GUSSET PLATES IN RAILROAD TRUSS BRIDGES FINITE ELEMENT ANALYSIS AND COMPARISON WITH WHITMORE TESTING

Transcription

1 GUSSET PLATES IN RAILROAD TRUSS BRIDGES FINITE ELEMENT ANALYSIS AND COMPARISON WITH WHITMORE TESTING Walid S. Najjar, Ph.D., P.E. 1, Frank DeOrtentiis, P.E. WSP SELLS 555 Pleasantville Road, Briarcliff Manor, New York, 10510, U.S.A (phone) / (fax) / (1) Member of AREMA Committees 8 and 15 ABSTRACT An effective simple procedure for two-dimensional finite element analysis of gusset plates in truss bridges is presented. Gusset and splice plates are modeled with plate elements and truss members are modeled with line or frame elements. While the focus is on gusset plates in railroad truss bridges, the procedure is applicable to gusset plates in highway truss bridges and other types of gusset plate connections. Results from finite element analysis validate key conclusions of the historic and still relevant experimental investigation of stresses in small-scale gusset plates by R. E. Whitmore, pending current large-scale gusset plate testing at the FHWA TFHRC Structures Laboratory. A distribution angle of 30 degrees, the basis of the so-called Whitmore section, provides reasonable estimate of stress. Maximum normal stress along a horizontal section immediately above a bottom chord is located slightly away from the edge of a gusset plate, instead of at the edge as

2 predicted by simple beam theory. And maximum shear stress is located near the center of the plate, consistent with that theory. A Mathcad work sheet has been developed for analysis and evaluation of gusset plates in railroad truss bridges, by modifying a NYSDOT template to include AREMA MRE applicable requirements and Committee 15 new requirements on gusset plates. Simple beam theory is assumed for calculating stresses on horizontal and vertical critical sections of a gusset plate. The Whitmore section is assumed for distribution of normal load from a truss member onto a gusset plate. Outlined in this paper is the development of the finite element analysis and the Mathcad analysis. Key Words: Gusset Plates, Railroad Bridge, Mathcad Analysis, Finite Element Analysis, Load Rating. INTRODUCTION Much has been said and written about truss gusset plates, since the unfortunate collapse of the I-35 Bridge over the Mississippi River. A list of key references (1 to 7) on gusset plate analysis, design, and evaluation are provided. The intent of this paper is to present a simple and effective methodology for analysis and evaluation of gusset plates in railroad truss bridges. While the Mathcad part of the methodology is partly focused on railroad bridges, the structural analysis part is entirely applicable to both railroad and highway truss bridges. Two-dimensional structural modeling and analysis of a typical Warren truss is presented. The truss model has realistic geometry and sectional properties for a railroad bridge and

3 consists of beam elements for truss members and finite elements for gusset plate connections. Analysis results are compared to experimental stress results from the landmark research by R. E. Whitmore (1). After almost 60 years, the Whitmore testing of a small-scale model of a truss joint connected by gusset plates continues to be the key reference in available literature on structural behavior of gusset plates. Currently at the FHWA TFHRC Structures Laboratory and as part of NCHRP Project (4), full-scale gusset plates are being tested. However at this time, there is no publicly available information on the progress of this testing. DEVELOPMENT OF THE STRUCTURAL MODEL A Warren type truss has been selected as the subject structure for this research, because it is considered one of the most common types of bridge trusses, and the Whitmore testing was done on a small model of a Warren truss joint. Shown in Figure 1 is the subject truss with an enlarged view of Joint L2 connection. This is a simple span truss, consisting of 14 equal panels, a span length of 542 feet and truss depth of 84 feet. Its geometry and sectional properties are consistent with the properties of a realistic railroad bridge. The enlarged view shows only perimeter rivets connecting the gusset plate to the bottom chord, two diagonal and vertical members of the truss. These rivets are shown to the exclusion of many other rivets, because their location is used to define the model width of truss members within the connection and for simplicity of presentation. Gusset plate dimensions are 124 inches for width, 103 inches for height, and 13/16 inches for thickness. There are two plates per joint.

