically are straightened,
the elastically deformed member should straighten itself.
Darwin (Darwin 1990) presents information on the design of beams with web holes, such
as might be necessary during construction for the placement ductwork or piping.
Because the paper is written from a design standpoint, it assumes the engineer will have
control over many details such as hole size and location, corner radius, and others.
Though this will not be the case when analyzing beams that have web holes due to
11
corrosion, many of the items present are still adaptable to the current situation. For
example, equations are provided for determining the ultimate moment capacity of a beam
with a web hole. By using approximate dimensions so the corrosion hole is assumed
rectangular, the given equations may be applicable. Also, multiple beam configurations
are addressed, including bare steel members and composite beams with varying slab
types.
Perhaps the most applicable piece of literature acquired was the report by Kulicki
(Kulicki, Prucz et al. 1990). This was a comprehensive report dealing with topics from
types and mechanisms of corrosion to how it affects many different elements of several
bridge types. One portion discusses material loss and provides useful equations. A
Vierendeel analysis is employed to analyze flexural stresses around a hole in a girder
web. These equations allow one to compare hand calculated stresses with those
generated by a finite element simulation, which proved to be especially valuable for the
current project. The equations provide predictions for a wide variety of rectangular hole
configurations and locations. Guidelines are given for transforming non-rectangular
shapes into rectangles for analytical purposes, resulting in a very versatile predictive
procedure.
Some articles reviewed were more pertinent to the finite element simulation aspect of the
current project. Olsson conducted a study on steel channel columns used in industrial
rack and shelving systems (Olsson, Sandberg et al. 1999). These columns are commonly
subjected to impact damage (such as from fork lifts or trucks). Though the channel
12
sections have significantly different geometry from bridge members, the side impact
damage is a similar scenario to over-height vehicle damage in bridge members. Thin-shell
finite elements were used to model the channel geometry, simulate impact damage
to the channels, then test the axial loading capacity. Finite element results were then
confirmed with laboratory results. Although axially loaded columns are not directly
relevant to the current research, this article provided an example of how finite elements
could be used to handle situations such as vehicular impact damage. Based on Olsson’s
work, it appeared that using similar elements would allow accurate modeling of vehicular
impact damage, and his laboratory verification helps confirm the validity of the
procedure (especially encouraging, since the current project is not able to include
laboratory testing).
Dinno and Birkemoe performed finite element analysis on plate girder web panels with
patches of localized corrosion damage (Dinno and Birkemoe 1997). The panels were not
entire girders, but were short sections. Dimensions varied from length being equal to
depth, to length twice the depth. The work was primarily a parametric study to determine
what variables cause the greatest decrease in strength (such as hole size, aspect ratio,
vertical or diagonal shift from panel center). Rectangular holes were the primary focus,
because results showed that rectangular holes had a greater influence on panel strength
than holes of other shapes with the same area. This finding influenced the use of square
and rectangular holes in the current research. Results showed that the extent of web
thinning was the most sensitive parameter in strength loss. This fact was considered in
the current research, when corrosion was modeled by holes in the web instead of thinned
13
sections. The sensitivity to thinning confirms the conservatism of using web holes. The
ratio of corroded patch area to the entire panel area was also a significant parameter.
General information on panel loading and model setup is provided, which makes this
another good example of how finite element analysis software can be used to address the
current project. Specific information on mesh size was not provided. Dinno and
Birkemoe used the Ansys software and the type shell43 element, described as “a four-noded
quadrilateral element that has large out-of-plane deflection and strain capabilities.”
This is similar to the ABAQUS S4R element used in the current research.
14
CHAPTER 3: LATERAL TORSIONAL BUCKLING OF CORROSION
DAMAGED MEMBERS
One of the limit states analyzed during the research was elastic lateral torsional buckling.
Beams with various forms of corrosion damage were analyzed for remaining lateral
torsional buckling (LTB) capacity. Damage parameters included holes in the web,
partial-width flange thinning for the full beam length, and full-width flange thinning for
part of the beam length. Analyses initially focused on the capacity of the beam alone and
later analyses included a composite slab.
3.1 Setup and Test Cases
A W27x94 was used as the standard test section. The majority of tests were run with the
W27x94 because the 27” depth is representative of the most commonly used rolled
shapes in Oklahoma Turnpike Authority bridges. Some tests were also run with a
hypothetical plate girder section, with flange dimensions 18” x 1” and web dimensions
60” x 0.375.” This section was created so results obtained with the W27x94 could be
compared to a significantly different beam geometry. The plate girder section has flanges
approximately twice as wide and twice as thick as the W27x94, while the girder web is
almost twice as deep and about 25% thinner than the W27x94 web. For most tests,
simply supported boundary conditions were applied at the bottom flange on both ends.
Also, rotation was restrained for all nodes in the cross section at each end (see Figure
15
3.1). This set of boundary conditions is referred to in this thesis as the “standard”
conditions for LTB tests. These boundary conditions were believed to closely match the
boundary conditions for which theoretical lateral torsional buckling equations were
derived. As shown in Table 3.1, finite element results using these boundary conditions
closely matched accepted theoretical results.
Theoretical results were calculated with procedures used by Timoshenko (Timoshenko
and Gere 1961). These procedures contain expressions for the critical load specific to
each loading condition (point load at midspan and uniformly distributed load). The AISC
Manual of Steel Construction (AISC 2001) results are from the commonly derived
expression (Salmon and Johnson 1996) as well as other texts. This expression is for the
critical moment, and is derived for constant moment along the beam’s entire length. A
scalar coefficient, Cb (AISC 2001) is introduced to modify the expression for cases of
non-constant moment. The manual provides Cb values of 1.32 for a point load at
midspan and 1.14 for a uniformly distributed load. Both Timoshenko and AISC Manual
results are included. Although the Timoshenko procedure consistently yields slightly
larger discrepancies from finite element results, it confirms the trends shown by the
newer AISC methods.
16
Figure 3.1: Standard Boundary Conditions
The following test cases were used to generate the critical loads (shown as q) in Table
3.1:
1) Simply supported W27x94, 30’, uniformly distributed load applied at neutral axis.
2) Simply supported W27x94, 30’, concentrated load at midspan, applied at neutral axis.
Uniformly Dist. Load
(Case 1)
Concentrated Load
(Case 2)
q crit, Finite Element 345.3 lb/in 70,306 lb
qcrit, Timoshenko* 357.1 lb/in 77,243 lb
qcrit, AISC ** 354.3 lb/in 73,850 lb
% Error, Finite Element vs.
Timoshenko 3.3% 9.0%
% Error, Finite Element vs. AISC 2.5% 4.8%
*(Timoshenko and Gere 1961)
**(AISC 2001)
Table 3.1: Test Cases with Standard Boundary Conditions, Load at Neutral Axis
1
2
3
End 1: Displacement restricted 1,2,3
End 2: Displacement restricted 1,2
Rotation Restricted
Along 3, Both Ends
17
However, beams in bridges are loaded along the top flange, not along the neutral axis.
The test cases were run again with the load applied at the top flange and checked for
agreement with theoretical results. Results are shown in Table 3.2.
Uniformly Dist. Load
(Case 1)
Concentrated Load
(Case 2)
q crit, Finite Element 245.6 lb/in 49,769 lb
qcrit, Timoshenko* 257.4 lb/in 51,410 lb
qcrit, modified AISC ** 253.1 lb/in 52,750 lb
% Error, Finite Element vs. Timoshenko 4.6% 3.2%
% Error, Finite Element vs. modified
AISC 3.0% 5.7%
*(Timoshenko and Gere 1961)
**(AISC 2001)
Table 3.2: Test Cases with Commonly Used Boundary Conditions, Load at Top Flange
Table 3.2 contains modified AISC results because the Timoshenko expressions include
coefficients to account for top flange loading versus neutral axis loading, while the AISC
expression assumes neutral axis loading. Another approach (Galambos 1998) modifies
the Cb factor to compensate for loading other than the neutral axis. For top loading, Mcr
is reduced by a factor of 1.4. The critical load decreases when applied at the top flange
because it will produce a tipping effect that destabilizes the beam. Applying the load at
the bottom flange would produce a stabilizing effect and increase the critical load by a
factor of 1.4. Tables 3.1 and 3.2 show finite element results consistently within 3-5% of
the theoretical results, which indicates a satisfactory model and boundary conditions have
been established.
18
Although the standard boundary conditions mirror those used in Timoshenko’s
derivations, it was theorized that beams in bridges may be subject to slightly different
conditions. The web may have nothing to restrain it, so it was decided to try test cases
reflecting this. The modified boundary conditions applied simple supported conditions to
the bottom flanges at the beam ends. Instead of restricting rotation throughout the cross
section, lateral motion was restricted at the top flange/web intersection (see Figure 3.2).
These boundary conditions were designed to model the lateral support provided by x-bracing
(with no slab present).
Figure 3.2: Modified Boundary Conditions
Since most tests would be run with loads at the top flange, the modified boundary
conditions were checked against the test cases involving loads along the top flange.
Results are shown in Table 3.3. As expected, the less restrictive boundary conditions
produced slightly lower critical loads. The difference in results between the two sets of
boundary conditions for both cases is less than 3%. Therefore, it was decided that the
1
2
3
End 1: Displacement restricted 1,2,3
End 2: Displacement restricted 1,2
Displacement
Restricted Along 1,
Both Ends
19
results obtained with the standard boundary conditions would provide an acceptable
model of beam behavior, even if actual beams did not have rotational constraints at the
supports.
Uniformly Dist. Load Concentrated Load
qcrit, Original Bound. Cond. 245.6 lb/in 49,769 lb
qcrit, Modified Bound. Cond. 239.1 lb/in 48,670 lb
% Difference,
Modified vs. Original 2.5% 2.2%
Table 3.2: Test Cases with Modified Boundary Conditions, Load at Top Flange
3.2 Holes in Web
The first type of beam damage analyzed was web damage due to corrosion. Corrosion
damage often consists of localized thinned sections in the web. In severe cases, corrosion
will completely penetrate the web. Figure 3.3 shows an example of corrosion which has
fully penetrated a girder web (Kulicki, Prucz et al. 1990). It also appears that holes have
been drilled to stop additional crack propagation.
