Guidelines for the Load and Resistance Factor Design and Rating of Riveted and Bolted Gusset-Plate Connections for Steel Bridges (2013)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

5 CHAPTER 1. BACKGROUND The collapse of the I-35W Bridge has focused significant attention on the reliability and safety of truss bridges in the U.S. The National Transportation Safety Board (NTSB) concluded the collapse was attributed to under-designed gusset plates at the U10 nodes. (1) Roughly the plates were about half as thick as they should have been. The under-sized plates resulted in a large portion of the plate yielded in shear, which in turn lead to a reduced capacity against out-of-plane sway. Ultimately, failure of the U10 connection occurred when the compression diagonal swayed out of the main truss plane and the connection destroyed itself as it then fell through the compression diagonal. A forensic design review of all gusset plates in the bridge indicated that the gusset plates at the U10 and L11 nodes were all under-sized, clearly as the result of an error in the design process. (2) The intended margin of safety against connection failure was far below that of a typical truss design. In its final report, the NTSB issued five recommendations to the FHWA and the American Association of State Highway Transportation Officials (AASHTO) to prevent such future catastrophes. These were: 1) Develop and implement a strategy to increase quality control measures for reviewing and approving bridge plans. 2) Develop guidance for owners regarding the placement of construction loads on bridges during construction and maintenance. 3) Require owners to assess regions where gusset plates cannot be inspected visually and recommend nondestructive techniques to assess hidden corrosion. 4) Revise inspection manuals and training materials to include guidance for gusset plates. 5) Require owners to include gusset plates as part of the load rating process. To address the fifth recommendation, FHWA issued Technical Advisory 5140.29 in January 2008 stressing the need to check connection capacity along with member capacity in the load rating process. (3) Based on feedback from many sources, it became apparent that there was no clear consensus on the specific procedures to follow for design or rating of gusset plates. The AASHTO code is vague on the subject, leaving room for considerable engineering discretion and judgment in the process. Load rating describes calculating how much live load an element can carry considering its current condition. To provide a uniform standard for load rating gusset plates, FHWA issued a guidance document in February 2009 based on the best available information on gusset plate design. (4) This document will herein be referred to as the “FHWA Guide” or “Guidance”. Early experience showed that some of the truss bridges in service will fail certain limit-state checks

6 when analyzed with respect to the FHWA Guide. This is not surprising considering that the original bridge designers had considerable discretion and probably did not follow or know of the exact procedures outlined in the FHWA Guide. These inconsistencies generate a fundamental question: Is the FHWA Guide overly conservative or are many of the existing truss bridges inadequately designed? Unfortunately, there is only limited research to support some of the provisions in the FHWA Guide. There has been limited experimental research on the ultimate strength of gusset plates, much of it directed to performance of tension members and their connections. There has been little experimental work on the compression capacity and stability of gusset plates and most of what exists focuses on bracing connections common in building structures. Only two references were found that report experimental results of ultimate capacity of large scale gusset-plated bridge connections. (5, 6) A meeting was held in April 2008 in Nashville, TN, to discuss gusset plate design provisions and to help refine the FHWA Guide provisions. This meeting was led by the Chairman (at the time) of the AASHTO T-14 Committee on Structural Steel Design, Ed Wasserman of Tennessee Department of Transportation (DOT), and included the following experts: • Robert Connor Purdue University • Rick Crawford Tennessee DOT • John Fisher Lehigh University • Theodore Galambos University of Minnesota • Joseph Hartmann FHWA • Firas Ibrahim FHWA • John Kulicki Modjeski & Masters • William Wright FHWA • Joseph Yura University of Texas/Austin It became apparent following the meeting that there was insufficient research to fully support refinement of many of the limit-state checks in the FHWA Guide. There was no clear consensus among the experts on what procedures were best to perform stress and stability checks for gusset plates in compression, and no clear consensus on which checks might typically govern design. While it was generally acknowledged that the current draft of the FHWA Guide represents the best available knowledge on gusset plate design, it was also acknowledged that it may be overly conservative for some limit-state checks. DISCUSSION ON GUSSET PLATE DESIGN The following discussion examines the individual design checks in the FHWA Guide and sets the framework for the issues addressed by this research. Although no formal minutes were generated at the Nashville meeting, opinions from this group will be discussed where appropriate. In addition, some results from FHWA work on the I-35W Bridge failure

7 investigation will be discussed as they pertain to the limit-states examined. The discussion will also reflect results from a literature review of gusset plate research. Throughout this document references will be made to the AASHTO LRFD Bridge Design Specifications herein referred to as the “BDS”.(7) The AASHTO Manual for Bridge Evaluation will herein be referred to as the “MBE”.(8) Resistance of Fasteners Numerous researchers have studied the performance of tension member connections for both bolted and riveted structures. The research has shown that connections are capable of redistributing load between fasteners as the connection nears ultimate capacity. Therefore, it is generally assumed that each fastener carries an equal share of the load (for non-eccentric connections less than 50 inches in length). Trusses built prior to 1965 typically have riveted gusset plates but more modern bridges typically use high-strength bolts. Rivet capacity is an area of concern, particularly for rating older bridges built prior to the ASTM A502 Specifications for rivet strength. This project will review all available literature involving rivet tests to determine what capacities are supported by research. Beyond that, no experimental work to determine bolt or rivet capacity will be included in the present study. It is possible, however, that the capacity of gusset plates will be somewhat different depending on whether rivets or bolts are used to assemble the connection. Bolted connections, particularly friction connections can be expected to be stiffer compared to riveted connections. This can possibly alter the way stress flows from the connected member into the gusset plate which can have an effect on the peak stress magnitude and location within the gusset plate. It can also alter the boundary conditions for stability analysis. This project will attempt to determine if this effect is significant and if different gusset plate design provisions are required for bolted versus riveted connections. Gusset Plates in Tension The basic capacity of gusset plates in tension is evaluated using the AASHTO provisions for the tension capacity of members (Section 6.8.2). The specific limit-state checks are for yielding on the gross section and fracture on the net section. It has become common practice to use the “Whitmore” method for determining both the gross and net section areas used for these two checks at member ends. Whitmore showed that this method provided a reasonable estimate of the elastic stress state observed in a series of scale model experiments.(9) Kulak and others have confirmed Whitmore’s conclusions by analyzing gusset plates using the finite element method.(10) There are several potential problems encountered in applying the Whitmore method to “tight” gusset connections where there is little space between members. It is common for the Whitmore