4 U1 U2 U3 U4 U5 U6 U7 U'6 U'5 U'4 U'3 U'2 U'1 52.3' 84' L0 L1 L2 L3 L4 L5 L6 L7 L'6 L'5 L'4 L'3 L'2 L'1 L'0 14 x 38.71' = 542' Figure 1: Subject Warren Truss and Joint L2 Gusset Plate (Shown Perimeter Rivets Only) In contrast, the tested Whitmore model connection was based on Joint 2 of a Warren truss, consisting of 10 equal panels, a span length of 295 feet and truss depth of 43.5 feet. Even though that truss was a highway bridge truss, comparison to the subject railroad truss in terms of structural modeling and analysis is valid. Structural behavior of a gusset plate in response to a moving load, modeled as a series of concentrated loads that are applied

5 strictly at truss joints, is a function of relative member stiffnesses and resulting internal forces, regardless of the load type. Figure 2 shows a two-dimensional model of the subject truss with Joint L2 gusset plate connection. Beam (line) elements represent all truss members (chords, diagonals and verticals), and plate elements represent the gusset plate as well as segments of the truss member that frame into the joint. The width of such segment is assumed to be the perpendicular distance between the two rows of edge rivets that are parallel to the framing truss member. The thickness of the segment is calculated by dividing the area of the truss member by the segment width. While only one joint is represented by plate elements in this particular model, more or all joints could be represented in the same manner and within the same model. Quadrilateral elements are used primarily for the framing segments of truss members and the horizontal zone of the gusset plate above the bottom chord. For the remaining areas of the gusset plate, a combination of quadrilateral and triangular (three-node) elements are used as necessary based on the local geometry. Structural modeling and analysis is done through the latest version of the software program STAAD.Pro (8). The mesh generation capability of this software is utilized, especially for the quadrilateral elements, which makes the modeling effort reasonably easy. The master-slave command of the program is used to model an assumed rigid transition from a beam element to plate elements. Edge nodes along the equivalent width of a framing truss member are assumed to be rigidly linked to the connecting interface node.

6 L2-U1 L2-U2 L2-U3 L1-L2 L2-L3 Figure 2: Structural Model of the Subject Truss and Joint L2 Gusset Plate ANALYSIS RESULTS AND COMPARISON WITH WHITMORE TEST RESULTS Live load forces in truss members at Joint L2 are shown in Table 1 for various load cases, including (i) maximum tension or compression in a particular member with corresponding force in other members and (ii) maximum tension or compression in each member. Tensile forces are shown as positive values, while compressive forces are shown as negative values. The first load case (LC 1) is the most critical because of concurrent maximum

7 tension in both chord members and close to maximum forces in the two diagonals. Gusset plate stresses at Joint L2 are shown in subsequent figures for only this dominant load case. Table 1: Live Load Member Forces at Truss Joint L2 Load Cases Truss Member Forces, kips L1-L2 L2-U1 L2-U2 L2-U3 L2-L3 LC LC LC LC LC Maximum Minimum It is worthy to compare member force ratios for this load case to the Whitmore force ratios, as shown in Table 2, in order to justify subsequent comparisons on stress distributions and locations of maximum stresses. Each force is divided by the maximum force in the group (or load case), which occurs in chord member L2-L3 for both groups; the ratio for this member is one. For the tension chord (L1-L2), the ratio is 0.53 for Load Case 1 and 0.58 for Whitmore testing. For the compression diagonal (L2-U3), the ratio is 0.34 for Load Case 1 and 0.30 for Whitmore testing. The force ratios for each of these members are close in value, except for the tension diagonal (L2-U1). Table 2: Comparison of Joint 2 Forces with Whitmore Test Forces Load Case 1 Whitmore Test L1-L2 L2-U1 L2-U2 L2-U3 L2-L3 Force Ratio Force Ratio

8 It is reasonable to state that most of the force ratios within Load Case 1 are approximately equal to the Whitmore testing forces ratios. This is significant considering the fact that a full scale truss and gusset plate model has been analyzed, whereas a small scale model of a gusset plate was tested by Whitmore, in addition to differences in truss geometry and relative axial rigidities of truss members. Principal tensile (major) stress contours at Joint L2 are shown in Figure 3. At the end of diagonal member L2-U1, there is a zone of high tensile stress in the gusset plate, shown as a green color. The location of this stress zone is consistent with the Whitmore test results. The highest tensile stresses, shown as yellow and red colors, are located at the interface corner zones between truss members especially chord L1-L2 and the gusset plate. L2-U1 L2-U2 L2-U KSI PSI L1-L2 High Stress Stress Concentration L2-L3 Figure 3: Principal Tensile Stress Contours at Joint L2