Figure 3.3: Photo of Corroded Web
20
All simulations conducted incorporated holes in the web instead of thinned sections,
because this more severe damage case should provide conservative data which can be
safely applied to thinned sections. Also, holes were modeled as square holes. The heavy
rectangle drawn on Figure 3.3 demonstrates how the actual damage could be
conservatively modeled by a rectangular hole. While square/rectangular holes may cause
issues with stress concentrations at the corners, they are conservative in that corrosion
holes will likely not have perfectly squared corners and stress concentrations will be less
severe. If corners were rounded to eliminate or reduce the stress concentration issue,
inspectors or engineers would have to determine whether or not holes in the field had
corners which were sharper than those modeled here. Hence, it was felt square holes
would conservatively approximate a worst-case scenario. Corrosion is most likely to
attack a beam web at supports, where there might be a joint in the deck. It is also
commonly seen just above the bottom flange, since the flange may retain moisture from
precipitation and condensation. Flanges may also collect moisture during wet weather as
vehicular traffic splashes water up onto bridge members. However, initial tests were
conducted with holes vertically centered at midspan. This configuration provided a good
starting point from which the model could easily be modified, and it was believed that
placing the hole at midspan (where the moment is highest) would have the most
detrimental affect on LTB capacity. Bearing-type failures such as web yielding or
buckling were not analyzed, since those type failures were explored in other work (Lindt
and Ahlborn 2004). The primary focus for this research was flexural failures. Two span
members were not tested, because the highest flexural stresses will occur over the
support. If a hole were introduced above the support, there would almost certainly be a
21
shear failure (see Table 3.4). Also, it is assumed that lateral bracing will be provided at a
support and LTB cannot occur.
3.2.1: Hole Depth
The first series of tests utilized a W27x94, length 30’, with standard boundary conditions.
A concentrated load is applied at midspan, on the top flange. These tests investigate the
affect on LTB capacity as hole depth increases. Several large holes were placed in the
model, with depths from 12” (45% of total beam depth) to 22” (82% of total beam depth).
Results are shown in Table 3.4. The plastic moment capacity (Mp) shown represents the
highest theoretical moment capacity. Mp calculations were based on the modified cross-sectional
geometry resulting from the presence of a hole. The losses in Mp and shear
capacity are included to provide a perspective on the relative importance of LTB losses.
It is possible that LTB capacity will never be the governing limit state. This is especially
true for cases in which the compression flange is fully laterally restrained, such as simple
span composite bridges.
Beam Hole
Length
Hole
Depth
LTB
Capacity
% loss
LTB
Capacity
% loss
Mp
% loss
Shear
Capacity
W27x94, 30’ -- -- 49769 lb -- -- --
W27x94, 30’ 12” 12” 49757 lb 0.02% 6.4% 44.6%
W27x94, 30’ 12” 18” 49748 lb 0.04% 14.4% 66.9%
W27x94, 30’ 12” 22” 49744 lb 0.05% 21.6% 81.8%
W27x94, 30’ 36” 12” 49669 lb 0.20% 6.4% 44.6%
W27x94, 30’ 36” 18” 49651 lb 0.24% 14.4% 66.9%
W27x94, 30’ 36” 22” 49644 lb 0.25% 21.6% 81.8%
Table 3.3: LTB Capacity with Increasing Hole Depth
22
As shown in Table 3.4, as the depth of the holes increases there is a non-linear decrease
in capacity. However, the last test in the series involves a hole with dimensions one-tenth
the total beam length and 82% of the total beam depth, and capacity is reduced only
0.25%. A hole this large clearly presents other problems, such as the 21.6% loss in
plastic moment capacity and the 82% loss in shear capacity. Therefore, it was decided
that developing extensive plots of LTB capacity vs. hole depth would not be of
significant value. It was also decided that if a vertically centered hole of 82% section
depth did not significantly affect LTB capacity, then there was no need to investigate the
effects of vertical hole location. The depth of holes in the web does not have a significant
impact on LTB capacity.
3.2.2: Hole Length
The next series of tests investigated the affects of increasing hole length. Again the beam
was a W27x94, length 30’. Boundary conditions were applied as shown in Figure 3.2. A
concentrated load is applied at midspan, on the top flange. Results are shown in Table
3.5.
Beam Hole
Length
Hole
Depth
Capacity % loss
LTB cap.
% loss
Mp
W27x94, 30’ -- -- 48670 lb -- --
W27x94, 30’ 12” 12” 48657 lb 0.03% 6.4%
W27x94, 30’ 36” 12” 48569 lb 0.21% 6.4%
W27x94, 30’ 60” 12” 48394 lb 0.57% 6.4%
W27x94, 30’ 120” 12” 47628 lb 2.14% 6.4%
Table 3.4: LTB Capacity with Increasing Hole Length
23
As with variable hole depth, there is a non-linear decrease in capacity as hole size
increases. However, the LTB capacity loss is very small. For the largest test case, the
hole was 1/3 the entire member length, and 45% the entire member depth. Yet the LTB
capacity was only reduced by approximately 2%. Plastic moment capacity reduction is
still of greater concern than LTB capacity.
The first two series of LTB tests all utilized beams with a concentrated load at midspan.
To further investigate LTB with web deterioration under a different loading
configuration, two more tests were run with a uniformly distributed load. A 30’ W27x94
with standard boundary conditions was used for the tests. Results are shown in Table 3.6.
Beam Hole
Length
Hole
Depth
Capacity % loss
LTB cap.
% loss
Mp
W27x94, 30’ -- -- 245.62 lb/in -- --
W27x94, 30’ 12” 6” 245.59 lb/in 0.01% 1.6%
W27x94, 30’ 36” 6” 245.27 lb/in 0.14% 1.6%
Table 3.5: LTB Capacity Under Uniformly Distributed Load
As with the concentrated loading configuration, LTB capacity is not significantly affected
by the web holes. With only a 6” deep hole vertically centered in the member, the plastic
moment capacity is only reduced 1.6%. Yet that is over 10 times higher than the loss in
LTB capacity for the member with a 36” x 6” hole in the web.
24
Based on the results of several tests with varying hole dimensions, it was concluded that
the presence of holes in beam webs does not significantly reduce lateral torsional
buckling capacity.
3.3 Flange Thinning
Corrosion commonly affects the flanges of steel bridge members, because they retain
moisture and debris. This retained moisture on the surface of the steel accelerates
corrosion. It is not feasible to model every potential flange deterioration possibility, so to
capture the effects of flange corrosion, tests were run with two basic configurations:
thinning part of the flange width for the full beam length, and thinning the full width of
the flange for part of the beam length. It was felt these two methods would provide
sufficient data for analysis of numerous corroded flanges encountered in the field.
3.3.1: Full Length, Partial Width Flange Thinning
The first series of tests for flange corrosion used the standard W27x94 test section.
Standard boundary conditions were applied, and a concentrated load was applied to the
top flange at midspan. Tests on a 30’ beam showed more significant losses than were
obtained while testing web holes, so beams of 25’ and 35’ length were also tested to see
how beam length affected sensitivity to damage. The damage was simulated by thinning
the bottom flange to ½ its original thickness for varying widths along the full length of
the beam. A typical view of the damaged cross section at the bottom flange is shown in
Figure 3.4, and test results are given in Table 3.7.
25
Figure 3.4: Full Length, Partial Width Flange Thinning
Beam Damage
Width
Capacity % loss
LTB cap.
% loss
Mp
W27x94, 25’ -- 76600 lb. -- --
W27x94, 25’ 0.125b 75777 lb. 1.1% 2.3%
W27x94, 25’ 0.25b 74878 lb. 2.2% 4.6%
W27x94, 25’ 0.375b 74032 lb. 3.4% 7.0%
W27x94, 25’ 0.5b 73168 lb. 4.5% 9.6%
W27x94, 25’ b 69360 lb. 9.5% 20.4%
W27x94, 30’ -- 49769 lb. -- --
W27x94, 30’ 0.125b 48973 lb. 1.6% 2.3%
W27x94, 30’ 0.25b 48282 lb. 3.0% 4.6%
W27x94, 30’ 0.375b 47611 lb. 4.3% 7.0%
W27x94, 30’ 0.5b 46923 lb. 5.7% 9.6%
W27x94, 30’ b 44059 lb. 11.5% 20.4%
W27x94, 35’ -- 34838 lb. -- --
W27x94, 35’ 0.125b 34327 lb. 1.5% 2.3%
W27x94, 35’ 0.25b 33759 lb. 3.1% 4.6%
W27x94, 35’ 0.375b 33219 lb. 4.6% 7.0%
W27x94, 35’ 0.5b 32658 lb. 6.3% 9.6%
W27x94, 35’ b 30391 lb. 12.8% 20.4%
Table 3.6: LTB Capacity with Full Length, Partial Width Flange Thinning
Removed
Sections
26
These results are plotted in Figure 3.5. Since a longer beam will have a greater
propensity to buckle than a shorter beam if all other variables are constant, it appears
reasonable that the results show increasing sensitivity to damage with increasing beam
length.
86%
88%
90%
92%
94%
96%
98%
100%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fraction of Flange Width Thinned
Remaining LTB Capacity
Length=25'
Length=30'
Length=35'
Figure 3.5: LTB Capacity for Three Beam Lengths, Full Length Partial Width Flange Damage
Flange thinning has significantly more effect on LTB capacity than holes in the web. As
the damage width increases, capacity decreases linearly. However, examination of the
results in Table 3.7 shows that LTB capacity may still not be the limiting criteria when
the flange is thinned. Losses in plastic moment capacity are still higher than losses in
LTB capacity.
27
3.3.2: Full Width, Partial Length Flange Thinning
The next series of flange thinning tests used a slightly different damage model. The
flange was again thinned to ½ its original thickness. However, this was done for the full
width of the flange for only a part of the beam length. The 30’ W27x94 was used with
standard boundary conditions. A 30’ W18x50 with standard boundary conditions was
also used, to see if a smaller beam would show more or less sensitivity to the flange
damage. Beams were loaded with a concentrated load on the top flange at midspan.
Results are shown in Table 3.8, and plotted in Figure 3.6.
Beam Damage
Length
Capacity % loss
LTB
cap.
% loss
Mp
W18x50, 30’ -- 69.96 lb/in -- --
W18x50, 30’ 3’ 69.73 lb/in 0.33% 21.9%
W18x50, 30’ 7.5’ 69.17 lb/in 1.1% 21.9%
W18x50, 30’ 15’ 66.79 lb/in 4.5% 21.9%
W18x50, 30’ 22.5’ 62.45 lb/in 10.7% 21.9%
W18x50, 30’ Full 58.32 lb/in 16.6% 21.9%
W27x94, 30’ -- 245.62 lb/in -- --
W27x94, 30’ 3’ 245.37 lb/in 0.10% 20.4%
W27x94, 30’ 7.5’ 244.06 lb/in 0.64% 20.4%
W27x94, 30’ 15’ 237.62 lb/in 3.3% 20.4%
W27x94, 30’ 22.5’ 226.74 lb/in 7.7% 20.4%
W27x94, 30’ Full 218.09 lb/in 11.2% 20.4%
Table 3.8: LTB Capacity with Full Width, Partial Length Flange Thinning
28
82%
84%
86%
88%
90%
92%
94%
96%
98%
100%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fraction of Beam Length Damaged
% Remaining LTB Capacity
W27x94
W18x50
Figure 3.6: Remaining LTB Capacity vs. Length of Full-Width Damage
For full width damage, capacity decreases non-linearly with increasing damage length.
