THE STRUCTURAL ANALYSIS, DESIGN, AND PROTOTYPE TESTING OF THREE-SIDED SIMALL-SPAN SKEWED BRIDGES

Transcription

1 FINAL REPORT THE STRUCTURAL ANALYSIS, DESIGN, AND PROTOTYPE TESTING OF THREE-SIDED SIMALL-SPAN SKEWED BRIDGES Prepared by Daniel Farhey, Manoochehr Zoghi, and Anis Gawandi Department of Civil and Environmental Engineering The University of Dayton 300 College Park Dayton, OH A report of research studies conducted under the sponsorship of The Ohio Department of Transportation and the U. S. Department of Transportation, Federal Highway Administration January 2002 State Job No (0)

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3 1. Report No. FHWAfOH-2002/ Title and subtitle. H 2. Government Accession No. 3. Recipient's Catalog No. 5. Report Date The Structural Analysis, Design, and Prototype Testing of Three- IApri19 2o02 Sided Small-Span Skewed Bridges 6. Performing Organization Code 7. Author(s) Dr. M Zoghi 8. Performing Organization Report No. 10. Work Unit No. (TRAIS) I 9. Performing Organization Name and Address University of Dayton 11. Contract or Grant No. Department of Civil 8 Environmental Engineering and Engineering /State Job No (0) Mechanical 300 College Park nu Sponsoring Agency Name and Address Ohio Department of Transportation 1980 W Broad Street Columbus, OH Supplementary Notes 13. Type of Repon and Period Covered Final Report 14. Sponsoring Agency Code I 6. Abstract An analytical study was carried out for the structural performance assessment of precast-concrete, short-span, skewed bridges with integral abutment walls. Typically, these structures are designed as simplified two-dimensional rigid portal frames, neglecting the degrading effects of the skew angle and laterally unsymmetrical vertical loading. This design practice produces under-designed bridges for certain aspect ratios, causing cracking and local deterioration symptoms, observed in some instances out in the field. To evaluate the limitations of this practice, three-dimensional finite-element models were developed and analyzed. Accordingly, these finite-element models simulate various geometric configuration parameters, as well as, laterally symmetrical and unsymmetrical vertical load conditions, capturing the amplification of the structural response. Field-testing was also performed on a bridge to substantiate and calibrate the finite-element results. The results of the simplified plane frame analyses and three-dimensional finite-element analyses were presented in correlation diagrams, enabling simple comparison and quantification. The correlation diagrams provide correction factors to amend the simple frame design. The response observations offer a qualitative insight into the actual behavior of the structure, allowing the performance assessment of existing bridges of the same type and a more reliable design in the future. '. Key Words bridges; experimental tests; finite-element analysis; integral abutments; precast concrete; reinforced concrete; skewed bridges; structural analysis.. %Cum ChSSif. (Of this NPOIt) lnclassified 20. Securlfy Claasif. (of this page) Unclassified!l. NO. of Pagw I 22. Prlce

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5 THE STRUCTURAL ANALYSIS, DESIGN, AND PROTOTYPE TESTING OF THREE-SIDED SMALL-SPAN SKEWED BRIDGES Daniel Farhey, Manoochehr Zoghi, and Anis Gawandi Department of Civil and Environmental Engineering The University of Dayton 300 College Park Dayton, Ohio Prepared in Cooperation with the Ohio Department of Transportation and the U.S. Department of Transportation, Federal Highway Administration January 2002 State Job No (0) The contents of this report reflect the views of the authors who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the Ohio Department of Transportation. This report does not constitute a standard, specification, or regulation.

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7 ABSTRACT An analytical study was carried out for the structural performance assessment of precastconcrete, short-span, skewed bridges with integral abutment walls. Typically, these structures are designed as simplified two-dimensional rigid portal frames, neglecting the degrading effects of the skew angle and laterally unsymmetrical vertical loading. This design practice produces underdesigned bridges for certain aspect ratios, causing cracking and local deterioration symptoms, observed in some instances out in the field. To evaluate the limitations of this practice, threedimensional finite-element models were developed and analyzed. Accordingly, these finite-element models simulate various geometric configuration parameters, as well as, laterally symmetrical and unsymmetrical vertical load conditions, capturing the amplification of the structural response. Fieldtesting was also performed on a bridge to substantiate and calibrate the finite-element results. The results of the simplified plane frame analyses and three-dimensional finite-element analyses were presented in correlation diagrams, enabling simple comparison and quantification. The correlation diagrams provide correction factors to amend the simple frame design. The response observations offer a qualitative insight into the actual behavior of the structure, allowing the performance assessment of existing bridges of the same type and a more reliable design in the future. Keywords: bridges; experimental tests; finite-element analysis; integral abutments; precast concrete; reinforced concrete; skewed bridges; structural analysis... 11

8 ACKNOWLEDGMENTS The present research study has been partially funded by the Ohio Department of Transportation (ODOT). This application has been a continuation of a practical research on the performance assessment of various short-span, skewed bridge systems integrating conceptual analysis with experimental field testing. The financial support of ODOT and the guidance of Mr. Roger Green and Mr. Kevin White are greatly appreciated. The national survey, initial structural modeling, and field test were conducted by Mr. David Niday, toward partial fulfillment for the Master of Science degree at the University of Dayton. The finite-element analysis for haunched models and statistical analysis were conducted by Mr. Anis Gawandi, toward partial fulfillment for the Master of Science degree at the University of Dayton. The field test was made possible by the Montgomery County Engineer s Office. The authors wish to express their gratitude to Montgomery County s Engineering Staff, especially Mr. Douglas Miller, for their cooperation and enormous assistance

9 TABLE OF CONTENTS ABSTRACT ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES 1. INTRODUCTION 1.1 BACKGROUND 1.2 OBJECTIVES AND SCOPE 2. STRUCTURAL MODEL 2.1 MODELING CONCEPT 2.2 SURVEY 2.3 MODELING 2.4 MODEL EVALUATION 3. EXPERIMENTAL FIELD TEST 3.1 TEST BF2DGE 3.2 TEST SETUP 3.3 LOAD TEST 3.4 TEST RESULTS 4. PRELIMINARY FINITE-ELEMENT ANALYSIS 4.1 BASIC PARAMETRIC MODELS 4.2 ANALYSIS RESULTS 5. FINITE-ELEMENT ANALYSIS 5.1 GENERAL 5.2 SCOPE 5.3 PARAMETRIC CHARACTERISTICS 5.4 ANALYSIS LEVELS Preliminary Analysis First Mesh Refinement Final Analysis 5.5 ANALYSIS RESULTS Effect of Haunches Comparison with No Haunches Comparison with Plane Frames 5.6 MAGNIFICATION FACTORS 6. STATISTICAL STUDY 6.1 GENERAL ii iii vi viii ' iv