9 There is high stress concentration in these zones, primarily due to simplifying assumptions in the model. For instance, rivets are not modeled and therefore stress is transferred totally at the interface from a truss member to the gusset plate, instead of gradually through the rivets in an actual joint or a three-dimensional model of a joint. Various interface modeling schemes have been tried to determine the extent of this stress concentration. When edge nodes of an interface line are removed from the master-slave list, for example at the bottom chord interface, the highest stress zone is moved away from the gusset plate and into the bottom chord segment within the joint. But when the interface master joint of chord L2-L3 is moved away from the edge of the gusset plate and into the chord, requiring additional plate elements to model the chord, a stress concentration zone remains at the same location as in Figure 3. Without further research, including laboratory test verification, a firm conclusion on the observed stress concentration could not be drawn. The Whitmore testing did not address this issue, but that does not necessarily mean it is a realistic concern or the other way around. Figure 4 shows the Whitmore section width at the end of diagonal member L2-U1. This width is determined by drawing a line starting from each corner rivet in the first row of the member, along an angle of 30 degrees with respect to member axis, and extending the two lines until they intersect a third line drawn through the bottom row of rivets. For this case however, the width is limited by the proximity of rivets in adjacent members. When the diagonal member is modeled to be shorter in length within the joint, angular divergence or distribution of tensile stress becomes even more obvious. It is interesting to note that based on the report of R. E. Whitmore (1), the angle of 30 degrees was commonly used at that time, implying that it was validated and not initiated by this historic testing.

10 30 o 30 o W Figure 4: Whitmore Section Width at the End of Diagonal Member L2-U1 The estimated Whitmore section width for the case shown in Figure 4 is 43.2 inches. A tensile distributed stress of 20.6 ksi on this section is calculated, based on the estimated width, total gusset plate thickness of inches and a tensile force of 1445 kips in diagonal member L2-U1. This stress value is approximately equal to the finite element stress of 20.1 ksi shown in green color in Figure 3. Principal compressive (minor) stress contours at Joint L2 are shown in Figure 5. A zone of high compressive stress in the gusset plate is shown at the edge of diagonal member L2- U3, as blue and dark blue colors. This observation is also consistent with the Whitmore testing. There is divergence of compressive stress; but given the proximity of the diagonal to the vertical and the tapering of the diagonal as shown in Figure 1, no conclusion could be drawn on a divergence angle.

11 L2-U1 L2-U2 Stress Concentration L2-U High Stress 4.4 KSI PSI L1-L2 L2-L3 Figure 5: Principal Compressive Stress Contours at Joint L2 However, when modifying the model by shortening the length of the diagonal segment within the gusset plate, stress distribution could be defined with a divergence angle of approximately 30 degrees, as suggested by Whitmore. Another key observation is the vertical zone of high compressive stress at the edge of the plate near diagonal member L2-U3. There is a steady decrease in compressive stress along the plate edge, starting at the top and moving downwards to chord member L2-L3. This is significant because when checking edge compressive stress for gusset plate design or evaluation, a maximum normal stress is calculated at the plate edge along a horizontal plane above the chord and the stress is assumed to exist along the vertical edge of the plate.

12 L2-U1 L2-U2 L2-U3 0.2 Stress Concentration 14.4 KSI PSI L1-L2 L2-L3 High Stress Figure 6: Principal Shear Stress Contours at Joint L2 Shown in Figure 6 are principal shear stress contours at Joint L2. There is a zone of maximum shear stress in the gusset plate, shown in yellow and green colors along a horizontal plane above the bottom chord, extending from plate edge above member L2-L3 to below diagonal member L2-U1. Again this observation is consistent with the Whitmore testing. The highest shear stress shown in red color is due to stress concentration, given the assumptions in the model and as explained for the principal tensile stress contours. Local shear stress along a horizontal section above the bottom chord is shown in Figure 7. For a typical design or evaluation of a gusset plate, this shear stress is calculated by assuming uniform distribution of the sum of the horizontal force components of the two diagonal members.