The lighter beam section showed a greater sensitivity to the beam damage, both in terms
of LTB capacity and plastic moment capacity. The greater sensitivity exhibited by the
W18x50 is somewhat expected, because the lower flange of the W18x50 accounts for
about 29% of the section’s total cross sectional area. The lower flange of a W27x94
accounts for less cross sectional area, making up 27% of the total area.
29
3.3.3: Web Holes and Flange Damage
Two more tests were run on beams which were subject to holes in the web and flange
damage. This was done to see if the presence of both damage types would compound the
effects. The tests used the standard beam and boundary conditions, and a uniformly
distributed load was applied at the top flange. Table 3.9 gives the results, and compares
them to results for undamaged sections and sections subjected to only one damage type.
Table 3.9: LTB Capacity When Web Holes and Flange Thinning Both Present
The test incorporating a 12”x 6” hole and flange thinning seems to have produced a small
numerical error, since it actually shows a slightly higher capacity than the beam with
thinning only. However, when looking at the final results for both beams, it can be seen
that the simultaneous presence of web holes and flange thinning does not compound the
damage effects. Although not exact, it would be more accurate to say that the effects of
the two damage types are approximately additive.
Beam Flange Thinning Hole
Dimensions
qcrit.,
lb/in
% loss
LTB cap.
% loss
Mp
W27x94, 30’ -- -- 245.62 -- --
W27x94, 30’ -- 12” x 6” 245.59 0.01% 1.6%
W27x94, 30’ ½ orig. thickness, ½
flange width, full length -- 238.93 2.72% 9.6%
W27x94, 30’ ½ original thickness, ½
flange width, full length 12” x 6” 238.96 2.71% 11.2%
W27x94, 30’ -- 36” x 6” 245.27 0.14% 1.6%
W27x94, 30’ ½ original thickness, ½
flange width, full length -- 238.93 2.72% 9.6%
W27x94, 30’ ½ original thickness, ½
flange width, full length 36” x 6” 238.68 2.83% 11.2%
30
3.3.4: Two Continuous Spans: Full Length, Partial Width Flange Thinning
Although the majority of the bridges dealt with by the Oklahoma Turnpike Authority
(OTA) are simple spans, some investigation was done for beams covering two continuous
spans. Because web holes had such little impact on simple spans, they were not
addressed for continuous spans. In addition to the standard section, a hypothetical plate
girder was used for some of the tests. The plate girder had flange dimensions 18” x 1”
and web dimensions 60” x 0.375.” A plate girder was tested to again see how different
sized sections would be affected by the same type of damage, and also to see if a thinner
web would have a significant impact on the results. Both sections were subjected to a
uniformly distributed load along the top flange and standard boundary conditions.
Lengths given are for the total of both spans. Because the girder was roughly twice as
deep as the rolled section, it was tested over a span twice as long as the rolled section.
Test results are provided in Table 3.10 and are plotted in Figure 3.7.
Beam Damage
Width
Capacity % loss LTB
cap.
% loss
Mp
W27x94, 60’ -- 360.34 lb/in -- --
W27x94, 60’ 0.25b 347.95 lb/in 3.4% 4.6%
W27x94, 60’ 0.5b 338.28 lb/in 6.1% 9.6%
W27x94, 60’ b 227.71 lb/in 36.8% 20.4%
Pl. Gird., 120’ -- 253.52 lb/in -- --
Pl. Gird., 120’ 0.25b 242.53 lb/in 4.3% 5.0%
Pl. Gird., 120’ 0.5b 235.96 lb/in 6.9% 10.5%
Pl. Gird., 120’ b 207.66 lb/in 18.1% 22.9%
Table 3.10: LTB Capacity for Two Continuous Spans, Full Length Partial Width Flange Thinning
31
50%
60%
70%
80%
90%
100%
0 0.25 0.5 0.75 1
Fraction of Flange Thinned
% Remaining LTB Capacity
W27x94
60" Plate Girder
Figure 3.7: Remaining LTB Capacity vs. Width of Full Length Flange Thinning
Figure 3.7 shows that the two beams lost capacity at nearly the same rate until over half
the flange width had been thinned. However, W27x94 clearly lost capacity much faster
as damage exceeded half the flange width. Although plate girders were not tested at
length, it is worth noting that they will likely retain their LTB capacity better than a
rolled section would when subjected to flange thinning.
Web holes will have very little impact on lateral torsional buckling. Flange thinning has
a more significant impact on lateral torsional buckling capacity, but neither damage type
is likely to make LTB the governing limit state. Plastic moment capacity or shear
capacity would likely be more critical.
32
CHAPTER 4: FLEXURAL STRESS DISTRIBUTION IN CORROSION-DAMAGED
MEMBERS
Another major subject of investigation was flexural stress distribution in the presence of
corrosion damage. Removing cross sectional area from a beam may affect the flexural
stress distribution, because less material is available to resist the applied loads. Non-linear
finite element analysis which assumed elastic perfectly-plastic behavior was used
to model distressed members, and stress distributions were studied to determine whether
corrosion damage could elevate stress to dangerous levels, perhaps causing yielding
under lower loads than anticipated during the design of the member. Topics addressed in
chapter 4 are summarized in Table 4.1.
33
Section Analysis Set Topic
4.1 -- Test cases, confirm finite element results match
known theoretical results.
4.2 -- Demonstrate application of Vierendeel method.
4.3.1 1 Initial model, Vierendeel vs. finite element stress
distributions at hole sides and center.
4.3.2 2 How does varying hole size affect the accuracy of
the Vierendeel method.
4.3.3 3 Vierendeel applicability in no-shear locations
(beams with multiple point loads).
4.3.4 4 Vierendeel applicability under distributed load.
4.3.5 5 How does beam geometry and h/tw ratio affect the
onset of stress shift.
4.3.6 --
In what circumstances will Vierendeel accurately
predict the magnitude of the maximum flexural
stress.
4.3.7 6
How does the flexural stress increase at the hole
corner which is not the location of maximum
stress.
4.3.8 7
How do the maximum stresses around vertically
eccentric holes compare to the maximum stresses
around vertically centered holes.
Table 4.1: Topics of Chapter 4
4.1 Setup and Test Cases
As with lateral torsional buckling, a W27x94 was used as the initial test section. Some
tests were also run with the large plate girder section, with flange dimensions 18” x 1”
and web dimensions 60” x 0.375.” Simply supported boundary conditions were applied
at the bottom flange on both ends. In order to verify the accuracy of the finite element
model, simple test cases were run on undamaged beams and finite element results were
compared to theoretical stress distributions.
34
4.1.1: Case 1, Concentrated Load
The first test case used a simply supported W27x94, 30’, with a concentrated load applied
to the top flange at midspan. Simply supported boundary conditions were applied at the
bottom flange of the beam. See Figures 4.1 for boundary conditions. Flexural stress
distributions over the depth of the cross section are shown in Figures 4.2 and 4.3.
Figure 4.1: Test Case 1, Simply Supported Conditions at Bottom Flange
1
2
3
End 1: Displacement restricted 1,2,3
End 2: Displacement restricted 1,2
35
-15
-10
-5
0
5
10
15
-600 -500 -400 -300 -200 -100 0 100 200 300 400 500
Flexural Normal Stress, psi.
Vertical Location, in.
Theoretical
Finite Element
Figure 4.2: Flexural Stress Distribution, Test Case 1, Midspan
As shown in Figure 4.2, the stress distribution at midspan has a noticeable deviation from
the theoretical stress distribution. The largest errors are present at the top of the beam
directly beneath the point load, where finite element stresses are about 32% higher than
the theoretical stresses. This may be the result of the point load in the finite element
model, which places the entire load on one node, which is an infinitely small area. The
increased stress could also be a result of contact stresses which are not accounted for in
the theoretical model. For example, a stress element at the top of the beam subject to a
vertical compressive load will try to expand horizontally as a result of Poisson’s effect.
The element will be unable to expand due to the flexural compression already present at
the top of the beam, causing increased horizontal (flexural) compressive stress in the
36
element. (Much as a fully restrained steel bar would experience compressive stress if
subjected to a temperature increase.) To investigate whether contact stresses were
causing the errors, the stress distribution at quarterspan was also checked (see Figure 4.3
below). At quarterspan, the topmost point still has 9% error. However, almost all of the
rest of the cross section has a 1% error. It was felt that the small aberration at the top
flange was not significant, and that for the point loaded simply supported case, the finite
element model yields satisfactory results. However, the stress distribution directly
beneath a concentrated load will be affected by contact stresses and will not precisely
match the theoretical stress distribution.
-15
-10
-5
0
5
10
15
-200 -150 -100 -50 0 50 100 150 200
Flexural Normal Stress, psi.
Vertical Location, in.
Theoretical
Finite Element
Figure 4.3: Flexural Stress Distribution, Test Case 1, Quarterspan
37
4.1.2: Case 2, Uniformly Distributed Load
The second test case used a W27x94, 30’ long, with a 100 lb/in. uniformly distributed
load applied along the top flange. Simply supported boundary conditions were applied at
the bottom flange. The ABAQUS software has a “line load” loading function with units
of force/length for beam elements, but it is not applicable for shell elements. In order to
simulate the distributed load, a concentrated load was applied to every node along the
centerline of the top flange. A 1” mesh was used, so a 100 lb. force was applied at every
inch along the beam in order to approximate the 100lb/in distributed loading. Results are
shown in Figure 4.4.
-15
-10
-5
0
5
10
15
-8000 -6000 -4000 -2000 0 2000 4000 6000 8000
Flexural Normal Stress, psi.
Vertical Location, in.
Finite Element
Theoretical
Figure 4.4: Flexural Stress Distribution, Test Case 2, Midspan
38
Figure 4.4 shows the flexural stress distribution at midspan. Here, the finite element
results differ from theoretical by 3% and 5% at the top and bottom nodes, respectively.
The average error for the rest of the nodes is less than 1%. It was felt this model yields
satisfactory results for the simply supported, uniformly distributed load configuration.
Applying many closely spaced concentrated loads is one way of approximating a
uniformly distributed load. Test case 2 demonstrates that a reasonable degree of accuracy
is obtained when the loads are spaced at 1” for this span (which is 0.28% of the span
length). After observing the results of test cases 1 and 2, it was felt that the finite element
results agreed with known theoretical results closely enough to validate the use of similar
modeling techniques on future analyses.