10 6.2 POSITIVE MOMENT STRESSES 6.3 NEGATIVE MOMENT STRESSES 7. DISCUSSION 7.1 GENERAL 7.2 BEHAVIOR 8. CONCLUSION AND RECOMMENDATIONS 8.1 CONCLUSIONS 8.2 RECOMMENDATIONS REFERENCES APPENDIX A - DOT SURVEY RESPONSES R- 1 A-1 APPENDIX B - STRUCT D FINITE-ELEMENT MODEL ' B-1 APPENDIX C - FINITE-ELEMENT STRESS RESULTS APPENDIX D - STATISTICAL CODE AND OUTPUT c-1 D-1 V

11 LIST OF TABLES Table 2-1. Range of Configuration Parameters of Surveyed Bridges Table 2-2. Configuration Parameters for The Parametric Bridge Models Table 5-1. Preliminary Analysis Results, B = 4 ft, H = 10 ft, S = 15 ft Table 5-2. Preliminary Analysis Results, B = 4 ft, H = 10 ft, S = 22.5 ft Table 5-3. Preliminary Analysis Results, B = 4 ft, H = 10 ft, S = 30 ft Table 5-4. Preliminary Analysis Results, B = 4 ft, H = 5 ft, S = 15 ft Table 5-5. Preliminary Analysis Results, B = 4 ft, H = 5 ft, S = 22.5 ft Table 5-6. Preliminary Analysis Results, B = 4 ft, H = 5 ft, S = 30 ft Table 5-7. Preliminary Analysis Results, B = 6 ft, H = 10 ft, S = 15 ft Table 5-8. Preliminary Analysis Results, B = 6 ft, H = 10 ft, S = 22.5 ft Table 5-9. Preliminary Analysis Results, B = 6 ft, H = 10 ft, S = 30 ft Table Preliminary Analysis Results, B = 6 ft, H = 5 ft, S = 15 ft Table Preliminary Analysis Results, B = 6 ft, H = 5 ft, S = 22.5 ft Table Preliminary Analysis Results, B = 6 ft, H = 5 ft, S = 30 ft Table First Mesh Refinement Analysis Results, B = 4 ft, H = 10 ft, S = 15 ft Table First Mesh Refinement Analysis Results, B = 4 ft, H = 10 ft, S = 22.5 ft Table First Mesh Refinement Analysis Results, B = 4 ft, H = 10 ft, S = 30 ft Table First Mesh Refinement Analysis Results, B = 4 ft, H = 5 ft, S = 15 ft Table First Mesh Refinement Analysis Results, B = 4 ft, H = 5 ft, S = 22.5 ft Table First Mesh Refinement Analysis Results, B = 4 ft, H = 5 ft, S = 30 ft Table First Mesh Refinement Analysis Results, B = 6 ft, H = 10 ft, S = 15 ft vi

12 Table First Mesh Refinement Analysis Results, B = 6 ft, H = 10 ft, S = 22.5 ft Table First Mesh Refinement Analysis Results, B = 6 ft, H = 10 ft, S = 30 ft Table First Mesh Refinement Analysis Results, B = 6 ft, H = 5 ft, S = 15 ft Table First Mesh Refinement Analysis Results, B = 6 ft, H = 5 ft, S = 22.5 ft Table First Mesh Refinement Analysis Results, B = 6 ft, H = 5 ft, S = 30 ft Table Final Analysis Results, B = 4 ft, H = 10 ft, S = 15 ft Table Final Analysis Results, B = 4 ft, H = 10 ft, S = 22.5 ft Table Final Analysis Results, B = 4 ft, H = 10 ft, S = 30 ft Table Final Analysis Results, B = 4 ft, H = 5 ft, S = 15 ft Table Final Analysis Results, B = 4 ft, H = 5 ft, S = 22.5 ft Table Final Analysis Results, B = 4 ft, H = 5 ft, S = 30 ft Table Final Analysis Results, B = 6 ft, H = 10 ft, S = 15 ft Table Final Analysis Results, B = 6 ft, H = 10 ft, S = 22.5 ft Table Final Analysis Results, B = 6 ft, H = 10 ft, S = 30 ft Table Final Analysis Results, B = 6 ft, H = 5 ft, S = 15 ft Table Final Analysis Results, B = 6 ft, H = 5 ft, S = 22.5 ft Table Final Analysis Results, B = 6 ft, H = 5 ft, S = 30 ft Table Summary of Negative Moment Stresses Table Summary of Positive Moment Stresses vii

13 LIST OF FIGURES Figure 2-1. Parametric Solid Model of a Modular Skewed Bridge Figure 2-2. Basic Prototype Bridge Models for Analysis Figure 3-1. Truck Load Test of the Bridge Figure 4-1. Preliminary Finite-Element Model with Unit Load Case Figure 4-2. Finite-Element to Frame Stress Ratios, B = 4 ft, H = 10 ft, Negative Moment Figure 4-3. Finite-Element to Frame Stress Ratios, B = 4 ft, H = 10 ft, Positive Moment Figure 4-4. Finite-Element to Frame Stress Ratios, B = 4 ft, H = 5 ft, Negative Moment Figure 4-5. Finite-Element to Frame Stress Ratios, B = 4 ft, H = 5 ft, Positive Moment Figure 4-6. Finite-Element to Frame Stress Ratios, B = 6 ft, H = 10 ft, Negative Moment Figure 4-7. Finite-Element to Frame Stress Ratios, B = 6 ft, H = 10 ft, Positive Moment Figure 4-8. Finite-Element to Frame Stress Ratios, B = 6 ft, H = 5 ft, Negative Moment Figure 4-9. Finite-Element to Frame Stress Ratios, B = 6 ft, H = 5 ft, Positive Moment Figure Finite-Element to Frame Deflection Ratios, B = 4 ft, H = 10 ft Figure Finite-Element to Frame Deflection Ratios, B = 4 ft, H = 5 ft Figure Finite-Element to Frame Deflection Ratios, B = 6 ft, H = 10 ft Figure Finite-Element to Frame Deflection Ratios, B = 6 ft, H = 5 ft Figure 5-1. Parametric Solid Model of a Haunched Modular Skewed Bridge Figure 5-2. Preliminary Haunched Finite-Element Model with Unit Load Case Figure 5-3. Preliminary Haunched FEYFrame Stress Ratios, B = 4 ft, H = 10 ft, Negative Moment Figure 5-4. Preliminary Haunched FE/Frame Stress Ratios, B = 4 ft, H = 10 ft, Positive Moment Figure 5-5. Preliminary Haunched FE/Frame Stress Ratios, B = 4 ft, H = 5 ft, Negative Moment viii,