13 KSI KSI Figure 7: Local Shear Stress on Horizontal Section above Bottom Chord at Joint L2 For the primary load case (LC 1), the sum of the horizontal force components is 1325 kips. The area of the horizontal section is the plate width (124 inches) multiplied by the plate thickness (13/16 inches). With two plates resisting member forces, the average shear stress on the horizontal plane is about 6600 psi, which is consistent with the shear stress shown in Figure 7. Local normal stress along the same horizontal section above the bottom chord is shown in Figure 8. Compressive and tensile stresses are shown respectively below and above the horizontal distance axes. Few observations are noteworthy. First, there is high tensile stress at the plate edge connected to chord member L2-L3, where compression is expected. This discrepancy is related to the observed concentration of principal tensile stress shown in Figure 3. The previous discussion of stress concentration applies to both principal and normal stresses. Second, there is tensile stress near chord member L1-L2 and along a horizontal projection of diagonal member L2-U1, as expected.

14 KSI KSI Figure 8: Local Normal Stress on Horizontal Section above Bottom Chord at Joint L2 The location of maximum stress is not at the edge, as predicted from beam theory, but inside the gusset plate area. This result is consistent with the Whitmore testing, except that the location of maximum tension is farther away from the edge and below the diagonal member. Third, a similar observation can be made for normal compressive stress within the plate, regarding its maximum value and location. MODELING AND ANALYSIS OF NON-TYPICAL GUSSET PLATE GEOMETRY The modeling and analysis procedure presented before is applicable to any gusset plate joint geometry. To demonstrate its versatility, the procedure is applied to a non-typical gusset plate connection, specifically Joint U1, which has non-rectangular plate geometry and non-symmetric configuration of four connecting truss members. Joint L2 in contrast has rectangular plate geometry and approximately symmetric configuration of five connecting members, which is typical for other lower chord joints such as L4 and L6, each

15 with two connecting diagonals. The same can be stated for upper chord joints with five connecting members, such as U5 and U7. Figure 9 shows a finite element model of gusset plates at upper chord Joint U1. The same modeling procedure is applied, whereby (i) rectangular plate elements are used primarily to represent truss member segments within the joint and (ii) triangular plate elements are used for other areas of the joint. For each of the four connecting truss members, a rigid link is provided at the interface between the beam element representing the member and the finite elements, as described before. U1-U2 U1-L0 U1-L1 U1-L2 Figure 9: Finite Element Model of Upper Chord Joint U1

16 Table 3: Live Load Member Forces at Truss Joint U1 Load Cases Truss Member Forces, kips U1-L0 U1-L1 U1-L2 U1-U2 LC LC LC LC LC Maximum Minimum Shown in Table 3 are live load forces in truss members at Joint U1 for various load cases, including maximum tension or compression force in a member with corresponding force in other members, and maximum tension or compression in each member. Positive values indicate tensile forces and negative values indicate compressive forces. Load Case 1 is dominant because of concurrent maximum compression in the upper chord and end post members and close to maximum forces in the vertical and diagonal members. The force in diagonal member U1-L2 is slightly different (0.5% less) than the force in the same member resulting from the analysis for Joint L2. The reason for this insignificant difference is that each joint is modeled separately, resulting in slightly different length and axial rigidity of this member. Gusset plate stresses at Joint U1 are presented for the dominant load case only. Figure 10 shows principal tensile stress contours in Joint U1. There is a high tension zone in the gusset plate, shown by the colors green and yellow, near the intersection of members U1- L0 and U1-U2 and above diagonal member U1-L2.

17 1.5 U1-U2 Stress Concentration High Stress Stress Concentration 12.9 KSI PSI U1-L0 U1-L1 U1-L2 Figure 10: Principal Tensile Stress Contours at Joint U1 This is consistent with the fact that the diagonal member has the highest tensile force, while members U1-L0 and U1-U2 have almost equal compressive forces that counteract the tensile force. The noted tensile zone is located between the end of the diagonal member and the intersection of the compression members. The maximum tensile principal stress in this zone is about 10 ksi. This value can be checked by assuming the diagonal force to be distributed over a section along its interior end rivets, which extends to neighboring rivets in the upper chord and end post and is limited by two lines making 30 degree angles with the axis of the diagonal. With an estimated distribution width of 50 inches, sum of gusset plate thickness of 2.75 inches, and tensile force of 1438 kips, a tensile stress of 10.5 ksi is calculated. Even though this is a normal stress, not a principal stress, it provides a rough check on the analysis results.