4.2 Basic Vierendeel Analysis
A literature review found that an approximate hand method had been derived to predict
the flexural stress distribution through the beam cross section when a hole is present.
Simple “hand” calculation methods could be especially useful to engineers who might
encounter damaged beams but do not have access to advanced analytical software. This
method, the Vierendeel method (Bower 1966; Kulicki, Prucz et al. 1990), was
investigated further. Several finite element tests were run to check the accuracy and
applicability of the Vierendeel method.
The Vierendeel method is named for the Vierendeel truss. A Vierendeel truss consists of
rigid upper and lower beams connected only by vertical members. These members are
considered to be rigidly attached as opposed to most truss analyses which used pinned
end connections. A beam with a hole in the web may be analyzed much like a Vierendeel
39
truss. The sections remaining above and below the hole are considered to be the upper
and lower chords of the truss, while the areas at either end of the hole act as vertical truss
members. Figure 4.5 shows the setup for a simple Vierendeel analysis with important
parameters included.
Figure 4.5: Basic Setup for Vierendeel Method
The principal of the Vierendeel method is a simple extension of the basic flexural stress
equation,
I
σ = M ⋅ y . The stress at a point above (or below) the hole is given by
Equation 4.1:
t n
x I
M h
I
σ = ±V ⋅ a ⋅ y ± ⋅
Equation 4.1
Equation 4.1 is composed of two terms; a component from the shear at the hole and a
component from the global moment in the beam. When the a cross section of the beam at
the hole edge is analyzed, a cut is taken through the center of the hole as shown in Figure
(Bower, 1966)
40
4.6. Figure 4.6 also shows the stress distributions caused by each component alone and
the resulting stress distribution when the two components are added together.
Figure 4.6: Stress Components of Vierendeel Method
The first component of Equation 4.1 arises from the shear at the cut, but still takes the
basic form of My/I. For the given beam (Figure 4.5) with a concentrated load of
magnitude 2R at midspan, the end reaction is equal to R, and the shear anywhere between
the end reaction and midspan is therefore R. The free body diagram in Figure 4.6 shows
resulting shear on each remaining tee is 0.5R. For the point “d” shown in Figure 4.6 at
the top left corner of the hole, the moment is equal to the resulting shear multiplied by the
distance to point d, or V*a. The “I” value for this component is “It,” which is the moment
of inertia of the remaining t-section above or below the hole. The “y” value is the
distance from the centroid of the t-section to the point where the stress is being
determined, as shown in Figure 4.7.
(Bower, 1966)
41
Figure 4.7: Cross Section of Top Tee Section
The second component of Equation 4.1 still takes the form of My/I, and it involves the
moment on the gross beam section (for example, M=PL/4 at the center of a simply
supported beam with a point load at midspan). Note that “M” is drawn in Figure 4.6 at
the center of the hole for clarity, but is actually the moment at the point where stress is
being determined. For the point d, it is the moment at the very edge of the hole. The “I”
value for this component of Equation 4.1 is “In,” which is the moment of inertia for the
net beam section (obtained from the gross cross-sectional area minus the cross-sectional
area of the hole). The “y” value in this component is “h,” which is the distance from the
centroid of the gross cross section to the point where the stress is being determined.
Therefore, the stress at point “d” shown in Figures 4.5 and 4.6 is given by
t n
x I
M h
I
σ = −V ⋅ a ⋅ y − ⋅ . Both terms are negative because both moments (the beam
moment and the shear-induced moment) cause point “d” to be in compression. Figure 4.6
shows the stress distributions resulting from each of the two components of Equation 4.1,
as well as the final stress distribution resulting from their superposition (Bower, 1966).
For clarification, an example calculation is provided in section 4.2.1.
Centroid of tee
section
Point “d”
y
42
Though the fundamental principles remain unchanged, the actual calculations become
more complicated when the web hole is not vertically centered in the web. This is due to
the fact that the shear is no longer carried proportionately (for example, if 1/3 of the
remaining cross sectional area is in the top t-section, it cannot necessarily be assumed
that 1/3 of the resulting shear is carried in the top section). Calculations to determine the
shear distribution through the cross section are given in Kulicki, 1990.
An advantage of the Vierendeel method is that it can be performed without advanced
software or a major time investment. It can be done by hand, or spreadsheets can be
written to perform the calculations. Several examples of Vierendeel stress predictions
were compared with finite element analysis, and these comparisons show that the
Vierendeel stresses are conservative at the extreme fibers of the beam, and are often
conservative next to the hole (this is not always the case, because sometimes the stress
concentrations naturally present at a hole corner will outweigh the conservatism of the
Vierendeel approach). A weakness of the procedure is that it cannot predict stress
concentrations, which will be present at the edges and especially the corners of a hole.
Examples of stress distributions generated by finite element analysis compared with those
predicted by the Vierendeel analysis are given in section 4.2.1. The Vierendeel stress
points were generated with an Excel spreadsheet. Note that “inside” hole edge refers to
the edge of the hole nearest midspan, while “outside” refers to the edge of the hole
nearest the end of the beam as illustrated in Figure 4.8.
43
Figure 4.8: Inside Hole Edge, Outside Hole Edge, and Hole Center Notation
beam(shell element) bending(hole moved)
4.2.1: Example of finite element results vs. Vierendeel Calculations
Setup: W27x94, length = 30’, with 12”x12” vertically centered hole in the web. Hole
center is 7.5’ from beam end. Beam is simply supported with 100 lb. point load at
midspan. Results at the inside and outside hole edges are plotted in Figures 4.9 and 4.10.
The stress distributions from this model show that the Vierendeel analysis does predict a
similar stress distribution to that generated by finite element analysis. There is some
discrepancy between Vierendeel and finite element results, and in places the percentage
error is significant. However, the Vierendeel analysis gives conservative results at the
locations of highest stress. The stress distribution through the center of the hole is not
shown. At the center of the hole, the Vierendeel stress was not conservative next to the
hole. Yet it was conservative at the extreme fibers of the beam, where the stress was
higher (the critical location). The stress that would be predicted by elementary beam
theory if no hole were present,
I
σ = My , is also plotted. Note that it provides a very poor
match to the stress distribution obtained from finite element analysis; the location of the
maximum stress is incorrect and the value is inaccurate.
Outside Edge Inside Edge
Hole Center
44
-15
-10
-5
0
5
10
15
-30 -20 -10 0 10 20 30
Flexural Normal Stress, psi.
Vertical Location, in.
Finite Element
Theoretical, Vierendeel
Theoretical, no hole
Figure 4.9: Example Flexural Stress Distribution, Inside Hole Edge
-15
-10
-5
0
5
10
15
-40 -30 -20 -10 0 10 20 30 40
Flexural Normal Stress, psi.
Vertical Location, in.
Finite Element
Theoretical, Vierendeel
Theoretical, no hole
Figure 4.10: Example Flexural Stress Distribution, Outside Hole Edge
A
45
The following calculations demonstrate how the Vierendeel stress at point “A,” indicated
in Figure 4.10, was obtained. It may be helpful to refer to Figures 4.5 and 4.6.
4.2.2: Example Vierendeel Stress Calculations
Section: W27x94, 12”x12” hole, centered vertically 7.5’ from end of beam (u = 7.5’)
Load: Point load at midspan = 100 lb.
Because of the location of Point A, the flexural normal stress will be given by the
expression:
t n
x I
M h
I
σ = −V ⋅ a ⋅ y + ⋅
Now fill in the pieces of the equation:
Ig = 3270 in.4 (from AISC manual)
yb = centroid of bottom t-section = 1.513 in. up from bottom surface
It = I for bottom t-section, = 44.28 in.4
In = 3199.4 in.4 ( = Ig-1/12(.49)(12)3; .49 and 12 are the cross-sectional dimensions
of a 12” hole in the web of a W27x94)
V= 25 lb. (from statics as shown in Figure 3.7)
a= 6”
y = distance between yb and A. Point A vertically located 13.06” down from beam
centerline, = 26.9/2 -13.06 = 0.39 in. up from bottom surface. The distance of
13.06” is a result of the mesh generated in ABAQUS; 26.9” is the actual depth
of a W27x94.
46
=1.513-.39 = 1.123 in.
M = 50lb(7’)(12”/1’)= 4200 lb-in.
h = 13.06 in.
= − + = −3.804 +17.144 = 13.34psi
3199.4
(4200)(13.06)
44.28
(25)(6)(1.123)
x σ
At the bottom extreme fiber at the outside hole edge (corresponding to point A in
Figure 4.10), finite element methods yield a stress of 14.09 psi. The Vierendeel
method predicts a flexural stress of 13.34 psi., which is about 5% less than the
finite element result.
4.3 Simple Spans with Holes in Webs
4.3.1: Set 1—12”x 12” Vertically Centered Hole at Quarterspan, Concentrated
Load at Midspan
The first stress analyses performed on damaged beams were done for simple spans with
holes in the web. The first model used was the one presented in section 4.2.1., which
involved a 12” x 12” hole placed in the web of a 30’ W27x94 (results from this model are
dealt with more fully here than in section 4.2). The hole depth was limited to 12”
because reference material suggested that the Vierendeel method was most applicable to
holes not exceeding half the web depth (Bower 1966). The hole center was located at the
neutral axis, at quarterspan. A concentrated load of 100 lbs. was applied at midspan.
Plots of stress distributions at the inside hole edge, hole center, and outside hole edge are
shown in Figures 4.11, 4.12, and 4.13. The “Theoretical, no hole” plot is included to
illustrate the distribution predicted by
I
σ = My if no hole is present. Throughout this
47
work, a “no hole” distribution was calculated using the moment of inertia of the gross
section (as if no hole were present, hence the name). This was done as a means of
comparing the finite element and Vierendeel methods, which attempt to compensate for
the presence of a hole, with beam theory that does not attempt to compensate for the hole.
At the inside edge, the stress distribution predicted by the Vierendeel method matches the
distribution well (see Figure 4.11). The Vierendeel stresses are conservative at all the
maximum stress points.
-15
-10
-5
0
5
10
15
-30 -20 -10 0 10 20 30
Flexural Normal Stress, psi.
Vertical Location, in.
Finite Element
Theoretical, Vierendeel
Theoretical, no hole
Figure 4.11: Flexural Stress Distribution at Inside Hole Edge
The average error is high at -50%, but the average error is somewhat misleading. At two
points in the distribution, the Vierendeel and finite element stresses have opposite signs
48
(these locations are indicated by highlighted ellipses on Figure 4.11). These stress points
caused very high errors which inflated the average error; if these two points are excluded
the average error throughout the distribution is -3%. These locations also happen to be at
the points where the magnitude of the stress is smallest. Therefore a high percentage
difference between finite element and Vierendeel stresses reflects a small difference in
the actual stress values (for example, at the vertical location +6 inches the finite element
stress is -0.90 psi and the finite element stress is 2.48 psi, a difference of 3.38 psi).