14 Figure 5-6. Preliminary Haunched FE/Frame Stress Ratios, B = 4 ft, H = 5 ft, Positive Moment Figure 5-7. Preliminary Haunched FE/Frame Stress Ratios, B = 6 ft, H = 10 ft, Negative Moment Figure 5-8. Preliminary Haunched FE/Frame Stress Ratios, B = 6 ft, H = 10 ft, Positive Moment Figure 5-9. Preliminary Haunched FEFrame Stress Ratios, B = 6 ft, H = 5 ft, Negative Moment Figure Preliminary Haunched FE/Frame Stress Ratios, B = 6 ft, H = 5 ft, Positive Moment Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 15 ft Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 22.5 ft Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 30'ft Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 15 ft Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 22.5 ft Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 30 ft Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 15 ft Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 22.5 ft Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 30 ft Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 15 ft Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 22.5 ft Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 30 ft Figure First Refinement Haunched Finite-Element Model Figure First Haunched FE/Frame Stress Ratios, B = 4 ft, H = 10 ft, Negative Moment Figure First Haunched FE/Frame Stress Ratios, B = 4 ft, H = 10 ft, Positive Moment Figure First Haunched FE/Frame Stress Ratios, B = 4 ft, H = 5 ft, Negative Moment Figure First Haunched FE/Frame Stress Ratios, B = 4 ft, H = 5 ft, Positive Moment Figure First Haunched FE/Frame Stress Ratios, B = 6 ft, H = 10 ft, Negative Moment ix

15 Figure First Haunched FE/Frame Stress Ratios, B = 6 ft, H = 10 ft, Positive Moment Figure First Haunched FE/Frame Stress Ratios, B = 6 ft, H = 5 ft, Negative Moment Figure First Haunched =/Frame Stress Ratios, B = 6 ft, H = 5 ft, Positive Moment Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 15 ft Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 22.5 ft Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 30 ft Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 15 ft I Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 22.5 ft Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 30 ft Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 15 ft Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 22.5 ft Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 30 ft Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 15 ft Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 22.5 ft Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 30 ft Figure Final Refinement Haunched Finite-Element Model Figure Final Haunched FE/Frame Stress Ratios, B = 4 ft, H = 10 ft, Negative Moment Figure Final Haunched FE/Frame Stress Ratios, B = 4 ft, H = 10 ft, Positive Moment Figure Final Haunched FE/Frame Stress Ratios, B = 4 ft, H = 5 ft, Negative Moment Figure Final Haunched FE/Frame Stress Ratios, B = 4 ft, H = 5 ft, Positive Moment Figure Final Haunched FE/Frame Stress Ratios, B = 6 ft, H = 10 ft, Negative Moment Figure Final Haunched FFdFrame Stress Ratios, B = 6 ft, H = 10 ft, Positive Moment Figure Final Haunched FE/Frame Stress Ratios, B = 6 ft, H = 5 ft, Negative Moment X

16 Figure Final Haunched FEFrame Stress Ratios, B = 6 ft, H = 5 ft, Positive Moment Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 15 ft Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 22.5 ft Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 30 ft Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 15 ft Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 22.5 ft Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 30 ft Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 15 ft Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 22.5 ft Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 30 ft Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 15 ft Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 22.5 ft Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 30 ft xi

17 1. INTRODUCTION 1.1 BACKGROUND An estimated 28% of the existing 590,984 bridges in the US are either structurally deficient andor functionally obsolete (Better Roads 2001). Nearly two thirds of the bridge inventory consists of short spans, with a clear span limited to 36.6 m (120 ft). Thus, short-span bridges that &-e in critical condition make up the largest population of bridges requiring intervention. When preservation is not a feasible option, replacement is the only solution. Often, due to pressing economical requirements, new bridge systems materialize before a comprehensive evaluation is made available. Objective information is essential for the reliable evaluation of a structural system. Identification of load resistance and failure mechanisms, through finite-element modeling and analysis, integrated with experimental field-testing, are necessary for the evaluation of structural performance and integrity. A comprehensive evaluation of the structural system should also consider the common design and construction practices that will follow until the final product is achieved. When the structural system is not designed essentially the same way it is evaluated, the final product is most likely to offer a different performance. Precast-concrete modular bridges with integral abutment walls often offer a preferable solution for short-span bridge replacements due to their low initial cost, rapid installation, and low maintenance. Typically, the bridge is composed of one or more modular units to accommodate the respective width of roadway. Often, construction details call for skewing of the ends of the deck to 1-1.

18 align the abutment walls with streams or roadways below the bridge that are not perpendicular to the roadway of the bridge. In design practice, these bridge structures are analyzed as individual modular units and further simplified to two-dimensional rigid portal frames, neglecting any three-dimensional phenomena and laterally unsymmetrical vertical loading effects. The skewed structures are however controlled by three-dimensional behavior that a two-dimensional frame analysis cannot reliably capture. Consequently, some of the existing short-span, skewed bridges have exhibited recurring symptoms of crackmg and problems of local deterioration. This study investigates the threedimensional characteristics of precast-concrete, short-span, skewed bridges with integral abutment walls and devises practical design tools for extrapolating the two-dimensional frame analysis to reflect the three-dimensional effects. Mahmoudzadeh (1984) noticed significant reductions in the longitudinal bending moments of skewed bridge decks, while investigating the parametric characteristics of 54 reinforced-concrete slab bridge structures. The stiffening of the ends by the skew is expected to have this effect to some extent, however, torsional effects caused by the skew were not considered in the analysis. Dagher et al. (1991a, 1991b) considered the skew effects of reinforced-concrete bridge decks with integral abutment walls. This study concluded that a two-dimensional analysis is inadequate for calculating the moments of skewed bridges. These researchers examined bridges of small aspect ratios (span to width), limited to 2.0. The bridges considered in the current study possess aspect ratios of 6.0 and higher. The study of Dagher et al. (1991a, 1991b) relies upon a linear finite-element model that utilizes thin plate elements and boundary elements to restrain degrees of freedom as desired. This approach for modeling the reinforcement, while apparently successful, requires the development of 1-2

19 unrealistic boundary conditions to yield a structurally consistent solution. Therefore, a solid model that uses the actual geometry and material characteristics of the structure is preferred. Such a model can identify geometric constraints and secondary effects implicitly. To examine typically used aspect ratios and get a more reliable representation of the bridge structure, this study evaluates the threedimensional behavior and quantifies the extent of non-conservatism involved in the two-dimensional frame analyses. Kankam et al. (1995) completed non-linear finite-element analysis of skewed bridges. This study relied upon the non-linear behavior of reinforced concrete. The finite-element model involved superposition of several steel and concrete layers into the same space to simulate the behavior of the reinforced-concrete section. The complexity of the development for the element occupies most of the reference study. The practical application of such a complex model is naturally limited. 1.2 OBJECTIVES AND SCOPE The principal objective of the present study was to assess the performance of precast- concrete, short-span, skewed bridges, with integral abutment walls. The scope of this study consisted Of 1. Formulation and performance of an extensive nationwide survey to identify the typical prototype parameters and their variation ranges in the most common precast-concrete, short-span, skewed bridges, with integral abutment walls. The survey also intended to identify the current design and analysis practices. 1-3

20 2. Preliminary conceptualization of prototype short-span bridge models with parametric dimensions and skew angles. 3. Three-dimensional finite-element modeling and analysis of the parametric bridge models. 4. Experimental testing of an actual bridge to evaluate the field performance and calibrate the finite-element modeling. 5. Comparison of plane frame analysis with three-dimensional finite-element analysis and quantification of the outcome through correlation diagrams. 1-4