8 section from one member to overlap another, raising the question of how to handle interaction effects between members. There is no evidence to suggest a problem exists due to the Whitmore method providing unconservative results, but additional verification of results from a wide range of geometries would be valuable. The AASHTO code also requires a check of block shear rupture resistance (Section 6.13.4). This failure mode has been studied by various researchers and various methodologies. Richard has proposed this type of check as a suitable approach for gusset plate design, but more study is needed to determine if it can be utilized to the exclusion of checking the Whitmore section.(11) Cai and Driver have recently provided recommendations that unify the block shear resistance calculation with other local limit-state yielding and rupture checks.(12) It is recommended in the “Design Guide for Bolted and Riveted Connections” that both the Whitmore and block shear checks should be performed with the lowest capacity controlling the gusset plate design.(10) The typical method of checking the general capacity of gusset plates is to section the plate along various planes and calculate the resulting moment and shear on the free body diagram. Finite element results performed at FHWA for the I-35W Bridge suggest that this may not accurately capture the magnitude and location of maximum tension stress in the plate. Figure 1 shows the U10 gusset plate layout from the I-35W Bridge and defines section A-A along the bottom of the top chord. The member forces are calculated by a finite element model considering the dead load and all live load on the bridge at the time of failure. Figure 2 shows the normal stress occurring across the horizontal plane below the chords for the U10 gusset plate assuming it was properly detailed with a 1.0 inch thick gusset plate (tension is positive). For reference the dashed line shows the theoretical stress state calculated by summing the moment and axial force from the free body diagram. The peak stress magnitude correlates fairly well between the theoretical prediction and the finite element results. However, there is a large discrepancy in the location of the peak stress.

9 Figure 1. U10 gusset plate from the I-35W Bridge showing the plane for plotting stress distribution (A-A) from the finite element model results at FHWA. Figure 2. Elastic normal stress distribution along section A-A showing a comparison between the finite element model results and beam theory of a 1 inch thick gusset.     Distance from North Edge (inch) 0 10 20 30 40 50 60 70 80 90 100 N or m al S tre ss (k si ) -80 -60 -40 -20 0 20 40 60 80 Beam Theory (P/A + My/I) Normal Stress Distribution

10 Gusset Plates in Shear The fundamental shear resistance of gusset plates is (BDS Section 6.13.5.3): Rn = vy(0.58AgFy) (Eq.1) The reduction factor  comes from BDS Section 6.14.2.8 and is taken as 1.0 if the shear stress distribution along the plane being checked is uniform. If the plate is not thought to be dominated by flexural shear then  is taken as 0.74. The theoretical parabolic shear distribution would result in Ω = 0.66 but this was considered to be an overly conservative reduction in shear resistance by the code authors. The factor of 0.74 represents a 10% relaxation of the theoretical parabolic shear distribution and is considered to be a more accurate basis to predict flexural shear resistance. LRFD defines the shear resistance factor as v=1.00 but the FHWA Guide recommends using vy=0.95 to be consistent with the resistance factor for tension yielding. It is not clear if this change is warranted, since the shear failure mode is not necessarily equivalent to the tension failure mode. The FHWA Guide also recommends checking the ultimate shear capacity along the net section: Rn = vu(0.58AnFu) (Eq. 2) The resistance factor for this check is recommended as to be consistent with the block shear provisions. Checking for shear rupture on the net section seems to be a logical extension of the block shear provisions, but this needs to be studied further. Many possible shear planes through gusset plates involve a significant net section. However, the AASHTO specification does not specifically require this check for shear in gusset plates. The need for this check will be explored further in the present study. If this check is needed, the appropriate reduction factor  needs to be determined. Since vu=0.80 already provides an added factor of safety against this mode of failure, adding an additional reduction factor may be overly conservative. One problem that arises when checking gusset plate shear is that neither the uniform nor parabolic shear assumptions apply in many cases. On most shear planes in properly designed gusset plates the amount of bending is minimal. This would suggest that the direct shear distribution should govern the design. However, the complicated load paths through the typical truss gusset plate result in a non-uniform stress distribution in which the peak shear stress substantially exceeds the uniform stress model. This peak shear is primarily caused by interaction between the direct stress paths for tension and compression through the plate and the average shear required for connection equilibrium. That is, there are local effects on the maximum stress that cannot be predicted well using beam theory. Figure 3 shows the finite element results for a shear check along the A-A plane through the gusset plate at location U10 in the I-35W Bridge. Again, the plate thickness was increased from 0.5 inch to 1.0 inch to represent a properly designed gusset plate. For reference, the dashed lines represent the theoretical uniform shear and parabolic shear distributions. It can be seen that the

11 peak shear stress exceeds that predicted by the uniform shear distribution and is slightly less than that predicted by the parabolic shear model. It may be fortuitous, but the AASHTO flexural shear provisions (Ω = 0.74) give a reasonable estimate of peak shear along the section for this example. However, it should be noted that the peak shear in the I-35W Bridge gusset plate is caused by stress concentrations at the ends of the diagonal members and not by bending along the horizontal plane. Figure 3. Elastic shear stress distribution along section A-A showing a comparison between the finite element model results and two different theoretical shear stress distributions (based on a correctly designed U10 gusset plate). Although  = 0.74 seems to work reasonably well for the I-35W Bridge example in predicting peak shear stress, more study is needed to determine if this is the correct factor to apply for shear resistance. Since the peak shear stress is not induced by flexure, it might be expected that different geometries and loading conditions will result in different reduction factors. More importantly, it needs to be determined if the peak shear is the most suitable basis for determining shear resistance. Setting the resistance level based on a localized stress concentration ignores the ability of the gusset plate to redistribute stress prior to reaching ultimate capacity. The present study will consider the inelastic redistribution of shear and determine if inelastic effects should be considered when predicting shear resistance. Inelastic action is already recognized in many parts of the AASHTO specifications when checking the capacity of members and connections. Preliminary feedback from state DOTs involved in checking truss gusset plates indicates that many connections will have a rating factor in excess of 1.0 when shear calculations are based on   Distance from North Edge (inch) 0 10 20 30 40 50 60 70 80 90 100 S he ar S tre ss (k si ) -60 -40 -20 0 20 Average Shear Stress () Beam Theory Shear Stress ( = 0.66) Finite Element Shear