At a cross section of the beam through the hole center (refer to Figure 4.8), the
Vierendeel method again provides a close match with the finite element results (see
Figure 4.12).
-15
-10
-5
0
5
10
15
-25 -20 -15 -10 -5 0 5 10 15 20 25
Flexural Normal Stress, psi. Vertical Location, in.
Finite Element
Theoretical, Vierendeel
Theoretical, no hole
Figure 4.12: Flexural Stress Distribution at Hole Center
49
It should be noted that the Vierendeel stress distribution is nearly the same as the “no
hole” distribution. Since the Vierendeel stress is derived from a cut taken at the center of
the hole, the “a” term (the moment arm over which the shear force “V” acts) goes to zero.
This eliminates the first term from the stress equation:
t n
x I
M h
I
σ = ±V ⋅ a ⋅ y ± ⋅ . For a
vertically centered hole, “h” is the same as “y” in the traditional flexural stress equation.
The only difference is in the moments of inertia; since the net moment of inertia is
slightly smaller than the gross moment of inertia the Vierendeel method will predict
slightly higher stresses than if the hole had been ignored.
At the outside edge of the hole, the results are somewhat similar to those at the inside
edge (see Figure 4.13). Although the shapes of the finite element and Vierendeel
distributions are somewhat different, inspection of Figure 4.13 shows the Vierendeel
method provides a better match with finite element results than would be obtained by
ignoring the hole. The Vierendeel stress is conservative at the points of highest stress,
which are next to the hole and not at the extreme fiber of the beam. At these locations,
the Vierendeel stress is about 22% higher than the finite element stress, while the “no
hole” theoretical stress is about 66% less than the finite element stress.
50
-15
-10
-5
0
5
10
15
-40 -30 -20 -10 0 10 20 30 40
Flexural Normal Stress, psi.
Vertical Location, in.
Finite Element
Theoretical, Vierendeel
Theoretical, no hole
Figure 4.13: Flexural Stress Distribution at Outside Hole Edge
It should be noted that the previous analysis was run entirely within the elastic range, so
the relatively small load used did not adversely affect results. Stresses will simply scale
up in proportion to the load as long as no observed stresses are outside the elastic range
(above 50 ksi).
Analysis set 1 indicated that the Vierendeel method had the potential to offer more
accurate flexural stress distributions around web holes than beam theory could provide.
51
4.3.2: Set 2—Preliminary Investigation of Vierendeel Applicability vs. Hole Size,
Holes at Quarterspan, Concentrated Load at Midspan
Analyses in Set 1 indicated that the Vierendeel method can provide accurate predictions
of flexural stress distributions. Reference material indicated that the Vierendeel method
was best used on holes which were less than half the member depth. Set 2 involved a
series of models with varying hole sizes. The deep plate girder described in section 3.1
(flanges 18” x 1”, web 60” x 0.375”) was used, because the deeper web allowed a wide
variety of hole depths to be modeled. Total member length was increased to 60’, and the
applied load was a concentrated load of 125,000 lb. applied on the top flange at midspan.
The load and length were increased to subject the larger member to a greater moment.
Holes were vertically centered at quarterspan, and were sized 10” x 10”, 20” x 20”, 30” x
30”, and 40” x 40”. The stress distributions at the inside hole edge, hole center, and
outside hole edge are shown for all cases.
10” x 10” Hole
The first model used in Set 2 involved a 10” x 10” hole at quarterspan. The flexural
stress distribution at the inside hole edge is shown in Figure 4.14.
52
-40
-30
-20
-10
0
10
20
30
40
-15000 -10000 -5000 0 5000 10000 15000
Stress, psi.
Vertical Location, in.
Finite Element
No hole
Vierendeel
Figure 4.14: Flexural Stress Distribution at Inside Hole Edge, 10” x 10”
Through most of the member cross section except very near the hole, stresses predicted
by elementary beam theory ignoring the hole provide a close match with finite element
results. At the top and bottom edges of the hole, the stress shows a steep increase. This
increase can be attributed to stress concentrations at the edge of the hole, which are not
accounted for by the Vierendeel method or the elementary beam theory. Although the
Vierendeel stresses at the top and bottom edges are closer to the finite element stress, the
plot shows that the “no hole” distribution has a better overall match with finite element
results. At the location of maximum stress, the beam theory stress differs from the finite
element stress by 3.1%, while the Vierendeel stress differs from the finite element stress
by 8.6%.
53
At the center of the hole, the finite element, Vierendeel, and “no hole” distributions are
all nearly the same (see Figure 4.15).
-40
-30
-20
-10
0
10
20
30
40
-10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
Figure 4.15: Flexural Stress Distribution at Hole Center, 10” x 10”
At the outside hole edge, the “no hole” distribution again matches finite element results
for most of the cross section (see Figure 4.16).
54
-40
-30
-20
-10
0
10
20
30
40
-10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
Figure 4.16: Flexural Stress Distribution at Outside Hole Edge, 10” x 10”
However, the deviations near the top and bottom of the hole due to stress concentrations
are more emphasized. For the 10” x 10” hole (16% total member depth), the flexural
stress distribution and the maximum flexural stress value are best predicted by the
elementary beam method, σ = My/I, with the exception of stress concentrations occurring
near the top and bottom hole edges. If a guideline with round numbers were to be used, it
appears that for holes sized less than or equal to 15% of the total member depth, the
flexural stress distribution is best calculated by ignoring the hole. (This will provide the
best general distribution, but it would still be prudent to note the effects of stress
concentrations around the hole.)
55
20” x 20” Hole
The next model used for Set 2 was identical to the first, but the hole size was increased to
20” x 20”. The flexural stress distribution at the inside hole edge is shown in Figure 4.17.
-40
-30
-20
-10
0
10
20
30
40
-15000 -10000 -5000 0 5000 10000 15000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
Figure 4.17: Flexural Stress Distribution at Inside Hole Edge, 20” x 20”
As demonstrated in Figure 4.17, increasing the hole size to 20” x 20” (32% total member
depth) causes the results to deviate further from the “no hole” stress distribution, and the
Vierendeel distribution showed a better correlation. At this point, neither method
produces a very accurate prediction of the finite element results, but inspection of the plot
shows the Vierendeel distribution matches slightly better than “no hole.” Although stress
concentrations at the top and bottom of the hole cause higher stresses than the Vierendeel
56
method predicts, the Vierendeel method is still conservative at the top and bottom fiber of
the beam, which is where the maximum stresses occur.
Figure 4.18 shows that for the 20” x 20” hole model, the stress distributions at the center
of the hole all match quite well.
-40
-30
-20
-10
0
10
20
30
40
-10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
span
Figure 4.18: Flexural Stress Distribution at Hole Center, 20” x 20”
At the outside hole edge (see Figure 4.19), there is a wide variation in the distributions.
57
-40
-30
-20
-10
0
10
20
30
40
-15000 -10000 -5000 0 5000 10000 15000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
Figure 4.19: Flexural Stress Distribution at Outside Hole Edge, 20” x 20”
When compared to the outside edge distributions for the 10” x 10” model and the 30” x
30” model (discussed next), it is apparent that the 20” hole depth is in “transition”—
ignoring the hole no longer provides a realistic stress distribution, but the Vierendeel
method doesn’t predict the distribution very well either. It is a similar situation to what is
seen at the inside edge, but the difference between the stress distributions is more
emphasized. The plot does show that the stress distribution is moving toward the
Vierendeel distribution and away from the “no hole” distribution. For this hole depth, the
maximum stress predicted by the Vierendeel method at the outside hole edge is about
29% less than the finite element stress. However, the Vierendeel method correctly
predicts the location of the maximum stress (next to the top and bottom hole edges),
58
which would not occur with a “no hole” distribution. It should be noted that although the
Vierendeel distribution does not match well for this hole depth, it comes closer to
predicting the magnitude and location of maximum stresses at all cross sections than does
the “no hole” distribution.
30” x 30” Hole
The third model incorporated in Set 2 had a 30” x 30” hole in the web; all other details
were the same as the previous two models. The flexural stress distribution at the inside
hole edge is shown in Figure 4.20.
-40
-30
-20
-10
0
10
20
30
40
-20000 -15000 -10000 -5000 0 5000 10000 15000 20000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
span
Figure 4.20: Flexural Stress Distribution at Inside Hole Edge, 30” x 30”
59
The plot in Figure 4.20 shows that for the 30” hole (48% of total member depth), the
Vierendeel method clearly provides a better method of approximating the stress
distribution than the “no hole” method at the inside edge. The Vierendeel method again
correctly identifies the locations of maximum stress, but it is not conservative at these
points (at the top and bottom hole edges). The Vierendeel stress at these points is
approximately 17% lower than the finite element predicted stress.
Stress distributions at the hole center continue to match very closely, as shown in Figure
4.21.
-40
-30
-20
-10
0
10
20
30
40
-10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
Figure 4.21: Flexural Stress Distribution at Hole Center, 30” x 30”
At the outside hole edge, the Vierendeel method provides a much closer match with finite
element results than it did for the outside edge of the 20” hole (see Figure 4.22).
60
-40
-30
-20
-10
0
10
20
30
40
-30000 -20000 -10000 0 10000 20000 30000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
y
Figure 4.22: Flexural Stress Distribution at Outside Hole Edge, 30” x 30”
As with the inside edge, the Vierendeel results are not conservative at the maximum
stress locations. The maximum Vierendeel stress is about 12% less than the maximum
finite element stress. For the 30” hole, the Vierendeel method provides a good
representation of the stress distribution, but does not conservatively predict maximum
stress at the hole edges.
40” x 40” Hole
The final model used in Set 2 incorporated a 40” x 40” hole (65% of total member depth).
The flexural stress distribution at the inside hole edge is shown in Figure 4.23.
61
-40
-30
-20
-10
0
10
20
30
40
-50000 -40000 -30000 -20000 -10000 0 10000 20000 30000 40000 50000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
Figure 4.23: Flexural Stress Distribution at Inside Hole Edge, 40” x 40”
At the inside hole edge, the Vierendeel stress distribution provides a reasonable match
with finite element results except for locations exactly at the top and bottom hole edges.
In this case the Vierendeel method and finite element do not agree on the location of the
maximum stress; finite element results show the maximum stress at the first node
above/below the hole edge instead of exactly at the hole edge. However, the Vierendeel
method is still conservative—the maximum Vierendeel stress is 16% higher than the
maximum finite element stress.