21 2. STRUCTURAL MODEL ' 2.1 MODELING CONCEPT In order to complete the analysis of any structure, an appropriate engineering model must be defined that adequately represents the structural system in question. The success of that model lies in how accurately and reliably it represents the behavior of the structure under the conditions in which the analysis is to be performed. With the objectives of this study stated, the next important step in the analysis is to define the geometry and loadings of typical bridges used in the industry. Once the bridge design parameters are defined, models in both two-dimensional and three-dimensional analysis methods can be created to complete the comparison studles of this study. Figure 2-1 shows a modular skewed bridge with integral abutment walls and details the relevant configuration parameters defined in this study. 2.2 SURVEY At the onset of the project, it became apparent that to represent the structural characteristics of typical short-span, skewed bridges, it was essential to define the prototype models considering the variation range of the individual parameters. To define the prototype parametric models, a nationwide survey was conducted (Niday 1997). The survey, summarized in Appendix A, included the fifty State Departments of Transportation (DOT) in the US, Washington, D.C., Puerto Rico, and 2-1

22 the Province of Ontario, Canada. The response percentage to the survey was over 64%. The range of configuration parameters identified from the responses to the survey is summarized in Table 2-1. Accordingly, based upon the survey, common bridge parameters consist of m (15-18 ft) span, 1.5 m (5 ft) width, m (6-8 ft) height, 300 mm (12 in.) uniform thickness for the deck and the abutment walls, and 30" or less skew angle. Furthermore, the survey aimed to identify the current practice methods for the analysis and design of these modular bridge units. Therefore, independent structural engineering design firms and precast-concrete manufacturers, involved with the design of this type of bridges, were polled in addition to the DOT'S. Most designers identified the two-dimensional portal frame as their common practice for modeling and the moment distribution method as their method of choice for analysis. The survey also revealed that the common engineering design practice to account for the effects of the skew angle is either to ignore it, or increase the span length slightly, or simply add additional reinforcement. Most practicing structural engineers recognize the significance of the skew effect on the overall behavior of the structure, however, stated that the difficulty and expense of a threedimensional analysis rules out its application. Also based on the survey, the effect of the haunches between the abutment walls and the deck are not typically incorporated in the common practice method of two-dimensional portal frame analysis. The survey also identified that these bridges are commonly designed according to AASHTO Standard's (1992) HS-20 or HS-25 truck series loads, or alternative military loads. 2-2

23 2.3 MODELING The success of any finite-element analysis hinges on the accuracy of the modeling. An efficient model is the one that simulates the actual structural conditions closely. The paramount importance is element choice and mesh density. As finite-element program computes the stresses at selected nodes, a denser mesh is preferable. At the same time, one needs to keep an eye on the convergence of results, as further refinement beyond a satisfactory convergence can be a redundant exercise. Based on the most common bridge configurations of the survey results, a representative series of parametric bridge models was formulated. The configuration parameters for the parametric bridge models are summarized in Table 2-2. This encompasses a combination of 60 models, having 12 basic prototypes, each with 5 different skew angles (Figure 2-2). The modeling and linear finite-element analyses are carried out using the program ANSYS 5.6, taking advantage of the elements that are capable of incorporating the reinforced-concrete behavior. Considering the relatively large number of bridge models to be generated and analyzed, a structured program code (Appendix B) was developed and used to automatically model the geometry and loading conditions, and also run the analyses, instead of the otherwise convenient graphical user interface. The program code is structured to entail the appropriate dimensions of the bridge as the key variables for creating a new geometry. This enables the user to run a multitude of analysis models without investing a considerable time in modeling with the graphical user interface. Furthermore, due to the skew, the model dimensions need the x and y components to be included in the program code. Figure 2-1 details the solid model of the bridge with the relevant configuration parameters defined in the finite-element analysis of this study. 2-3.

24 The preliminary stage intended to evaluate the model and identify the most appropriate boundary and loading conditions. Each model was analyzed for various load cases in an attempt to identify the most critical and representative loading configuration. Since the intent of this study was to evaluate the comparative relationship between two-dimensional and three-dimensional analysis results, the load regime was kept linear and elastic within the service range. Using linear isotropic solid structural elements, three-dimensional finite-element models were developed for each parametric bridge structure. In general, the modular bridge units are set in place by a crane while the abutment walls rest on cast-in-place concrete strip footings and grouted in a full keyway. This type footing restraint being considered as partially fixed, both pinned and fixed supports were simulated separately at the lower edges of the abutment walls. The preliminary analysis intended to evaluate the extent of the effect of I both of these restraint options on the structural response. Another special feature is to consider the edge conditions. These bridges are constructed by placing several modular units adjacent to one another. The edges of two adjacent units are connected by one of several types of joints. Typical connections use either interlocking edges with compressed sealant, cover strip with low-pressure foam filler sealant, cast in place post-tensioning ducts, or keyway edges with grout (some with longitudinal reinforcement). These methods rely upon the bond of the sealant or grout to the concrete and the integrity of the former to transfer loads from one modular unit to another. Using available methods, the side edges of the deck are structurally connected to the adjacent modular units with a detail that does not include continuous reinforcement in direction perpendicular to the connected edge. Since no significant load transfer was anticipated from this type of connection, the boundary conditions at the sides of the deck were considered as free in the preliminary model. This behavior was later confirmed in the experimental field test. 2-4

25 The structure is typically backfilled using a well-graded granular soil material, compacted according to the specifications. Upon loading the structure, the deck deflects and the abutment walls tend to flex outward, mobilizing the passive resistance of the backfill soil. A comprehensive performance assessment should therefore incorporate the soil-structure interaction characteristics of the bridge. This phenomenon was modeled at this stage by linear translational springs with stiffnesses representing the passive soil compression. The preliminary conceptualization and the experimental field test intended to evaluate the extent of the soil-structure interaction. 2.4 MODEL EVALUATION The parametric models were analyzed to characterize a typical bridge structure and identify its critical responses. Material properties for use as input in the finite-element model were obtained from design specifications. To simulate a common truck load on the structure, both finite-element and frame models were analyzed under various loading conditions. The AASHTO Standard's (1992) contact area for a design truck wheel was used to load the bridge models to enable comparison with the two-dimensional frame method. The convergence of the finite-element analysis results was determined by running a series of mesh refinements for the same model geometry, loads, restraints, and material properties. The meshing of the model was refined until optimized convergence was achieved (Niday 1997). To reconcile the local singularity at the 90" comer between the deck and the abutment walls, a model with fillet elements along the comers was developed and compared to the convergence models without the fillet comers. This provided a further evaluation of the optimized mesh density. 2-5.