12  = 1.0, but will be less than 1.0 when  = 0.74. Therefore, the method used to calculate shear resistance can have a substantial impact on evaluating bridge safety and determining the need for strengthening retrofits. This project will attempt to determine if the current AASHTO shear strength model applies to gusset plates and what reduction factor, if any, is needed to ensure safety. The project will also investigate alternate methods of checking shear resistance. One method that was discussed at the Nashville meeting was to calculate uniform shear on an effective area. In this case, areas of the shear plane where the capacity is compromised due to interaction effects or stability would be subtracted from the gross area for shear checks. Such a method will be explored in the current research. Overall, the goal is to determine a straightforward method of checking shear resistance that will define the minimum resistance required to ensure bridge safety. Gusset Plates in Compression There are a number of questions concerning the methodology used to evaluate the strength of gusset plates in compression. The AASHTO specifications provide little guidance in this area and it is essentially left to the designer to develop some sort of reasonable column or plate analogy to evaluate buckling capacity. Given the wide range of geometries encountered in gusset plates, it is unlikely that a single analogy will work for all cases. The current AASHTO specifications do, however, provide an edge slenderness limit for unstiffened edges that presumably ensure that the gusset plates perform as compact elements (i.e., this limit presumably ensures that resistance is not degraded by local plate buckling failure modes when the gusset plate is modeled to fail in a column buckling fashion). Crush Capacity The crushing capacity of the gusset plate in compression is usually assessed by evaluating the compressive stress at the base of the compression member using the Whitmore method. Reaching this limit-state obviously assumes that stability failure does not occur first. Experiments performed by Brown on gusset plates typical to building construction in compression showed that buckling of the free edge of the gusset always preceded crushing failure for the geometries tested.(13) If this conclusion can be verified for a comprehensive range of bridge truss gusset plates, then checking the crush resistance may not be required. Stability The methodology used to determine the critical buckling load for gusset plates is inexact at best. Given the complexity of the gusset geometry and loading, no exact solutions exist for this limit- state. Considerable engineering judgment is required to apply existing column or plate buckling solutions to the problem. The Nashville meeting showed that there is no clear consensus among experts about the best way to address this issue. A review of literature shows few studies of gusset plates in compression have been performed and those that exist were conducted on building connections. Modern finite element methods are capable of providing accurate estimates of buckling capacity but this requires considerable effort to develop connection-specific models.

13 Clearly an uncomplicated and consistent method of checking stability, even if the results are conservative, is a needed addition to the specifications. Initially, to address stability, the model that best predicts buckling in gusset plates needs to be determined. Brown investigated both plate and column buckling analogies and concluded that the column analogy seemed to work best based on the experimental data. Lacking specific guidance, the FHWA Guide advises using the procedures in BDS Sections 6.9.2.1 and 6.9.4.1 for axial resistance of compression members. It is left up to the engineer, however, to determine the effective slenderness ratio, KL/r , to apply the equations to gusset plates. All three terms in this ratio can vary widely depending on assumptions made by the engineer. Calculation of the terms “L” and “r” involves determining the shape of an equivalent rectangular column in the triangular area of the gusset plate. The FHWA Guide recommends using the Whitmore section to set the column width. The unbraced length is determined as the distance from the Whitmore plane to the closest adjacent member parallel to the axis of the compression member. This approach seems reasonable, but there is little research evidence to support the accuracy. The problem of applying the Whitmore section to tight connections still exists since it will typically overlap adjacent members or adjacent equivalent struts in the gusset plate. Brown observed that edge buckling always preceded buckling or crushing of the triangular area at the base of the compression member. If this can be verified for a wide range of gusset geometries, a stability check of the gusset at the base of the compression member might not be required. The other inherent problem is determination of the K-factor for the gusset plate geometry. The FHWA Guide indicates that K ranges from 1.2 to 2.1 for gusset plates subject to sidesway, depending on the anticipated buckled shape. This topic was discussed at the Nashville meeting and it became clear that this is a judgment issue. The I-35W Bridge gusset plate at U10 appears to have failed in a sidesway buckling mode which suggests 1.2< K< 2.1 bounds the correct value. Finite element modeling performed by the FHWA suggests K = 1.3 when the FHWA Guide model is applied, assuming a gusset plate with adequate thickness. The FHWA Guide model is also suggesting that K varies between 0.2 and 0.5 when the model is braced to prevent sidesway buckling. The FHWA Guide recommends using 0.65< K <1.0 for braced gusset plates, again depending on the mode shape assumed by the designer. Given that both the unbraced length and the K-factor have equal influence on buckling capacity and given that both terms are subject to engineering judgment, it is difficult to refine the value of either term to the exclusion of the other. Possibly the most important provisions in the AASHTO specifications for the stability of gusset plates are the edge slenderness limits. As previously mentioned, Brown found that the buckling limit-state was associated predominantly with out-of-plane bending along the free edge of the gusset with relatively little change in axial load. This plate edge buckling was identified as the most important parameter affecting connection capacity. The FHWA modeling of the I-35W Bridge gusset plate failure shows that once the critical buckling load is reached there is a large change in lateral displacement at approximately constant load. This is consistent with the