62
At the center of the hole, the Vierendeel and “no hole” deviate from finite element results
more than they have in the previous models. However, neither is significantly better than
the other at matching finite element results; they still match each other very closely. Both
are still conservative at the extreme fiber of the beam (location of maximum stress), as
shown in Figure 4.24.
-40
-30
-20
-10
0
10
20
30
40
-10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
Figure 4.24: Flexural Stress Distribution at Hole Center, 40” x 40”
At the outside hole edge, the Vierendeel method matches finite element results closely
near the extreme fibers. However, Figure 4.25 shows that the stresses differ near the
hole. Since the Vierendeel stresses near the hole are near 50 ksi, it appears that the
section is yielding near the hole. For this beam geometry and loading condition, a hole
through 65% of the member depth creates stresses beyond the elastic range. The
63
Vierendeel method does not have any inherent stress limit; it is up to one performing the
calculations to recognize when the method is predicting stresses beyond the yield stess.
-40
-30
-20
-10
0
10
20
30
40
-60000 -40000 -20000 0 20000 40000 60000
Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
Figure 4.25: Flexural Stress Distribution at Outside Hole Edge, 40” x 40”
Reference material suggested the Vierendeel method was best employed for holes less
than half the total member depth, and the final model of Set 2 confirms that. However,
Figures 4.23 and 4.25 indicate that for a hole at 65% of member depth, the Vierendeel
method will accurately predict the stress distribution until stresses approach the yield
point. If stresses were reduced by application of a smaller load, it is likely that the
64
Vierendeel method would provide accurate results around a hole as deep as 65% of
member depth.
Analysis set 2 shows that for small holes, flexural stress distributions are best predicted
by beam theory. As holes get larger, the maximum flexural stress will be found next to
the hole instead of at the extreme fiber of the beam. At this point, the magnitude and
location of the maximum flexural stress appear to be better predicted by the Vierendeel
method. This will be dealt with more in sections 4.3.5 and 4.3.6.
4.3.3: Set 3—Four-Point Loading, Hole in No-Shear Region
Sets 1 and 2 focused on simply supported beams with a single concentrated load. Set 3
was another configuration designed to address the possibility of multiple concentrated
loads. Figure 4.26 shows such a configuration. The response at A would be analyzed as
previously done, however notice point B is an area of no shear. This precludes the use of
the Vierendeel method, as shear is the parameter which alters the flexural stress response.
Figure 4.26: Four-Point Loading
Set 3 used the standard 30’ W27x94, simply supported with concentrated loads at the
third points. A 6” x 6” hole was vertically centered at midspan. Since there is no shear
A B
65
force present, the Vierendeel procedure would match elementary beam theory for a
section with no hole, i.e. σ = My/I. The stress distribution obtained at the hole edges is
provided in Figure 4.27 (the stress distributions at both hole edges are the same because
the center of the hole was at midspan and the beam was symmetrically loaded). The
finite element distribution in the presence of the hole appears to match almost perfectly
with the elementary beam theory distribution with no hole present. This is due to the
extremely small change in moment of inertia. A 6” hole vertically centered in a W27x94
will reduce the moment of inertia by about 0.3%.
-15
-10
-5
0
5
10
15
-20000 -15000 -10000 -5000 0 5000 10000 15000 20000
Stress, psi.
Vertical Location, in.
Finite Element
Elementary Beam
Theory
Figure 4.27: Flexural Stress Distribution at Hole Edge, 4 pt. Loading
66
Figure 4.28 is a plot of the stress distribution through the center of the hole at midspan.
Finite element results and elementary beam theory match well, except for a small
deviation next to the hole. The Vierendeel stress is about 22% lower than the finite
element stress. However, this is at the lowest stress location, so the actual stress
difference is not extremely large.
Figure 4.28: Flexural Stress Distribution at Midspan (Hole Center), 4 pt. Loading
Figure 4.29 is a plot of the stress distribution two feet from the hole edge. This location
is beyond the hole, but still in the no-shear region.
-15
-10
-5
0
5
10
15
-20000 -15000 -10000 -5000 0 5000 10000 15000 20000
Stress, psi.
Vertical Location, in.
Finite Element
Elementary Beam
Theory
67
-15
-10
-5
0
5
10
15
-20000 -15000 -10000 -5000 0 5000 10000 15000 20000
Stress, psi.
Vertical Location, in.
Finite Element
Elementary Beam
Theory
Figure 4.29: Flexural Stress Distribution 2’ From Hole Edge, 4 pt. Loading
The stress distributions for Set 3 show that elementary beam theory matches the
prediction of the flexural stress distribution given by finite element analysis anywhere in
the no-shear region, except for slight deviations next to the hole shown in Figure 4.28.
These deviations may still be a result of stress concentrations. Set 3 only modeled a hole
at midspan, but the results did not suggest a reason to model a hole at any other location
in the no-shear region. Results indicated that elementary beam theory will adequately
predict the stress at various locations at and around the hole in the no-shear region.
Moving the hole within the region will not create any shear force, so this pattern should
continue regardless of hole location within the region.
68
4.3.4: Set 4— Partial Length Uniformly Distributed Load, Hole at Midspan
The Vierendeel method was derived for a case with constant shear over the length of the
hole. Set 4 was run to investigate the accuracy of the Vierendeel method if a distributed
load is placed over the hole in the web, thus causing a variable shear over the length of
the hole. The test beam was the standard W27x94, 30’, simply supported. A uniformly
distributed load of 10,000 lb/ft. was applied to the center 7.5’ of the beam (1/4 of the total
beam length). The flexural stress distribution at the hole edges is shown in Figure 4.30
(again, the beam and hole are symmetrical, so the stress distribution at both sides of the
hole is the same).
-15
-10
-5
0
5
10
15
-30000 -20000 -10000 0 10000 20000 30000
Flexural Normal Stress, psi.
Vertical Location, in.
Finite Element
Vierendeel
No hole
Figure 4.30: Flexural Stress Distribution at Hole Edge, Partial Uniformly Distributed Loading
69
The Vierendeel method in its original form is not applicable to a distributed load with
variable shear across the hole. In an attempt to extend the application of the Vierendeel
method, Vierendeel calculations were conducted by replacing the distributed load with an
equivalent point load. As Figure 4.30 shows, this results in an extremely inaccurate
stress distribution. The “no hole” distribution does provides an accurate prediction of the
stress distribution. Closer examination of the scenario reveals that the results make
sense. The balanced nature of the distributed load and end reaction leave no shear to act
on the remaining tee sections when the beam is cut for analysis. Therefore, the “no-shear”
conditions of Set 3 were effectively re-created.
70
4.3.5: Investigation of Stress Shift
In many practical situations, the magnitude and location of the maximum stress will be
more important than predicting the entire stress distribution. Previous analyses for holes
of varying sizes indicate that the maximum flexural stress will either occur at the inside
hole edge at the extreme fiber (point A in Figure 4.31), or at the outside hole edge at the
hole corner (point B in Figure 4.31). The maximum stress will be found at point A for
smaller holes, then as the hole size is increased the maximum stress shifts to point B.
Figure 4.31: Stress Shift
For a simply supported beam with a point load at midspan, elementary beam theory states
that the maximum flexural stress will occur at the extreme fiber of the beam at midspan.
If a hole is present and beam cross sections at the inside and outside hole edges are
considered, elementary beam theory would still predict the maximum stress occurs at the
extreme fiber where the moment is highest (point A in Figure 4.31). As the hole size
increases, the shear induced moments (see Figure 4.32) become more and more
significant, and eventually the location of the maximum normal stress shifts to point B.
71
Figure 4. 32: Global and Shear-Induced Moments
The increased significance of the shear induced moments can be more clearly understood
if the Vierendeel stress equation and a sample beam are examined. If the statics
presented in Figure 4.32 are considered, the stress at both points A and B (Figure 4.31) is
given by the expression
t n
x I
Mh
I
σ = −Vay − . The reason for the shift comes from the “y”
term corresponding to the shear-induced moment, as illustrated in Figure 4.33.
Figure 4.33: “y” Term for Shear-Induced Moment
Centroid of tee
section
Point “B”
yB
Point “A”
yA
72
This term is the distance between the neutral axis of the tee section and the point in
question. For the top tee where points A and B are found, the neutral axis will be very
near the flange, if not in the flange, depending on beam geometry. Therefore, the “y”
term for point B will be larger than that for point A.
For small holes, this is offset by the fact that the global moment at point A is higher.
However, as the hole size increases, the value of “y” at point A decreases faster than it
does at point B. This is effectively an increase in the value of “y” at point B as hole size
increases. For example, see Table 4.2. In this table, “y” values are calculated at points A
and B for increasing hole sizes in a beam with 18” x 1” flanges and a 60” x 0.375” web.
When the relative size of the “y” values at points A and B are considered, it is apparent
that the shear-induced moment is going to play a larger role as hole size increases.
Hole Size Point A Point B
Relative Size of “y”
(Point B “y”/Point A “y”)
10”x10” 4.45 21.05 4.73
20”x20” 3.09 17.41 5.64
40”x40” 0.95 9.55 10.08
Table 4.2: “y” Values for Shear-Induced Moment in Sample Beam
In an effort to determine when stress shift occurs, analysis set 5 modeled several beams
with finite element analysis. Preliminary tests which compared a W27x94 to a built-up
plate girder suggested that the h/tw ratio (web depth: web thickness) of a beam might
affect the hole size at which stress shift occurs, so analyses focused on the effects of
73
varying the h/tw ratio. The first series of beams analyzed is described in Table 4.3 (this
beam series is referenced at other times, and will be referred to from this point on as
beam Set A). The h/tw ratios were varied by holding the web thickness constant and
changing the web depth.
h/tw ratio Flanges Web Length Length/Depth
ratio Load (lb)
76 14” x 1” 0.375” x 28.5” 408” 13.38 75,963
107 14” x 1” 0.375” x 40” 552” 13.14 82,350
144 14” x 1” 0.375” x 54” 744” 13.29 86,853
155 14” x 1” 0.375” x 58” 804” 13.40 87,570
160 14” x 1” 0.375” x 60” 828” 13.35 88,589
165 14” x 1” 0.375” x 62” 852” 13.31 89,592
176 14” x 1” 0.375” x 66” 888” 13.06 92,790
211 14” x 1” 0.375” x 79” 1080” 13.33 95,433
213 14” x 1” 0.375” x 80” 1080” 13.17 96,961
Table 4.3: First Series of Beams Tested for Stress Shift (Set A)
The beam lengths were varied in order to maintain a roughly constant length-to-depth
ratio. (Whole foot increments were easiest to use in the Vierendeel spreadsheet, so there
is some slight variation in the length-to-depth ratio. However, the largest ratio is only
2.6% larger than the smallest ratio.) This ratio is important because it dictates how much
of the flexural stress is caused by shear-induced (Vierendeel) moments and how much is
caused by the global moment. For example, one might consider a cross section at
quarterspan in a simply supported beam with a point load at midspan. The global
moment at that cross section will be given by M = PL/8, so if the length of the beam is
74
doubled but P is held constant, the global moment will double. However, the shear-induced
moment is unaffected by beam length (again, see Figures 4.5 and 4.6).