26 Table 2-1. Range of Configuration Parameters of Surveyed Bridges Configuration Parameter Range of Dimensions Span length Height Width Thickness Skew angle m (5-45 ft) m (4-20 ft) m (2-10 ft) m (6-36 in.) 0"-61"

27 Table 2-2. Configuration Parameters for The Parametric Bridge Models Configuration Parameter Span length (1) Discrete Dimensions Modeled (2) 4.57 m (15 ft) 6.86 m (22.5 ft) 9.14 m (30 ft) Height Width 1.52 m (5 ft) 1.22 m (4 ft) 3.05 m (10 ft) 1.83 m (6 ft)

28 Q cu 0

29 6 W 0 F I W m I m

30 3. EXPERIMENTAL FIELD TEST 3.1 TEST BRIDGE An experimental load test was performed, to evaluate the field performance of a representative short-span, skewed bridge and also calibrate the finite-element model. The full-scale load test involved a typical modular bridge selected from available local bridge inventory to represent the range of configuration parameters identified from the survey responses (Table 2-1). Bridge design data, drawings, and specifications were obtained for an initial finite-element modeling and analysis of the bridge structure prior to field testing. The two-unit test bridge comprised a 4.83 m (15 5/6 ft) long span, 1.5 m (5 ft) width, 1.5 m (5 ft) height, 300 mm (12 in.) uniform thickness for the deck and the abutment walls, and 15' skew angle. 3.2 TEST SETUP The test bridge was instrumented to measure displacements at various locations to determine the global response, load distribution characteristics, degree of integration of the abutment walls subjected to passive soil pressure, and load transfer between adjacent modular units. The bridge was subjected to vertical, quasi-static, controlled, and gradually increasing truck loads, considering various load cases. A single-axle truck with a Gross Vehicle Weight Rating of -200 kn (44 kip) was selected to simulate AASHTO Standard S (1992) HS design truck series of vehicular loading 3-1

31 (Figure 3-1). The load truck was positioned in twelve load cases, to provide the most probable critical loading patterns, given the geometry of the bridge. 3.3 LOAD TEST After the sensors were zeroed for reference, the first load case was applied. Once the load stabilized (3 to 4 minutes), steady-state measurements were recorded. The position of the truck was then modified for the next load case and repeated until all twelve of the load cases were reached. Then, the truck load was removed and the test ended. Later, the tested bridge unit was inspected for its post-test condition. The testing procedure was completed in one day. The instrumentation readings during each, load case verified the proper functioning of the sensors. The recorded responses were checked against engineering logic and correlated with their analytically estimated counterparts for safety. Since the load cases were designed to apply gradually increasing loads to the test bridge, a verification of acceptable structural response at each load step was mandatory before proceeding to the next. Measurement consistency was ensured by correlating symmetrical sensor locations. 3.4 TEST RESULTS In accordance with the experimental data obtained during the field performance assessment, the analytical model could be calibrated to more accurately simulate the behavior of the bridge. 3-2

32 Based on the experimental field test and the preliminary conceptualization analyses, the following findings were concluded and consequently considered in the subsequent finite-element analysis of the parametric models. The possible fixed or pinned restraint boundary conditions at the grouted footing keyway in the lower edge of the abutment walls have insignificant effect on the response of the bridge. The general behavior is essentially controlled by the skew phenomenon. Therefore, a pin support condition at the bridge foundations deemed appropriate for the parametric analysis models. To model this condition, the nodes running along the centerline of the abutment wall thickness were restrained in all coordinate directions. The added stiffness of adjacent modular bridge units did neither significantly reduce the deflections of the loaded test unit nor added to its resistance. Because the modular bridge units were structurally connected with a non-continuous connection detail, no significant load transfer was neither expected nor observed to the adjacent units. For the type of backfill material used at the test site, i.e., a well-graded granular soil that is typical of most such installations, the test revealed that the backfill does not significantly reduce deflections nor add to the resistance of the bridge. It may be observed from the results that the passive backfill pressure contributes some 6% change in the distribution of moments under normal operating loads. The preliminary finite-element model that incorporated modeling of the passive soil compression also predicted the field performance analytically. In conclusion, the backfill restraint effects due to the deflection-based passive soil pressure profile do not significantly affect the behavior of the bridge. This correlation justified the elimination of the soil modeling from the parametric models for an efficient and still reliable performance assessment of the bridge structure. 3-3

33 8 s ccr 0

34 4. PRELIMINARY FINITE-ELEMENT ANALYSIS 4.1 BASIC PARAMETRIC MODELS Based on the results of the structural model conceptualization and field test, the finiteelement model was calibrated to provide a more realistic simulation. Subsequently, an error estimation analysis was performed on representative parametric models to evaluate the meshing density. Many parametric studies were needed for this project with several hundreds of runs to be analyzed, evaluated, and compared. The model was eventually optimized and simplified for adequate efficiency, considering reasonable analysis effort versus relevant accuracy of the solutions. Engineering judgment was exercised for evaluating the mathematical accuracy. To present a comparative study between the finite-element and the simple frame results, the preliminary basic parametric models did not include the haunches between the abutment walls and the deck, since the survey identified that the haunches are not typically incorporated in common practical design calculations. A series of parametric studies was then conducted by varying the associated geometric properties of the prototype bridge models and performing their respective finite-element analyses. To enable a comparative study among the various parametric models and their corresponding frame models, only two simple load cases applied at key points on the bridge were considered. The first load case represents a symmetrically positioned concentrated vertical load, as laterally centered at the mid-span of the bridge. The second load case intended to capture the additional torsional effects and evaluate the stress amplification due to a laterally unsymmetrical concentrated vertical load at the 4-1,-A.

35 edge of the deck. The study employs a unit load application in both load cases. The comparison of analysis results between the three-dimensional finite-element models and the plane frame models is further simplified by incorporating unit loads in the analysis. This approach also offers the advantage of enabling the development of correlation diagrams and their usage by easier modification of bridge design moments. To avoid singularity problems, the unit load was applied as a distributed pressure load over the surface area of the respective finite-elements at the designated load case positions, as shown in Figure 4-1. The analysis results of each parametric model were individually examined to study the global behavior of the bridge system and obtain practical observations. The finite-element results were successively entered into the correlation diagrams and compared with the twodimensional frame analysis results. The finite-element modeling of the bridge configuration used three-dimensional, solid, 20- node elements (Solid 95) with three degrees of freedom per node, which are the translations in three coordinate directions, and linear orthotropic material properties. This element has stress stiffening, large deflection, and large strain capabilities, to model reinforced-concrete behavior more accurately. The following material properties were assigned for the reinforced-concrete elements of the finiteelement models, based on the survey s most common characteristics: Modulus of elasticity of concrete, Ec = Pa (4,000 ksi). Poisson s ratio of concrete, = 0.2 (per AASHTO Standard 1992). Unit weight of concrete, kg/m3 (145 pcf). 4-2