14 behavior observed by Brown.(13) The FHWA model also shows that adding initial geometric imperfections along the gusset plate edge produces a significant reduction in the critical buckling load. Results show about a 40% capacity reduction when imperfections are added to the U10 node model within expected tolerances. This suggests that edge stiffening may be a very effective means to increase or ensure gusset plate capacity. The AASHTO specifications provide the following edge slenderness limit for unstiffened gusset plates in compression (Section 6.14.2.8): 1/2 2.06 y l E t F       (Eq. 3) The exact origin of this limit has not yet been determined. However, this limit was present in the 1963 AASHO Specifications and the 1985 AASHTO Guide Specifications for Strength Design of Truss Bridges (Load Factor Design), and it has been carried forward into the current LRFD Specifications. Brown speculates that this limit was derived based on a plate buckling model for gusset plates. The origin and applicability of this limit needs to be thoroughly investigated in the proposed research. Presumably the intent of this limit is to ensure that gusset plates behave as compact sections (i.e., no degradation of the equivalent column flexural buckling resistance) prior to developing their intended strength. The finite element modeling performed at FHWA indicates that meeting this limit alone would not have prevented buckling as the mode of failure when the U10 gusset plates reached their ultimate capacity. The FHWA model investigated the effect of gusset plate thickness on the behavior of the U10 connection in I-35W, varying the thickness from 3/8 inch to 1.5 inches. The failure mode, sidesway buckling, remained essentially the same even down to a plate edge slenderness of l/t=20 . It is possible that the current requirement was derived assuming sidesway was prevented in the connection. An edge slenderness requirement, if one can be determined, that prevents gusset “local” buckling would be a very straightforward check in the evaluation of gusset plates. In design, the most cost effective solution for plates failing this requirement is probably to increase plate thickness. For existing bridges, addition of edge stiffening provides a relatively cost effective retrofit. For these reasons, the research project will thoroughly investigate the edge slenderness limit and the effect of this limit on different potential gusset plate failure modes. The FHWA Guide currently states that the AASHTO edge slenderness limit should not be checked as part of the rating process. This needs to be examined closely in this study. Some sort of compactness criteria will be needed to enable consideration of inelastic capacity for other strength checks of gusset plates. Inspection and Retrofit The primary purpose of inspection is to look for changes and deterioration in a bridge’s condition that would affect its ability to perform in service. For gusset plates, the two applicable

15 deterioration mechanisms are corrosion and fatigue. Of the two, corrosion is clearly more important from a strength perspective. General experience has shown that fatigue cracking has not been a significant problem in truss bridge gusset plates. Normally the live load stress range is below the fatigue threshold so infinite life is expected. This conclusion may change if significant pitting or pack-out corrosion exists as these forms of deterioration can amplify stresses significantly. Since corrosion will probably always precede the onset of fatigue, evaluating the impact of corrosion is the primary need for the present study. Corrosion can have a significant effect on the compression and tension capacity of gusset plates. This was clearly demonstrated by the failure of the gusset plate in the I-90 Bridge over Grand River in Ohio.(14) Significant corrosion loss reduced the capacity of the gusset plate at a lower truss connection adjacent to one of the main piers, allowing the plate to buckle. In this case, the compression diagonal went into bearing and prevented collapse even though the critical buckling load for the gusset plate was exceeded. The U10 gusset plate in the I-35W Bridge also failed by buckling although inadequate design, not corrosion, was the primary cause. Assessment of the capacity of gusset plates with corrosion damage is not always a simple procedure. Prucz and Kulicki indicate that when the corrosion is moderate and uniformly distributed it is appropriate to analyze the gusset plate as a member with reduced thickness.(15) This approach probably works well for checking stress in the gusset, particularly for tension members, as long as there is no severe pitting or other localized section loss. Severe pitting will induce local stress concentration effects that will create a more severe condition than that predicted by average section loss calculations. The effect of section loss on compression stability is more difficult to analyze. Corrosion in gusset plates is typically more severe in the localized areas along the edge of members where debris accumulates. This localized weakening of the plate can significantly change the boundary conditions assumed in buckling models. Since the buckling models used for gusset plates already involve many assumptions, it is difficult to assess this boundary effect without further research. It is also common to have conditions where crevice corrosion and pack rust induce out-of-plane deformation in the gusset plate. This again will alter the fundamental assumptions of the buckling model. The presence of corrosion in gusset plates further complicates the already complicated models used to evaluate stability. However, this is a very real concern for engineers faced with rating existing truss bridges. The second experimental phase of this study will try to address this issue by performing tests on gusset plates with simulated corrosion damage. Better guidance will be developed for assessing the effect of corrosion on the compression capacity of gussets. It is probably too ambitious to develop comprehensive gusset retrofit guidelines under this project. The primary focus will concentrate on methods to determine gusset plate capacity. If the capacity is determined to be inadequate, the option always exists to replace or add doubler plates

16 to the gusset. While effective, this is typically a very costly and time consuming operation and is best avoided unless absolutely necessary. The FHWA modeling of the U10 gusset plates on the I-35W Bridge showed that edge stiffening will provide a significant strength increase to gusset plates in compression. Compared to the alternatives, edge stiffening is a relatively simple and inexpensive retrofit to perform. Therefore, this research will include experiments to determine the effect of edge stiffening. An attempt will be made to develop guidelines for evaluation of edge stiffened gussets under this project. There are very few cases where in-service truss bridges have failed. The I-35W Bridge failed in a catastrophic manner, but a severe design error made this structure artificially weak. There are several examples of partial bridge failures where gusset plates were considerably weakened by section loss due to corrosion. For example, four deteriorating corroded steel plates supporting the eastbound I-90 Grand River Bridge in Perry Township, Ohio, buckled in 1996 while work crews were preparing them for painting. The partial collapse dropped part of the span 3 inches. No one was hurt, but the bridge was closed nearly six months for repairs that cost $1.6 million.(1, 14) However, while deterioration always needs to be considered with regard to the performance of existing structures, these failures are primarily a maintenance issue and not a gusset plate design issue. In addition, very few new truss bridges are currently being built around the country since other design types are generally more cost effective. Therefore, the dominant issue with truss bridges is rating of the existing inventory, including bridges that have some degree of corrosion damage. SELECTION OF REPRESENTATIVE JOINTS Before the larger experimental and analytical studies were undertaken, the NCHRP panel requested the research team to model typical gusset-plated connections to help guide the research project, and to possibly offer immediate relief to the provisions of the FHWA Guide. This section will briefly describe those modeling efforts and some of the results derived from them. It must be pointed out that this was the project’s first attempt at modeling joints and as the project evolved, so did the finite element modeling methodology. The modeling methodology is described more in Chapter 2 and Appendix I, but exact details for these joints can be found in the work performed by Mentez.(16) This portion of the work is only meant to convey a broad overview of this early modeling effort and not necessarily convey more detailed aspects of how the models were created. Five joints were selected by the research team for further study. One joint was the I-35W U10 joint because FHWA had performed the forensic analysis and it was an obvious joint to include. The other four joints are all based on the original design drawings without considering any changes that occurred during the life of the bridge. The forces on some members are changed from the original design, which give only envelope values, to obtain equilibrium in the FEA truss model. In addition, assumptions for the material and fastener models had to be made. Specific attributes of each selected connection are explained in detail below:

17 a) Original I-35W Bridge over Mississippi River (MN) This joint (shown in Figure 4) was pinpointed as the failure trigger to the collapse of this bridge and actually represents a true failure point. The joint was located two panel points off the pier and was approximately located at the theoretical inflection point of the continuous structure. As such, the joint was subjected to a large horizontal shear force. At failure, much of the horizontal plane was yielded and ultimately the gusset plate buckled and the compression diagonal swayed out-of-plane leading to an overall rupture of the connection. Some of the modeling details were first reported in Ocel and Wright (17), those not covered in that report were completed as part of this project. Figure 4. I-35W U10 joint geometry. b) I-94 Bridge over the Little Calumet River (IL) The L2 joint (shown in Figure 5) from the I-94 Bridge over the Little Calumet River Bridge in Cook County, IL, is selected. This bridge is a deep truss with a substantially steep (60° from horizontal) angle for the diagonal members. The diagonals have a relatively large distance from the work point of the joint resulting in a relatively long gusset plate free edge between the compression diagonal and the chord member. This joint is a relatively modern bridge design using high-strength bolts (1990). Figure 5. I-94 joint geometry.

18 c) HW-23 Bridge over the Mississippi River (MN) The U2 joint (shown in Figure 6) from the Highway 23 Bridge over the Mississippi River in St. Cloud, MN, is selected since it has relatively flat angles for the diagonal members (37° from horizontal), resulting in a relatively long gusset plate free edge between the diagonal members and the vertical member. The Minnesota Department of Transportation found “buckled” gusset plates in this bridge and it was decommissioned after rating calculations indicated a possible unsafe condition. This bridge is a representative design from the 1950s utilizing welded members and riveted gusset plate connections. Figure 6. HW-23 joint geometry. d) I-40 Bridge over the French Broad River (TN) The U8 joint (shown in Figure 7) from the I-40 Bridge over the French Broad River in Jefferson County, TN, is selected since it provides an example with shingle (double) gusset plates. Similar to the HW-23 Bridge, this bridge also represents a 1950s design with welded members and riveted gusset plate connections. Figure 7. I-40 joint geometry. e) I-80 Bridge over the Clarion River (PA) The last connection selected for the preliminary studies is the hypothetical joint used in the FHWA Guide as an example. The connection is a modification of the L3 joint (shown

19 in Figure 8) in the I-80 Bridge over the Clarion River in Pennsylvania. FHWA made changes to the joint to facilitate the illustration of the guidance provisions. These changes include the angle of chord members, the reduction in gusset plate strength from 50 to 36 ksi, and an increase in member forces as compared to the original design. Figure 8. I-80 joint geometry. Analysis Procedure Nonlinear finite element analysis of the four joints was performed using the Abaqus analysis engine and the nodal loads calculated from the original design drawings.(18) Both dead load (DC) and live load plus dynamic load allowance or impact (LL+I) are imposed on the truss members using the appropriate load factors for LRFD Strength I load combination. The member forces in the model are set so that the forces exactly match the envelope values for the compression diagonal and the chord member nearest to it. Adjustments are required for the forces in the remaining three members relative to the values on the engineering drawings to satisfy the joint equilibrium (the forces on the engineering drawings are envelope values). The main reason for the equilibrium mismatch is the force in the vertical member of the truss. For trusses with parallel top and bottom chords, this is ideally a zero force member or it is subjected to relatively small loads directly applied from the deck system of the bridge. The analysis is implemented in two steps. First, the factored dead load is applied to the truss nodes. Second, the factored live load plus impact load is incrementally applied. From the analysis, the total factored load at any stage of the loading for the first five joint analyses can be calculated using Equation 4. A wearing surface component is included in the equation because that information was available for the I-80 joint. For the other joints, all dead load was assumed to be from components (i.e., DC).

20  )()()( IMLLALFDWDCP ILLDWDFEA   (Eq. 4) The term “ALF” used in the equation represents the Applied Load Fraction (i.e., the multiple) of the factored LL+IM loads applied at a given stage of the finite element analysis. The load factors are taken as D = 1.25, DW = 1.50 and LL+IM =1.75 corresponding to the Strength I limit-state. Failure Determination With the exception of the I-35W joint, members had to be modeled with elastic properties to identify the failure mode of the gusset. Had they been modeled with inelastic properties the members would fail before the gusset plate. This means four of the joints represented properly designed connections that were stronger than the members they connected. The primary mode of failure in the finite element model is difficult to define in most cases since multiple failure modes are often occurring simultaneously. However, the analyses show that a significant amount of the gusset plate material reaches the yield strength limit based on the Von Mises yield criterion prior to the ultimate load. It is important to decide the load level at which the gusset plate reaches its analytical resistance. However, the gusset plate may reach a “limit of maximum useful resistance” prior to achieving a limit load in the analysis due to other factors such as significant plasticity or excessive fastener displacement. Therefore, the following three load levels are considered in assessing the joint response; the first one reached determines the resistance of the joint. 1) 4% equivalent plastic strain (PEEQ) at mid-thickness In the finite element models, the actual bolt holes are not modeled and fracture of the steel is not incorporated into the nonlinear material model. Hence, the models cannot capture net section limit-states. Typically the gusset plates undergo a significant amount of plasticity beyond the Strength I limit. This criterion is indicative of when ductile fracture may start to be of concern. This value is arbitrarily set at 4%, but the experience of the research team has found this to be an indicative strain level where faith in the model may begin to become unreliable (i.e., when net section failures may occur considering net sections are not modeled. 2) 0.2 inch fastener shear displacement The fastener models used in the analyses are also nonlinear. The ALF value at a fastener shear displacement of 0.2 inches is decided to be a practical limit of fastener useful resistance. Fastener shear fracture may start to be of concern for shear displacements beyond this limit. 3) Peak of the load versus displacement curve