Loads were also varied for the sake of consistency; all of the given loads will yield the
same maximum flexural stress in their respective beams when no hole is present. All
beams were simply supported with point loads at midspan.
Vertically centered square holes of varying sizes were placed at quarterspan on the
beams. The hole size required to cause the stress shift was not determined exactly, but
was bracketed in a 2% window. Because of the variation in beam geometry, hole size
was dealt with as a percentage of total member depth (a 15” hole in a 30” deep beam is
considered the same size as a 20” hole in a 40” deep beam).
Figure 4.34 is a plot of the hole sizes (as a percentage of beam depth) required to cause
stress shift as determined by finite element analysis and predicted by the Vierendeel
method.
75
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
50 75 100 125 150 175 200 225
h/tw ratio
Hole Depth
Finite Element
Vierendeel
Difference
Linear (Difference)
Figure 4.34: Hole Size Required to Cause Stress Shift vs. h/tw Ratio
Although there is a 10-20% discrepancy between the finite element and Vierendeel
results, both trends are approximately linear. The fact that hole sizes weren’t determined
exactly but were bracketed to within ±2% probably accounts for the deviation from the
linear trend line. This bracketing procedure also explains how there can be a deviation
between points plotted very close together, such as those for h/tw ratios of 211 and 213.
For example, the beam with an h/tw ratio of 211 showed stress shift at a hole depth of
36”, or 44.4%, according to the Vierendeel method. The previous hole size checked was
35”, or 43.2%. Since the difference was less than 2%, the shift was considered to be
bracketed and the result was plotted as 44.4%. For the beam with an h/tw ratio of 213,
the Vierendeel method showed stress shift at a hole depth of 36”, or 43.9%, while the
76
previous hole checked was 35”, or 42.7%. The true stress shift points could lie anywhere
in the bracketed windows. The true stress shift for h/tw=211 could be anywhere between
43.2% and 44.4%, while the true stress shift for h/tw=213 could be anywhere between
42.7% and 43.9%. This means it is possible for the shift point of h/tw=213 to be slightly
higher than that for h/tw=211, even though the current plot shows it slightly lower.
To investigate whether the initially observed trends held true for other beam geometries,
a second and third series of beams were modeled. In these two series, beams having
approximately the same h/tw ratios as those of the first series were tested. However, the
flange geometry and web geometry were altered. The second series of beams, listed in
Table 4.4 and referred to from this point on as Set B, still had 0.375” thick webs but had
wider flanges. The third series of beams, listed in Table 4.5, had the same 14” x 1”
flanges as Set A. It is referred to from here on as Set C. In Set C, the webs were thinned
and the overall section depths were reduced.
h/tw ratio Flanges Web Length Length/Depth
ratio Load (lb)
76 18” x 1” 0.375” x 28.5” 408” 13.38 95,364
101 18” x 1” 0.375” x 38” 540” 13.50 98,900
128 18” x 1” 0.375” x 48” 672” 13.44 103,447
156 18” x 1” 0.375” x 58” 804” 13.40 107,583
181 18” x 1” 0.375” x 68” 936” 13.37 111,479
208 18” x 1” 0.375” x 78” 1068” 13.35 115,225
Table 4.4: Second Series of Beams Tested for Stress Shift (Set B)
77
h/tw ratio Flanges Web Length Length/Depth
ratio Load (lb)
76 14” x 1” 0.25” x 19” 280” 13.33 69,451
100 14” x 1” 0.25” x 25” 360” 13.33 72,179
128 14” x 1” 0.25” x 32” 454” 13.35 74,628
156 14” x 1” 0.25” x 39” 546” 13.32 77,040
180 14” x 1” 0.25” x 45” 627” 13.34 78,636
208 14” x 1” 0.25” x 52” 720” 13.33 80,547
Table 4.5: Third Series of Beams Tested for Stress Shift (Set C)
As with Set A, hole size was not determined exactly but was bracketed to within ±2%.
The finite element results for all three beam sets are plotted in Figure 4.35. Appendix A
contains similar to 4.34 for beam sets B and C, comparing Vierendeel and finite element
predictions for stress shift.
78
0%
5%
10%
15%
20%
25%
75 95 115 135 155 175 195 215
h/tw ratio
Hole Size
14" Flanges, 0.375" Webs (A)
18" Flange, 0.375" Webs (B)
14" Flanges, 0.25" Webs (C)
Figure 4.35: Hole Size Required to Cause Stress Shift in Three Sets of Beams
Figure 4.35 shows that altering beam geometry does impact the point of stress shift. The
top line plotted in Figure 4.35 corresponds to the beam Set A. The lower two lines show
consistently lower points of stress shift for all h/tw ratios in the modified beam
geometries. Also, the plots for all three beam series demonstrate a trend to level out
around h/tw=155. Plots for Vierendeel predictions do not do this; as shown in Figure 4.34
they remain linear throughout the h/tw range. Since the main thing the Vierendeel cannot
account for is stress concentrations, it appears that stress concentrations become
especially significant around h/tw=155. After this point, stress concentrations may
govern the stress shift (instead of the shear-induced moment).
79
Figure 4.36 addresses the question of how well Vierendeel predictions for stress shift
match finite element results. The differences between Vierendeel and finite element
results for all three beam sets are plotted in Figure 4.36.
5%
7%
9%
11%
13%
15%
17%
19%
21%
23%
50 75 100 125 150 175 200 225
h/tw ratio
Difference
14" flange, 0.375" web (A)
18" flange, 0.375" web (B)
14" flange, 0.25" web (C)
Figure 4.36: Differences Between Finite Element and Vierendeel Hole Sizes to Cause Stress Shift
Note that the percent difference is still expressed in terms of hole size. For example, if
Vierendeel predicted stress shift at a hole depth of 30%, and finite element results
indicated shift at a hole depth of 12%, then the difference is plotted as 18%. Figure 4.36
demonstrates that altering flange geometry has little to do with how well Vierendeel and
finite element results match. However, the section with the thinner web showed
consistently less difference between Vierendeel and finite element predictions for stress
shift.
80
Two main trends were illustrated in Section 4.3.5. The first is that the hole size required
to cause stress shift tends to increase with h/tw ratio, until a point at which stress
concentrations govern over the shear-induced moment. For the three beam geometries
examined, this was around h/tw=155. The second trend is that the difference between
finite element and Vierendeel predictions for stress shift tends to increase linearly over
the h/tw range observed. The differences were similar for all three beam geometries
studied, with the thinned flange geometry (Set C) showing slightly less discrepancy
between Vierendeel and finite element predictions for stress shift.
The difference between Vierendeel and finite element predictions appears to increase
linearly with h/tw ratio regardless of beam geometry.
It should also be noted that besides the beams modeled in Sets A, B, and C, the W27x94
was also modeled. This section has an h/tw ratio of only 49.5. Finite element analysis
predicted stress shift at a hole size of 35%, while Vierendeel predicted stress shift at a
hole size of 41%. This 6% difference is in keeping with results shown in Figure 4.36,
which demonstrate that there is less discrepancy between finite element and Vierendeel
predictions at lower h/tw ratios.
4.3.6: Vierendeel Method Applicability
Section 4.3.2 examined the Vierendeel technique in approximating flexural stress
distributions. Results showed that the stress distribution varies with hole size, and for
some configurations the stress distribution is still best predicted by beam theory (σ =
My/I). However, since the maximum stress in a cross section (and the location of that
81
stress) is usually more important, efforts were made to establish ranges for which the
Vierendeel method could be used to determine the magnitude and location of the
maximum stress around the hole.
Before Stress Shift—As discussed in section 4.3.5, the maximum flexural stress will be
found at the extreme fiber of the inside hole edge until the hole is large enough to induce
stress shift (see point A, Figure 4.37). This is because the global beam moment
dominates the shear-induced moments for small holes.
Figure 4.37: Maximum Stress Locations Before and After Stress Shift
Figure 4.38 examines the difference between maximum stresses predicted by the
Vierendeel method and finite element analysis. The hole sizes plotted were too small to
have induced stress shift, so the maximum stresses are all at the inside hole edge. Beam
Set A (14”x1” flanges, 0.375” webs) was used to generate Figure 4.38.
82
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
5% 7% 9% 11% 13% 15% 17% 19% 21% 23% 25%
Hole Depth
Discrepancy
h/tw = 76
h/tw = 107
h/tw = 144
h/tw = 176
Figure 4.38: Discrepancy Between Finite Element and Vierendeel Max Stresses (Before Stress Shift)
In Figure 4.38, few data points are plotted for each beam because data were obtained
while attempting to determine hole size required for stress shift. Analysis was begun
with hole sizes estimated to be near the stress shift size. The last hole size plotted is the
size at which stress shift occurred. Figure 4.38 demonstrates that as hole sizes increase,
the discrepancy between Vierendeel and finite element maximum stresses increases.
However, the maximum discrepancy for all beam geometries plotted remained around
8% and the Vierendeel stresses were always conservative when compared to finite
element stresses. Similar plots for the beam Sets B and C are included in Appendix B,
with similar results.
83
Figure 4.39 plots the difference between maximum stresses predicted by beam theory and
the finite element method at the inside hole edge (before stress shift) for the beams of Set
A. The largest hole size plotted for each beam was the largest hole modeled before finite
element analysis predicted stress shift to occur. Errors for hole sizes between 5% and
30% of member depth are less than 4%. When compared with Figure 4.38, which often
shows discrepancies in the 8-9% range, it is apparent that before stress shift occurs beam
theory does an adequate job of predicting the maximum flexural stress.
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
4.5%
0% 5% 10% 15% 20% 25% 30% 35%
Hole Size
Difference Between Beam Theory and
Finite Element Max. Stress
h/tw=76
h/tw=107
h/tw=144
h/tw=155
h/tw=165
h/tw=176
h/tw=213
Figure 4.39: Difference Between Beam Theory and Finite Element Maximum Stresses
Before Stress Shift
84
After Stress Shift—For holes large enough to have caused stress shift, the maximum
flexural stress will be located next to the hole at the outside hole edge (point B in Figure
4.37). In this case, beam theory will never predict the correct location for the maximum
stress, because it always yields highest stresses at the extreme fibers. The Vierendeel
method is better able to predict the correct location of maximum stress after stress shift
has occurred. Figure 4.40 plots the discrepancy between maximum stresses predicted by
the Vierendeel method and finite element analysis after stress shift has occurred. The
beams modeled were those from Set A.