36 4.2 ANALYSIS RESULTS The principal parameters investigated in this study are longitudinal stresses for the positive and negative bending moments, and the maximum deflection along with maximum lateral stresses at mid-span and supports. These are compared with those obtained from the simplified twodimensional frame analysis to address the question of adequacy of three-dimensional analysis from the perspective of conventional design practice. The results of the parametric analyses are summarized in Figures 4-2 through 4-9. The negative stresses are defined as stresses developing at the location of the negative moment (tension on top of the cross section), close to the abutment walls. The positive stresses are defined as stresses developing at the location of the positive moment (tension on bottom of the cross section), around mid-span. These diagrams depict the stress ratios between the three-dimensional finite-element method and the two-dimensional portal frame method versus the skew angle of the bridge. Therefore, the line representing the unit stress ratio describes the value where the results of the two methods coincide. The three individual curves exhibit the results from three different clear spans, respectively 4.57,6.86, and 9.14 m (15,22.5, and 30 ft), for which the parametric models were analyzed. Each curve displays the values calculated for a certain parametric model of the bridge with various skew angles (O", 15", 30, 45O, and 60"). Any curve or part of a curve positioned above the unit stress ratio line reveals that the bridge is experiencing stresses higher than calculated by the simple frame method, and therefore is under-designed. Any curve or part of a curve positioned below the unit stress ratio line exhibits that the bridge is over-designed. The results of the finite-element to frame deflection ratios are summarized in Figures 4-10 through

37 Besides evaluating the effect of the skew on the behavior of the bridge, this study intends to come up with design modification factors. These factors enable the designer to modify the moments obtained from the conventional simplified plane frame analysis method, and thus, reflect more accurately the actual response of the bridge. Consequently, the above analysis results are formulated and developed into practical correlation diagrams of the stress ratios between a three-dimensional finite-element model and a two-dimensional frame model. Accordingly, structural engineers will be able to adjust the plane frame analysis and compensate for the three-dimensional stress effects of the skew angle and laterally unsymmetrical vertical loading. The adjustment is achieved by multiplying the plane frame design results with the corresponding value of the stress ratio in the correlation diagrams, which in effect serve as magnification factors that enable the usage of the simple frame design method.

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40 2 I -&- Clear Span = 15 ft 1 v) v) F 5 a E s LL Skew Angle (deg.) FIG Finite-Element to Frame Stress Ratios, B = 4 ft, H = 5 ft, Negative Moment.- 0 ti [r 1.2 +Clear Span = 15 ft +Clear Span = 22.5 ft Clear Span = 30 ft ' Skew Angle (deg.) FIG Finite-Element to Frame Stress Ratios, B = 4 ft, H = 5 ft, Positive Moment

41 E v) G t 1.4 E 1.2 \ k i- lu 1.2 n fn fn 2 1 G Q) E!! 0.8 % LL 0.6 I I I I I 0.4!, I Skew Angle (deg.) FIG Finite-Element to Frame Stress Ratios, B = 6 ft, H = 10 ft, Positive Moment

42 oc u) cn g 1.4 z 1.2 5LL Skew Angle (deg.) FIG Finite-Element to Frame Stress Ratios, B = 6 ft, H = 5 ft, Negative Moment 1.4 '= a [r u) u) Skew Angle (deg.) FIG Finite-Element to Frame Stress Ratios, B = 6 ft, H = 5 ft, Positive Moment

43 1.1 - I d- Clear Span = 15 ft -0- Clear Span = 22.5 ft.- 0 c K.- 0 c = a, 0.7 E Skew Angle (deg.) FIG Finite-Element to Frame Deflection Ratios, B = 4 ft, H = 10 ft I d- Clear Span = 15 ft +Clear Span = 22.5 ft = Skew Angle (deg.) FIG Finite-Element to Frame Deflection Ratios, B =4ft, H =5ft

44 I d- Clear Span = 15 ft 1 6! 0.8 C 0 a % 0.6 n E 0.4 Clear Span = 22.5 ft Skew Angle (deg.) FIG Finite-Element to Frame Deflection Ratios, B = 6 ft, H = 10 ft D- Clear Span = 22.5 ft + Clear Span = 30 ft Skew Angle (deg.) FIG Finite-Element to Frame Deflection Ratios, B = 6ft, H = 5 ft

45 5. FINITE-ELEMENT ANALYSIS 5.1 GENERAL As stated earlier, the conventional design approach fails to account for the effect of skew making it unsafe in many instances. This is corroborated by the fact that a multitude of skewed bridges, designed ignoring the effect of skew, have shown deterioration. Often, the cracks are observed at the abutment wall-bridge deck junction. In general, haunches are provided at these comers to enhance the stiffness of the structure. Figure 5-1 shows a modular skewed bridge with integral abutment walls and haunches, and details the relevant configuration parameters defined in this study. The haunch is supposed to enhance the stiffness of the support and provide enough stability against potential structural problems. According to the conducted national survey, the typical haunch configuration consists of 17 inch by 17 inch (0.43 m by 0.43 m) in height and length, respectively. Other dimensions are also employed depending upon the availability of precast units for the construction. The principal objective of this phase of the project is to investigate the influence of haunches on the overall structural behavior, utilizing rigorous three-dimensional finite-element analysis. The relevant background along with the specific objectives of the project and scope of work are delineated first. Then, the preliminary finite-element modeling, consisting of boundary conditions and loading applications are presented. Finite-element mesh refinement and parametric studies will follow, signifying the influence of haunches on the skewed structures of different spadrise configurations along with statistical analyses. 5-1.

46 5.2SCOPE This phase of the current research study will focus on the effect of haunches on the response of precast-concrete, short-span, skewed bridges with integral wall abutments. The study aims at examining the effect of haunches in terms of whether a haunch provides enough stiffness to the joint so as to alleviate complex stress condition in the skewed bridges, and hence justifies its provision to account for the unknown skew effects. As in the previous chapter, design magnification factors are developed to help engineers modify the design moments obtained from two-dimensional frame analysis. This will eliminate the need to employ more expensive and elaborate analysis tools like three-dimensional finite-element method. The actual finite-element analysis is based on the preceding models discussed in the previous chapters of this manuscript. Considering that it was the load case with the load at the edge of the bridge deck at the mid-span that yielded more critical stresses than the load case with the same load at the center, loading is applied only at the edge. Also, the skew angle of 60 degrees will not be included in the modeling since its use is not common practice. It is proposed to use the same angle only for the verification of the trend in the results. The 0-degree skew is included for the reference as this case is closest to the two-dimensional frame analysis. The study begins with preliminary finite-element analysis runs, utilizing the same mesh density generated previously. The purpose of the preliminary runs is to determine whether the mesh density needs refinement in order to obtain a better convergence of results and to observe a general pattern for the bridge behavior. The next few stages in the study include mesh refinement and the correspondmg analysis runs and data generation. Correlation diagrams are plotted for all configurations. The ratios of positive 5-2