18 There are stress concentrations, shown in Figure 10 in the colors red and yellow, at the interface between members U1-L2 and U1-U2 and the gusset plate. A discussion of stress concentration and the effects of modeling assumptions on this behavior are presented before. Figure 11 shows principal compressive stress contours in Joint U1. Stress concentrations are shown in this figure in pink and brown colors, at the gusset plate interface with members U1-L0 and U1-U2. An elongated zone of maximum compression is shown along the inside edges of members U1-L0 and U1-U U1-U2 Stress Concentration High Stress 2.5 KSI PSI U1-L0 U1-L1 U1-L2 Figure 11: Principal Compressive Stress Contours at Joint U1

19 0.1 U1-U2 High Stress 8.3 KSI PSI U1-L0 U1-L1 U1-L2 Figure 12: Principal Shear Stress Contours at Joint U1 Principal shear stress contours are shown in Figure 12. The highest shear stress zone in the gusset plate occurs near the intersection of members U1-L0 and U1-U2 and above diagonal member U1-L2, and is shown by the colors red and brown. This zone is approximately at the same location as the zone for high principal tensile stress. MATHCAD ANALYSIS AND DESIGN CHECKS A Mathcad work sheet is developed for analysis and evaluation of gusset plates in railroad truss bridges. It has been achieved by modifying a NYSDOT work sheet template (3) to include applicable requirements from AREMA Manual of Railway Engineering (9) and new requirements on gusset plates by AREMA Committee 15. Stresses on horizontal and vertical critical sections of a gusset plate are calculated by assuming simple beam theory.

20 Truss member forces are distributed along a Whitmore section, as previously described in this paper. Figure 13 shows the assumed critical sections on Joint L2. A horizontal section (A-A) is located above the bottom chord at the location of the lowest rivet or line of rivets. A vertical section (B-B) is located on one side of the vertical member, along the line of perimeter rivets. On the other side of the vertical member, another vertical section (C-C) is assumed in a similar manner. C B A A C B Figure 13: Critical Sections for Mathcad Analysis The analysis part of the Mathcad procedure consists of the following steps: 1. Resolve member forces into horizontal and vertical force components

21 2. Calculate properties of each section, specifically area, moment of inertia, neural axis, section modulus and moment arms 3. Add separately components of axial force, shear force and bending moment, 4. Calculate stresses, specifically axial stress (force divided by area), flexural stress (moment divided by section modulus), and average shear stress (force divided by area) 5. Combine axial stress with flexural stress to calculate maximum compression and tension at plate edges. Calculations for Load Case 1 on Joint L2 result in the following stresses on the horizontal section (A-A): an axial tension stress of 1.6 ksi, a flexural stress of 5.3 ksi, and a shear stress of 6.5 ksi. The axial and flexural stresses are combined (added or subtracted) to calculate a tension stress of 6.9 ksi at the left edge of the plate (near member L2-L1) and a compressive stress of 3.7 ksi at the right edge of the plate (near member L2-L3). In comparison and as shown in Figure 8, the finite element analysis results in the following local stresses on the critical horizontal section a tension stress of 2 ksi at plate edge, maximum tension stress of 8 ksi within the plate, and maximum compressive stress of 5.5 ksi within the plate. The stress concentration at the interface of the gusset plate and member L2-L3 is not considered, considering previous discussion. A key difference between the two sets of results is the location of maximum normal stress. Aside from this, the values of maximum stress calculated through Mathcad are 14% less for tension and 33% less for compression, as compared to the finite element analysis.