-50%
-40%
-30%
-20%
-10%
0%
10%
20%
0% 10% 20% 30% 40% 50% 60% 70% 80%
Hole Depth
Discrepancy
h/tw= 76
h/tw= 107
h/tw=144
h/tw= 176
Figure 4.40: Discrepancy Between Finite Element and Vierendeel Max Stress (After Stress Shift)
85
The discrepancy between Vierendeel and finite element maximum stresses tends to
decrease as hole size increases. In some cases, the discrepancy is positive, meaning the
Vierendeel stress is conservative.
Figure 4.41 brackets the ranges of Vierendeel applicability for beam sets A, B, and
C (three series of beams are included to illustrate the effects of varying beam geometry).
Applicability was defined as predicting a stress which was no more than 5% less than the
finite element stress.
0%
10%
20%
30%
40%
50%
60%
70%
75 85 95 105 115 125 135 145
h/tw ratio
Hole Depth
Set A (upper)
Set A (lower)
Set B (upper)
Set B (lower)
Set C (upper)
Set C (lower)
Figure 4.41: Upper and Lower Bounds on Vierendeel Applicability
86
Two lines are plotted for each beam geometry. The uppermost line is the upper bound of
applicability; for hole sizes above this line the Vierendeel procedure should yield
maximum flexural stresses within 5% of finite element maximums. For example, a beam
might fit the geometry of the Set C (0.25” thick web) and have an h/tw ratio of 105 and a
square hole depth of 50%. This point would fall above the upper bound line for that
beam geometry (the upper red line), so the Vierendeel method and finite element analysis
should agree within 5% on the maximum flexural stress. Likewise, the lower line is the
lower bound of applicability; for hole sizes below this line the Vierendeel procedure
should yield maximum flexural stresses within 5% of finite element maximums. The
lower bounds are not particularly useful, because they are typically below the hole size
required for stress shift, and maximum stresses can be adequately predicted by beam
theory. The heavy dashed lines in Figure 4.41 are the linear trend lines for each bound.
Changing the flange geometry had little affect on the upper bound, but thinning the web
significantly lowered the upper bound line (meaning the Vierendeel method has a wider
range of application for beams with thinner webs).
Although all three beam sets included members with h/tw ratios into the lower 200’s,
Figure 4.41 only includes a few data points. This is because the hole size is becoming
quite large for the upper bounds of the last h/tw points plotted. As hole size continues to
increase, yielding is occurring next to the hole edge. If the error is not less than 5% when
yielding begins to occur, it will never be less than 5%—finite element stress will remain
at yield stress, while Vierendeel stresses will continue to increase. For the higher h/tw
ratios, Vierendeel stresses were never applicable (within 5% of finite element values).
87
4.3.7: Stress Increase at Hole Corner, Inside Hole Edge
As discussed in previous sections, the maximum stress around a hole will be at the
extreme fiber of the inside hole edge before stress shift occurs. After the shift occurs, the
maximum stress is located next to the hole on the outside hole edge. At no point is the
maximum stress located at the inside hole edge next to the hole. However, that does not
necessarily mean that there are not significant stress increases at that location (marked as
point “x” in Figure 4.42).
Figure 4.42: Inside Hole Edge, Next to Hole
Although yielding will not be a concern, elevated stress levels could be high enough to
exceed threshold levels for fatigue. Figure 4.43 plots the results of analysis set 6, which
utilized the beams from Set A to investigate the difference between stresses yielded by
finite element analysis and those predicted by beam theory.
88
0%
100%
200%
300%
400%
500%
600%
0% 10% 20% 30% 40% 50% 60% 70%
Hole Depth
Percent Difference Between Stresses
h/tw=76
h/tw=107
h/tw=144
h/tw=160
h/tw=176
h/tw=211
Figure 4.43: Difference Between Beam Theory and Finite Element Stresses,
Inside Edge Next to Hole
It is important to note that the finite element stress was higher than the beam theory stress
in all cases. The minimum discrepancy between beam theory and finite element stresses
typically occurred at hole sizes around 30% of member depth. For small holes, the
location in question (shown in Figure 4.42) is near the neutral axis. Because of the low
flexural stresses near the neutral axis, the large discrepancies encountered for hole sizes
below 30% of member depth may or may not be significant. However, the increased
stresses for holes larger than 30% are likely more significant since the hole edge is
further from the neutral axis. Similar plots were created for beam sets B and C to
confirm similar trends in members with varied geometries. These plots are attached as
Appendix C. Although the maximum flexural stress is not at the inside edge next to the
89
hole, stresses at that location may elevate significantly, necessitating inspection for small
cracks/tears in the web that could initiate a fatigue failure.
4.3.8: Vertically Shifted Holes
Previous sections have all focused on holes which are vertically centered on the web.
This was a simple place to begin analysis, but it is possible that holes encountered in the
field would not be vertically centered in the web. The trends developed for vertically
centered webs required several dozen tests to be run with finite element analysis, and
time constraints did not allow as many tests to be run again for members with vertically
shifted holes. Analysis set 7 employed finite element analysis to make some basic
comparisons between maximum stresses in vertically centered and vertically eccentric
holes. Beam set A was employed.
Before Stress Shift: It has been established that while holes are smaller than a certain
size, the maximum flexural stress around the hole is still found at the extreme fiber of the
cross section on the inside hole edge. Figure 4.44 demonstrates the trend in the
maximum flexural stress at the inside hole edge as vertical eccentricity increases.
90
0
2000
4000
6000
8000
10000
12000
14000
16000
50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0%
Eccentricity
Flexural Stress, psi.
h/tw=76, 5" hole
h/tw=144, 10" hole
h/tw=211, 14" hole
Figure 4.44: Maximum Stress at Inside Hole Edge with Increasing Vertical Eccentricity
Three widely varied h/tw ratios were used, and each case employed a hole depth about
17% of total member depth (this hole was smaller than the stress shift size in each case).
Vertical eccentricity is increasing from left to right across the x-axis; eccentricity is
defined here as a ratio of the web area below the hole divided by the total remaining web
area, a/(a+b). See Figure 4.45 for “a” and “b.” Note that 50% will be vertically centered,
and decreasing percentages correspond to downward shifting of the hole.
Figure 4.45: "a" and "b" in Eccentricity Definition
91
Figure 4.44 demonstrates that depending on h/tw ratio, the hole can be shifted down until
only 1/3 to 1/4 of the remaining web area is below the hole without causing significantly
higher stresses than those caused by vertically centered holes. For greater amounts of
eccentricity, the maximum stresses begin to increase at roughly 2% for each additional
1% eccentricity.
After Stress Shift: After stress shift has occurred, the maximum flexural stress will
occur next to the hole at the cross section on the outside hole edge. Figure 4.46 is similar
to Figure 4.44; it utilizes the same three beams and displays the maximum stresses at the
outside hole edge as vertical eccentricity increases.
92
0
5000
10000
15000
20000
25000
50% 45% 40% 35% 30% 25% 20% 15% 10% 5% 0%
Eccentricity
Flexural Stress, psi.
h/tw=76, 12" hole
h/tw=144, 22" hole
h/tw=211, 32" hole
Figure 4.46: Maximum Flexural Stress at Outside Hole Edge with Increasing Vertical Eccentricity
The holes used in Figure 4.46 are all about 39% of total member depth, and were large
enough that stress shift has already occurred. At the outside hole edge after stress shift,
eccentricity immediately causes an increase in maximum flexural stress. The stress
increases at a rate of approximately 1% for each 1% of vertical eccentricity (0.84%/1%
for h/tw=76, 1.08%/1% for h/tw=144, and 1.23%/1% for h/tw=211). This may continue
until 1/3 to 1/5 of the remaining web area is below the hole (depending on h/tw ratio), at
which point the maximum stresses begin to decrease again.
Chapter 4 dealt with several issues regarding flexural stress distributions around web
holes. Analysis set 2 showed that for small holes, flexural stress distributions are best
predicted by beam theory. As holes get larger, the maximum flexural stress will be found
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next to the hole instead of at the extreme fiber of the beam. At this point, the magnitude
and location of the maximum flexural stress appear to be better predicted by the
Vierendeel method. Analysis set 5 demonstrated that the hole size required to cause
stress shift tends to increase with h/tw ratio, until a point at which stress concentrations
govern over the shear-induced moment. Also, the difference between finite element and
Vierendeel predictions for stress shift tends to increase linearly over the h/tw range
observed. Analysis sets 3 and 4 showed that for holes in no-shear regions of beams,
beam theory adequately predicts the location and magnitude of the maximum flexural
stress. Analysis set 6 revealed that even at hole corners where flexural stress is not at a
maximum level, stresses can be significantly elevated from beam theory predictions.
This stress elevation is pertinent because it could lead to fatigue failures. Finally,
analysis set 7 dealt with maximum flexural stresses around vertically eccentric holes.
Before stress shift, holes can be shifted so that only 1/3 to 1/4 of the remaining cross
section is below the hole before maximum stress is significantly higher than for a
vertically centered hole. After stress shift, maximum stress begins to elevate immediately
with vertical eccentricity. For the three geometries studied, stress increases at a rate of
approximately 1% for each 1% of eccentricity, until a maximum stress is reached and
maximum stresses begin to decline again.
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CHAPTER 5: PLASTIC MOMENT CAPACITY
Bridge design often utilizes the plastic capacity of members as a limit state. The plastic
capacity (or the plastic moment) is defined as the moment which causes (theoretically)
every fiber in the member to yield. This requires a member with elements compact
enough to fully yield before buckling. Chapter 2 demonstrated that for many members
with thinned flanges or holes in the web, the loss in lateral torsional buckling capacity
was overshadowed by the loss in plastic moment capacity. Chapter 3 focused largely on
the location of maximum flexural stresses, which can be a concern for first yielding and
fatigue. Chapter 5 considers the plastic moment capacity of sections with web holes in
greater depth. For a beam with a hole in the web, the plastic moment (Mp) can easily be
calculated without the help of the finite element method. A spreadsheet was used to
calculate Mp by simply performing a force balance to determine the location of the
plastic neutral axis (PNA), then summing moments about the PNA to determine capacity.
5.1 Vertically Centered Holes
Figure 5.1 demonstrates the loss in plastic moment capacity versus hole size fo