47 and negative moment stresses from finite-element analyses to those from the frame analyses are plotted versus the skew angles. Maximum deflections from the two methods are compared on the same graphs. When a satisfactory convergence for the results from the mesh refinements is established, statistical analyses of the data obtained is carried out. Statistical study is particularly useful due to the large number of models analyzed generating a considerable volume of data. The software SAS is employed to execute the statistical analyses. SAS is a useful software that offers an insight into how different parameters including the span, rise, skew angle, etc. or variables affect a measured response such as the stress, strain, and deflection. For example, one is able to verify statistically whether the impact of width values, incorporated in the study, are significantly different to affect the response quantity. Also the software can determine if some of the variables interact (for example span and skew) to affect the stresses, which can explain any inconsistencies in the results. 5.3 PARAMETRIC CHARACTERISTICS The dimensions required to develop the finite-element model of this stage include the bridge clear span, height (rise), width, skew angle, and haunch configuration. As mentioned previously, the typical haunch configuration consists of 17 inch by 17 inch. The other relevant bridge dimensions used for the finite-element modeling were kept the same as in the previous stage. 5-3

48 The above array of parameters, produce a total of 60 different combinations of models to be run. The smallest aspect ratio considered herein is 2.5 and the largest is 7.5. The zero-skewed configuration is used in the analysis for reference only. 5.4 ANALYSIS LEVELS Finite-element analysis often requires mesh refinement for the convergence of the results. Mesh refinement means redefining the nodes and elements and more analysis time required. In view of the large number of models to be run in this study, it was decided to start the analysis with a preliminary finite-element model. This part of the study will use a relatively coarse mesh and, therefore, provide the extent to which mesh refinement is needed. Accordingly, the mesh refinement and further analyses runs and data collection would be undertaken Preliminary Analysis The preliminary runs were designed to provide the basis of the investigation of the structural behavior of the bridge models. As this was a preliminary run, one can have liberty of using a relatively coarse mesh. The preliminary model with haunches used 245 elements. All elements, except the ones at haunches, were quadrangular. The haunch was modeled using triangular wedge shaped elements. Figure 5-2 shows the preliminary finite-element model with haunches. Tables 5-1 through 5-12 include the preliminary analysis results. Tabulated here are the positive and negative moment stresses (longitudinal), lateral stresses, and the maximum deflection, from the finite-element and frame analyses. The finite-elemendframe ratios are also computed. Note

49 that the two-dimensional frame analysis is not capable of computing the lateral direction stresses. Figures 5-3 through 5-10 include the correlation diagrams for longitudinal stresses. Positive stresses are for maximum positive flexural moment at mid-span. Negative stresses are for maximum negative flexural moments in or near the haunch. Figures 5-11 through 5-22 are the correlation diagrams for maximum deflection. The preliminary finite-element analysis yields quite a steady pattern for positive moment stresses. For all model configurations the stress decreases with the increasing skew angle. The correlation diagrams show that for B4HlOS (15,22.5,30) class actual three-dimensional stresses are less than the two-dimensional stresses. The same is true for the B4H5S (15,22.5,30) class. For the B6HlOS15 model, the finite-element stresses are up to 1518% more than the plane frame stresses for a straight bridge without skew. The difference between the stresses however narrows down as the skew angle increases. The 45-degree skew models for all models of B6HlO category show less actual stresses than given by frame analysis. The same pattern is observed for the other class with B6H5 configuration. It is the negative moment stresses where the preliminary analysis does not offer definitive results. The correlation diagrams exhibit an inconsistent trend. For example, the model B6HlOS15 with 45-degree skew shows an increase in the stress from the 30-degree model whereas all other model with B = 6 ft and S = 15 ft show decrease in the stress for the 45-degree model. Configuration B4HlOS15 shows an increase in stress from 0" to 15" and again a decrease for 30" and 45" skews. Overall, the inconsistent nature of the results makes a case for the further refinement of the mesh. The frame analysis yields the maximum deflection that is same for all skew angles for a particular configuration. The finite-element analysis on the other hand gives deflection values for all configurations and skew angles. The results show that for majority of cases the frame analysis 5-5

50 overestimates the deflections. Only exception is the B6HlOS22.5 configuration, which has finite- element deflections greater than the frame analyses deflections. This, to be confirmed by the further refinement means a design based on the frame stresses might produce an over safe structure First Mesh Refinement The inconsistencies in the results from the preliminary finite-element analysis (especially in the stresses for the negative bending moments) meant a refinement was needed. Though mesh refinement means an increase in the number of the elements, one should be mindful of the fact that a refined model has to be inclusive of the previous model for the refinement will not be correct otherwise in the mathematical sense. To refine the mesh the generic code was modified with new node and element positions and designations. The refined mesh would have a layered look, that is the bridge slab and the abutment walls will have two layers of elements. The refined mesh ended up with 1380 elements. Like the preliminary mesh the refined one had quadrangular elements except for those modeling the haunches, which had, wedge shaped elements. Figure 5-23 shows the first refined finite-element model at this stage. Tables 5-13 through 5-24 contain the results from the analyses runs with first refinement. Figures 5-24 through 5-43 represent the correlation diagrams. Most models show less stress than given by two-dimensional frame analysis. It is only the model configurations B6H10S15 and B6H5S 15 (all skew angles) that show approximately 20% higher stresses than given by the frame analysis. Correlation diagrams show a similar pattern to that obtained in the preliminary runs, indicating that the mesh is good enough to compute the positive moment stresses accurately. For each configuration of span, width, and height, the stresses decrease smoothly from 0-degree to

51 degree skew. The finite-elemendframe stress ratio for all models with width 4 ft is less than 1, which indicates that the frame analysis overestimates the stress. On the other hand all models with B = 6 ft show the above mentioned ratio to be more than 1 which means the frame analysis underestimates the stress for these models. The actual stresses as given by the finite-element analyses are approximately 20% higher than those given by the frame analyses. The refined mesh does yield a more definitive stress pattern as seen in the correlation diagrams for the same. At the same time the results are not consistent enough to shed light on the actual behavior of the structure. For example, the model B6HlOS15 with the 45-degree skew now shows an increase in the stress from the 30-degree model unlike in the preliminary runs. Similarly the model B6H5S15 with 45-degree skew shows a steep increase in the stress from the 30-degree model unlike in the preliminary run when it showed a sharp decrease. Overall there is an increase in the stress values from the preliminary runs. For each configuration of the span, width, and height, the stresses increase or decrease without a consistent pattern from 0-degree to 45-degree skew. For most models, the finite-element maximum deflection is less than the corresponding frame deflection. For configuration B6HlOS22.5, the deflections of the finite-element analysis are more than those from the frame analysis. The results support the conclusion of the preliminary finiteelement analysis that overall the two-dimensional approach overestimates the deflections Final Analysis The second refinement became necessary after the evaluation of the results of the first refinement runs. The overall increase in stress magnitudes and the inconclusive nature of the negative moment stresses were the two areas the second refinement tried to iron out. The main feature of this refinement was the uniform layered modeling of the whole structure. 5-7.