21 In general, the load versus displacement curve from the analysis has a loading and a post- peak unloading path. The peak ALF value from this plot is another important limit for the joint response. RESULTS OF TYPICAL JOINTS This section discusses the observations from the nonlinear finite element analysis performed on each of the joints. A brief summary on the initial imperfection sensitivity and stress and strain distributions within the gusset plates at the Strength I and ultimate resistance levels are provided respectively. As the FEA model does not always create output exactly at ALF = 1.0 , the load step closest to 1.0 is chosen as the Strength I load level for output of the stresses and strains. All stress contours in this section are presented as the Von Mises stress to show the combined effect of all the stress components toward yielding. In addition, the equivalent plastic strain contours (PEEQ contours) are presented to show the plasticity patterns and degree of yielding in the gusset plates. I-35W U10 Joint Analysis The U10 connection utilized 0.5 inch thick gusset plates fabricated from grade 50 material connected together with 1 inch A502 Grade II rivets. The NTSB concluded this connection failed due to a design error leading to gusset plate thickness roughly half as thick as necessary to carry the design loads. (1) The ultimate failure of the connection was inelastic sidesway buckling of the gusset plate after a significant portion of the plates reached yield along the horizontal plane and at the base of the compression diagonal. The finite element load-displacement plot shown in Figure 9 indicates the connection failed at 95% of the 1.25DL portion of the Strength I load combination. This is not surprising given the under-designed condition of the gusset plates. The primary failure mode of the model was sidesway buckling of the gusset plate that displaces the compression diagonal out-of-plane. Simultaneous gross yielding occurred along the A-A plane. Figure 10 shows the normalized Von Mises stress contours at the point when the model became unstable. The red contour indicates areas very close or just exceeding the yield strength. The equivalent plastic strain contour shown in Figure 11 more clearly shows the areas where plastic deformation is occurring (any contour other than dark blue is actively yielding). This is most pronounced around the base of the compression diagonal and along the horizontal plane along the chords. Figure 12 shows the normal and shear stress distributions along the horizontal plane at peak load. Each of the stress plots has been normalized by the factored design stress for shear and bending, respectively. Both distributions are disturbed in the area around the compression diagonal (70- 104 inches on the x-axis). This is caused by out-of-plane bending and the resulting P- effects on gusset plate stress. The plot clearly shows that the shear capacity has been reached along much of the horizontal plane. This effectively softens the plate and reduces the out-of-plane stiffness resisting sidesway.

22 Figure 13 shows the shear and normal stresses occurring along the B-B plane. The B-B plane stresses are harder to interpret because of the out-of plane bending effects around the compression diagonal. The normal stress certainly does not follow a linear bending stress distribution as it decreases toward the bottom edge of the plate and it is very uniform along the top 30 inches. The chord members are roughly 29 inches deep and the uniform normal stress distribution shows that the gusset plate is primarily acting as a chord splice in this region. Superposition of bending and axial stress distributions cannot be assumed valid across the entire plane due to yielding. Figure 9. Load-Displacement plot for 0.5 inch gusset plates. Vertical Deflection of L9 (inch) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Am pl ifi ca tio n of 1 .2 5D L C om bi na tio n (A F) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

23 Figure 10. Von Mises stress contour normalized to the yield stress at the peak load for I-35W U10 connection. Figure 11. Equivalent plastic strain plot at failure for I-35W U10 connection.

24 Distance From Left Edge of Gusset (inch) 0 10 20 30 40 50 60 70 80 90 100 110 N or m al iz ed S tre ss -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Shear (fv/vy0.58Fy) Normal (fb/yFy) Figure 12. Section A-A mid-thickness shear and normal stresses at connection failure.

25 Figure 13. Section B-B mid-thickness shear and normal stresses at failure. I-94 Bridge (Joint L2) Based on the criteria defined at the beginning of this chapter, important load levels are identified on the ALF vs. out-of-plane displacement plot as shown in Figure 14. The Strength I load level occurs at an ALF=1.05. The 4% PEEQ limit is reached just after the peak load. Without showing a figure, the maximum stress in the gusset plate at the Strength I limit is approximately 25 ksi, or about half of its yield strength. Therefore, this connection has a considerable amount of reserve capacity beyond the Strength I load level. The Von Mises stress state in the gusset plate corresponding to the limit load at ALF = 9.72 is shown in Figure 15. The plate has reached its yield strength at its mid-thickness at areas shown with the grey contours. Figure 16 shows the corresponding equivalent plastic strain contours. The material that is still elastic is shown as dark blue, while the light blue and green areas indicate the variation of higher magnitudes of plastic deformation. The largest amount of plastic deformation occurs in the region between the compression diagonal and the chord member. The displaced shape, shown in both figures with a deformation scale factor of 5.0, confirms that the final joint mode of failure is out-of-plane compression buckling of the gusset plate. Normalized Stress -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 D is ta nc e fro m T op E dg e of G us se t ( in ch ) -80 -70 -60 -50 -40 -30 -20 -10 0 Shear (fv/vy0.58Fy) Normal (fb/yFy)

26 Figure 14. Plot of out-of-plane displacement motion of Point 6 versus the ALF. Figure 15. Von Mises stress response contours at the limit load (ALF=9.72). 1.05 9.729.68 0 2 4 6 8 10 12 -4 -3 -2 -1 0 A pp lie d Lo ad F ra ct io n of 1 .7 5( L +I ) Out of Plane Displacement at Point 6 w/ 1/4" Maximum Initial Imperfection (in.) STRENGTH I PEAK LOAD 4% PEEQ @ Mid-Thickness

27 Figure 16. Equivalent plastic strain response contours at the limit load (ALF=9.72). HW-23 Bridge (Joint U2) Figure 17 shows the important load levels on the ALF versus out-of-plane displacement plot. At the Strength I load the ALF=0.97 at which point the gusset plate was entirely elastic. Similar to the previous case the usefulness of the connection is determined by the peak load at an ALF=2.70 and the 4% PEEQ occur well beyond the peak load. This also demonstrates that the connection has reserve capacity to extend almost three times beyond the Strength I load level. The Von Mises stresses in the gusset plate corresponding to the peak load at an ALF = 2.70 is shown in Figure 18. The plate reaches its yield strength at areas shown with grey contours. Figure 19 shows the corresponding equivalent plastic strain contours. The largest amount of plastic deformation is occurring in the splice region and the horizontal plane just below the chord members. The displaced shape observed in both contour plots with a deformation scale factor of 5.0 also confirms that out-of-plane compression buckling of the gusset plate does occur. However, these two plots show the difficulty in interpreting the results as likely this connection would first suffer failure of the chord splice (though it is a compression chord splice that could go into a bearing condition and suppress that failure mode), followed by horizontal shear and/or buckling around the compression diagonal.