22 Referring to Figure 7, the local shear stress on the horizontal section as a result of the finite element analysis is as follows: a maximum value of 12.5 ksi near member L2-L3, and two peak values of 9.5 ksi and 10.5 ksi within the gusset plate. The maximum value is influenced by stress concentration at the interface between the plate and the truss member. In comparison, an average shear stress of 6.6 ksi is calculated through Mathcad, which is consistent with manual calculations. Based on simple beam theory, a maximum shear stress of 1.5 times the average stress (9.8 ksi) is assumed to occur at the middle of a section. This maximum value is approximately equal to the peak shear stresses calculated through finite element analysis. Design or Evaluation Checks The second part of the Mathcad procedure consists of checking various design requirements for gusset plates in the AREMA Manual of Railway Engineering, including new requirements by AREMA Committee 15, for each of three critical sections (A-A, B-B, and C-C). This part consists of the following steps: 1. Design checks consistent with draft Article b: Normal stresses on the Whitmore section Compressive stress at plate edge Tensile stress at plate edge Average shear stress Unsupported edge length requirement Rivet or bolt shear capacity Bolt slip-capacity Bearing capacity Block shear capacity 2. Allowable stresses per Table :

23 Shear on gross section (0.35F y ) Tension on gross section (0.55F y ) Compression on gross section (< 0.55F y ) based on buckling of unsupported edge and buckling factor and average buckling length for Whitmore section as per draft Article b Shear on rivet or bolt Bearing on rivet or bolt 3. Allowable block shear per Article e 4. Unsupported edge length to thickness ratio of less than or equal to 2.06 E/F y as per draft Articles c and c 5. Slenderness ratio (ratio of length to least radius of gyration) of less than or equal to 100 per draft Article b A detailed presentation of an example with design checks is beyond the scope of this paper. CONCLUSIONS Presented in this paper is a simple and effective procedure for performing two-dimensional structural analysis of gusset plates in railroad truss bridges, through combined use of beam elements for truss members and plate elements for gusset plate. A typical lower chord joint is modeled and analyzed, and stress results are compared to the historic Whitmore testing. Locations of peak values of principal stresses as determined by the analysis are consistent with those reported by Whitmore. The same is true for the normal stress on a horizontal section, whereby peak values of either tensile or compressive stresses are located within the gusset plate, not at its edges as predicted by simple beam theory. Further, the distribution section for truss member forces is consistent with the Whitmore section, defined by two lines making 30 degrees with the axis of a member, starting at its outer

24 corner rivets and intersecting a third line drawn along its inner row of rivets. Versatility of the finite element procedure is demonstrated by modeling and analyzing a non-typical gusset plate at an upper chord joint. A Mathcad work sheet is developed for analysis and evaluation of gusset plates. The analysis part of the worksheet consists of calculating normal and shear stresses on horizontal and vertical sections. For the typical lower chord joint, calculated stresses on a horizontal section are compared with those from the finite element analysis. Maximum normal stress values from Mathcad are 14% to 33% less than those from the finite element analysis. An average shear stress from Mathcad is converted to maximum shear stress, which is approximately equal to peak shear stresses from the finite element analysis. Listed in the design and evaluation part of the Mathcad work sheet are various checks on gusset plate design, as per the AREMA Manual of Railway Engineering, including draft requirements by AREMA Committee 15. REFERENCES 1. Whitmore, R.E., Experimental Investigation of Stresses in Gusset Plates, May FHWA Report H-08-1, Adequacy of the U10 & L11 Gusset Plate Designs for the Minnesota Bridge 9340 (I-35W over the Mississippi River), January NYSDOT Technical Advisory , Truss Gusset Plate Analysis, May NCHRP Guidelines for the Load and Resistance Factor Design and Rating of Riveted, Bolted and Welded Gusset-Plate Connections for Steel Bridges, June 2008 Present (In-progress Testing at the Federal Highway Administration (FHWA), Turner-Fairbanks Highway Research Center (TFHRC))

25 5. FHWA, Finite Element Modeling of I-35W Bridge Collapse Final Report, TFHRC Report, October FHWA-IF , Load Rating Guidance and Examples for Bolted and Riveted Gusset Plates in Truss Bridges, February NCHRP, Guidelines for the Load and Resistance Factor Design and Rating of Riveted, Bolted, and Welded Gusset-Plate Connections for Steel Bridges, Project 12-84, 2008-Present 8. Bentley Systems, Inc., STAAD.Pro Structural Software, Version V8i, AREMA, Manual of Railway Engineering, American Railway Engineering and Maintenance-of-Way Association (AREMA), 2010