28 Figure 17. Plot of out-of-plane displacement motion of Point 8 versus the ALF. Figure 18. Von Mises stress response contours at the limit load (ALF=2.70). 0.97 2.70 2.21 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 A pp lie d Lo ad F ra ct io n of 1 .7 5( L +I ) Out of Plane Displacement at Point 8 w/ 1/4 " Maximum Initial Imperfection (in.) STRENGTH I PEAK LOAD 4% PEEQ @ Mid-Thickness

29 Figure 19. Equivalent plastic strain response contours at the limit load (ALF=2.70). I-40 Bridge (Joint U8) Important load levels on the ALF versus the out-of-plane displacement plot are shown in Figure 20. The Strength I load occurs at an ALF = 1.04 where the entire connection was elastic. For this connection, the maximum usefulness of the connection is dictated by the strength of the fasteners in the chords as the 0.2 inch fastener deformation limit first occurs at an ALF=2.16, which still shows this connection has considerable capacity beyond the Strength I limit. The shear rupture of the critical fastener is precipitated by extensive tension yielding of an inside web splice plate in the chord, resulting in redistribution of shear forces to the plane between the chord and the gusset. The Von Mises stresses at an ALF = 2.16 is shown in Figure 21 where grey contours show areas exceeding yield which are limited to the primary gusset plate. Figure 22 shows the corresponding equivalent plastic strain contours at the same load level. The shingle plate does not have plastic strains although the plasticity is significant in the gusset plate at this load level. The displaced shape observed in the contour plots with a deformation scale factor of 5.0 also confirms that buckling of the gusset plate does not occur for this joint.

30 Figure 20. Plot of out-of-plane displacement motion of Point 7 versus the ALF. Figure 21. Von Mises stress response contours at limit load (ALF=2.16). (Left) Showing with the shingle plate in place. (Right) Showing with the shingle plate removed. Figure 22. Equivalent plastic strain response contours at limit load (ALF=2.16). (Left) Showing with the shingle plate in place. (Right) Showing with the shingle plate removed. 1.04 3.03 2.74 2.16 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0 0.25 0.5 0.75 1 A pp lie d L oa d Fr ac tio n of 1 .7 5( L +I ) Out of Plane Displacement at Point 7 w/ 1/4" Maximum Initial Imperfection (in.) STRENGTH I PEAK LOAD 4% PEEQ @ Mid-Thickness 0.2" FASTENER DISPLACEMENT

31 I-80 Bridge (Joint L3) Important load levels on the ALF versus out-of-plane displacement plot are shown in Figure 23. The Strength I load is attained at an ALF=1.03. Shown in Figure 24 the connection is not elastic at the Strength I load level, though it must be restated that this connection was originally designed using 50 ksi yield properties, though 36 ksi yield strengths were used in the FHWA Guide. The maximum usefulness of the connection is determined using the 4% PEEQ criterion at an ALF=1.76. The Von Mises stress and equivalent plastic strain contours at an ALF=1.76 are shown in Figure 25. Much of the gusset plate has yielded, though the likely failure mode of the connection would be failure of the tension chord splice. There is also evidence that block shear of the right-hand chord could also occur. Figure 23. Plot of out-of-plane displacement motion of Point 6 versus the ALF. 1.03 2.89 1.76 2.12 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 A pp lie d Lo ad F ra ct io n of 1 .7 5( L +I ) Out of Plane Displacement at Point 6 w/ 1/4" Maximum Initial Imperfection (in.) STRENGTH I PEAK LOAD 4% PEEQ @ Mid-Thickness 0.2" FASTENER DISPLACEMENT

32 Figure 24. Response contours at Strength-1 level (ALF=1.03). (Left) Von Mises stress contours. (Right) Equivalent plastic strain contours. Figure 25. Response contours at limit load (ALF=1.76). (Left) Von Mises stress contours. (Right) Equivalent plastic strain contours. SUMMARY OF THE RESULTS For each of the five connections, the FHWA Guide was used to calculate the LRFR Inventory Rating Factor (RF) for each member. The RF is calculated using Equation 2, where Rn is the factored nominal resistance, DL is the dead load resisted by the element, and LL+IM is the live load amplified by the dynamic load allowance (impact) resisted by the element. The load modifier, η, was assumed to be 1.0 and the condition and systems factors specified in LRFR are also assumed to be 1.0.     1.25 1.75 nR DLRF LL IM    (Eq. 5) + 36.0 + 33.0 + 30.0 + 27.0 + 24.0 + 21.0 + 18.0 + 15.0 + 12.0 + 6.0 + 0.0 Von Mises Stress unit: ksi + 9.0 + 3.0 + 42.5 0.040 0.036 0.032 0.028 0.024 0.020 0.016 0.012 0.008 1.00 e-05 0.00 0.004 1.00 e-10 Equiv. Plastic Strain unit: in/in 0.040 0.036 0.032 0.028 0.024 0.020 0.016 0.012 0.008 1.00 e-05 0.00 0.004 1.00 e-10 Equiv. Plastic Strain unit: in/in 0.041

33 Table 1 shows the resulting LRFR rating factors for each of the five connections. The I-94 connections used high-strength A325 bolts and therefore the slip capacity of the fasteners was checked, but not on the other joints which used rivets. The Whitmore buckling checks for compression diagonals all assumed the equivalent column length factor (i.e., K-factor) was equal to 1.2. For the Whitmore buckling checks of compression chords a K-factor of 0.65 was used and it was assumed the force was proportioned between the gusset plate and additional chord splice plates by their area. The shaded cells in the table indicate the most likely failure modes observed from the analysis. In all the joints, except for I-35W, the smallest rating factor was 1.87 with the majority of them greater than 2.00. Note these favorable rating factors are attained using LRFR despite none of the connections being designed to an LRFD philosophy. The I-35W joint is recognized to be an under-designed gusset plate and as expected it has either negative or rating factor less than 0.50. The data shown in Table 2 is in a different form as a professional factor, which is the unfactored failure load from the model divided by the unfactored nominal resistance check (i.e., RFEA/Rnominal). Professional factors greater than 1.0 represent conservative predictions and are favorable over ratios less than 1.0 that are unconservative. As will be further described in Chapters 3 and 4, it assesses the accuracy of the nominal strength check and plays a role in the LRFD calibration. As done in Table 1, the failure modes identified to be likely are shaded in Table 2. It is important that the professional factors presented in unshaded cells are studied carefully as more often than not they are less than 1.0. This is not to mean that the resistance equation is unconservative because it must be tempered that the connection did not fail in this limit-state and it would be an unfair judgment. Only professional factors in shaded cells should be used to judge the conservatism of a particular resistance equation. After performing the detailed analysis of the five joints, the following anomalies were noted by the research team in comparing the results of the finite element simulation versus the predictions of the FHWA Guide. These items were given full consideration when developing the experimental and paramedic finite element studies. 1) The application of the Whitmore section approach for both tension and compression chords is questionable. Since all five of the gusseted connections also performed as chord splices, a distribution of force between the gusset plate and the additional chord splice plates had to be assumed. Since the I-80 and HW-23 joints likely failed in the chord splice, the professional factors for the Whitmore checks for these two connections were an unconservative 0.91 and 0.37. This shows that chord splice failure is not well predicted by the Whitmore section checks. This likely has to do with the assumptions of how stress is distributed between the gusset plate and alternate chord splice plates. 2) No glaring issues could be identified with tension member checks. However, it was noted that Whitmore tension and block shear checks did produce about the same resistance and

34 there may be the possibility that the two checks are redundant. The sensitivity of the Whitmore section crossing over multiple members should also be evaluated. 3) There is a great amount of uncertainty with the buckling checks of compression members. In the three connections that may have buckled, the Whitmore buckling professional factors varied greatly. There are four unknowns that exist with the resistance check. First, what K-factor should be used because considering boundary conditions, K=1.2 appears to be most appropriate, though it produced very conservative professional factors for some connections. Second, what equivalent column length should be used, because in many cases the Whitmore width overlaps adjoining fastener lines and a zero length is attained of which the sensitivity needs to be addressed. Third, what is the sensitivity of the Whitmore section overlapping neighboring members and should it be truncated. Fourth, does the free edge slenderness have a role in the buckling resistance as the current limit does not appear to be a good predictor of buckling. 4) In the evaluation of shear, it is critical that the -factor is explored because as shown for the two connections that possibly failed in shear, the professional factors using =1.00 produce the best results. Based on these five analyzes, the -factor is somewhere in- between the current limits, likely closer to 0.90. 5) The load sharing of multi-layered (i.e., shingled) gusset plates needs to be verified. The model of the I-40 connection had multi-layered gusset plates, but it is important to evaluate the validity of the fastener models to ensure that the results of load sharing are true. 6) The role of section loss within all the limit-states needs to be addressed. Frequently, field inspections show that gusset plates often suffer from section loss due to corrosion and it is not well understood how it should be integrated into the various resistance checks.

35 Table 1 Rating Factors for Five Representative Joints Type of Check I-80 I-40 I-94 HW-23 I-35W TENSION CHORD Whitmore Section 2.60 3.71 9.75 N/A 5.62 Block Shear 2.30 2.96 9.92 N/A 5.63 Fastener Shear Capacity 2.75 4.73 9.67 N/A 3.38 Fastener Slip Capacity N/A N/A 4.28 N/A N/A COMPRESSION CHORD Whitmore Buckling N/A N/A N/A 3.27 9.43 Fastener Shear Capacity N/A N/A N/A 4.38 7.58 TENSION DIAGONAL Whitmore Section 2.86 6.93 14.87 4.41 2.25 Block Shear 2.66 3.82 6.17 3.58 1.38 Fastener Shear Capacity 2.31 2.32 9.42 2.66 2.54 Fastener Slip Capacity N/A N/A 4.30 N/A N/A COMPRESSION DIAGONAL Whitmore Buckling 3.88 5.83 8.84 2.98 0.29 Edge Slenderness(a) 37.3 (58.5) 54.7 (49.6) 45.3 (49.6) 77.1 (49.6) 60.4 (49.6) Fastener Shear Capacity 3.55 3.52(b) 7.24 2.32 1.62 Fastener Slip Capacity N/A N/A 3.05 N/A N/A A-A SECTION SHEAR Gross Yield (Ω = 0.74) 1.87 2.28 8.27 2.15 -0.33 Gross Yield (Ω = 1.0) 2.91 3.80 12.24 3.14 0.45 Net Section Fracture 3.05 2.96 10.40 2.50 -0.15 B-B SECTION SHEAR Gross Yield (Ω = 0.74) 1.87 3.21 8.62 2.92 -0.19 Gross Yield (Ω = 1.0) 2.99 5.05 12.72 4.26 0.44 Net Section Fracture 3.03 4.41 11.45 3.63 0.10 N/A = Not Applicable a - Edge slenderness (Slenderness limit) b - Includes angles along top and bottom of compression diagonal

36 Table 2 Professional Factors for Five Connections, (RFEA/Rnominal) Type of Check I-80 I-40 I-94 HW-23 I-35W TENSION CHORD Whitmore Section 0.91 0.95 0.91 N/A 0.26 Block Shear 0.82 1.01 0.88 N/A 0.26 Fastener Shear Capacity 0.77 0.91 1.34 N/A 0.46 Fastener Slip Capacity N/A N/A 3.10 N/A N/A COMPRESSION CHORD Whitmore Buckling N/A N/A N/A 0.37 0.06 Fastener Shear Capacity N/A N/A N/A 0.63 0.09 TENSION DIAGONAL Whitmore Section 0.94 0.69 2.09 0.73 0.66 Block Shear 0.83 0.90 3.42 0.74 0.66 Fastener Shear Capacity 1.14 1.51 1.10 1.21 0.67 Fastener Slip Capacity N/A N/A 4.11 N/A N/A COMPRESSION DIAGONAL Whitmore Buckling 0.63 0.85 3.63 1.29 1.00 Edge Slenderness Pass Pass Pass Fail Fail Fastener Shear Capacity 0.74 1.18(a) 1.32 1.60 0.71 Fastener Slip Capacity N/A N/A 4.91 N/A N/A A-A SECTION SHEAR Gross Yield (Ω = 0.74) 1.19 1.44 3.29 1.52 1.325 Gross Yield (Ω = 1.0) 0.88 1.07 2.43 1.12 0.98 Net Section Fracture 0.72 1.05 2.33 1.14 1.03 B-B SECTION SHEAR Gross Yield (Ω = 0.74) 1.23 1.19 3.21 0.69 0.96 Gross Yield (Ω = 1.0) 1.10 1.14 2.38 0.51 0.71 Net Section Fracture 0.76 0.82 2.18 0.49 0.72 N/A = Not Applicable a - Includes angles along top and bottom of compression diagonal