52 Previous refinement did not have layered element arrangement for the bridge slab. In the second refinement the bridge model has 4 layers of elements for the walls and the slab in the vertical plane along the span and 20 layers in the vertical plane along the width of the bridge. Figure 5-44 shows the final refined model with haunches. The mesh refinement ended up with 10,880 elements, which is a considerably large number of elements. Thus, each run in ANSYS now took a significantly long time. As a consequence, the run time taken by the final refinement models turned out to be approximately 17 to 18 times as much as the previous refinement. 5.5 ANALYSIS RESULTS The effect of haunches on the structural behavior of skewed bridges can be evaluated from the correlation diagrams and the stress contour plots obtained from the finite-element analyses. The stresses and maximum deflection showed a considerable increase in the magnitudes with the refinement of the mesh. Overall, the stress plots obtained from the finite-element analyses exhibit consistently higher stress values than those obtained from the frame analyses. Tables 5-25 through 5-36 provide results from the finite-element analyses. Figures 5-45 through 5-8 show the correlation diagrams for the longitudinal stresses whereas Figures 5-9 through 5-20 present correlation diagrams for maximum deflection. Compendium of stresses obtained from all finite-element refinements, along with those from two-dimensional frame analyses, are tabulated in Tables 5-37 and Following is the detailed discussion of the results from the finite-element analyses. 5-8.

53 5.5.1 Effect of Haunches A haunch is expected to stiffen the wall-slab joint of the bridge, which means it will not allow as much moment carry-over to the nearby joints. Thus one expected to find maximum stresses at the haunch region. Sure enough the maximum stress for negative moment in all models with the skew angle greater than 0" occurred in the haunch region at the obtuse corner of the bridge. The negative moment stresses are the yardstick to evaluate the haunch influence in the bridge behavior. The most appropriate method to study whether the haunch can provide enough stiffness so as to account for the unknown skew effects if the bridge were to be analyzed by two-dimensional frame method is to compare the negative moment stresses with those in the models without the haunches Comparison with No Haunches As expected, the negative moment stresses for almost all models are greater than those without haunches. The difference is even more considerable for higher skews of 45" and 60". The positive moment stresses are also higher as compared with the stresses in the previous study. It should be noted that a relatively coarse mesh was employed in the precedmg study; therefore, the actual stresses may be somewhat different from what they appear to be. Nonetheless, the comparison confirms the overall increase in stresses and the maximum deflections with the addition of haunches Comparison with Plane Frames The most important aspect of this study is to evaluate whether the provision of the haunches at the abutment wall-slab joint negates any need for knowing the effect of skew not captured by the two-d~mensional frame analysis. The ratios of stresses from the finite-element analyses to those from 579.

54 the frame analyses help us resolve this issue. Following is the summary of longitudinal stresses for the negative and the positive moments. Negative Moment Stresses B4HlOS15: The negative moment stress increases with the increase in the skew angle. The stress for the 0"skew angle, which is the reference model, is nearly equal to the corresponding stress given by the frame analysis, which confirms the adequacy of the mesh density. For all the other skew angles finite-element analyses yield considerably higher stresses than those from the frame analyses. Especially with the skew angle of 60" the finite-element to frame stress ratio is as high as 2.7. B4HlOS22.5: like the earlier configuration the stresses increase with the skew angle with the stress for the 60" skew almost 11 1% greater than that from the frame analysis. B4HlOS30: Here the pattern of increase in the stress with the skew angle is similar to the above configurations, except the stress in the 60" model for the configuration deviates by 70% from the frame stress, which is less than the deviation observed in the above configurations. B4H5S15: Stresses are higher in the range of 57 to 138% compared to the frame stresses. B4H5S22.5: The stress for the 0" skew angle is almost equal to the one from the two-dimensional analysis, whereas the stress for the highest skew of 60" is almost twice as much as the corresponding frame stress. 5-10

55 B4H5S30: Like the configuration B4HlOS30 this one shows least overall deviation from the results of frame analyses. The maximum deviation is approximately 69% observed for the 60" skew angle. B6HlOS 15: This configuration yields high finite-element to frame stress ratios for all skew angles with the 45", and 60" skew stresses approximately 180% higher than the frame stresses. B6HlOS22.5: The stresses are higher 50 to 233% higher than the frame stresses B6HlOS30: Though this is the maximum span with other parameters unchanged the stresses do not deviate from the frame stresses as much with the increase in the skew angle. The maximum deviation is approximately 87% for 60" skew angle. B6H5S 15: this configuration perhaps shows the greatest difference observed between the finite- element and frame stresses. The finite-element stresses are higher in the range of 40 to 257%. B6H5S22.5: Stresses increase considerably from one skew angle to the next one. But the overall deviation from the frame stresses is less than observed in the 15 ft span above. B6H5S30: As observed before, the stresses though increase as the result of the increase in the span, the finite-element to frame stress ratio is not as high. With all other parameters unchanged except the span, stress for each skew angle is less than that observed in the smaller spans

56 Positive Moment Stresses For all models analyzed the maximum positive moment stresses were observed at the area of load application, that is the edge of the bridge at the mid-span. In general the positive moment stresses increase with the increase in the skew angle. Results for the positive moment stresses for the configurations used are as follows, B4HlOS 15: The stresses are consistently higher than given by the frame analyses. The stresses for the skew angles from 0" to 30" are approximately equal in magnitudes, and 34% greater than the corresponding frame stresses. Stresses for the skew angles 45", and 60" are 39 and 49% higher respectively, than the frame stresses. B4HlOS22.5: For this configuration, the stresses are moderately higher than the corresponding frame stresses. Models with the skew angles 0" to 45" exhibit stress magnitudes approximately equal to the frame stresses, whereas the model with skew angles 60" has the stress approximately 37% higher than the corresponding frame stress. B4HlOS30: Stresses are almost equal to those from the frame analyses of this configuration. Unlike the above-mentioned configurations, the finite-element stresses are slightly less than the frame stresses except for the skew angle of 45". B4H5S 15: This configuration yielded stresses considerably higher than the corresponding frame stresses with the 60" skew model showing a stress as much as 70% greater than the frame stress. Even the 0" skew model gave 36% higher stress than the frame stress. 5-12

57 B4H5S22.5: Positive moment stresses for 0", 15", and 30" skew angles are not significantly different from the frame stresses as indicated by the finite-elemendframe stress ratios for the models. The remaining models have stresses approximately 32% higher than the frame stresses. B4H5S30: Maximum stress for the positive moment for O", 15", and 30" models are almost equal to those from the frame analyses. The 45" model has the highest deviation of 22% from the frame stress whereas surprisingly the 60" model shows less deviation of 13% and thus a less stress value than the 45" model. B6HlOS15: Among the cases discussed so far this configuration shows the highest deviation from the corresponding frame stresses. All skew angles yield stresses 55% or more than the frame stresses. Moreover the stresses decrease although very slightly with the increasing skew angle up to 45" and again drop for the 60" case. B6HlOS22.5: Stresses are approximately equal for the 0", 15", and 30" models, and are almost 24% higher than the corresponding frame stresses. The 60" skew stress is 50% higher than the frame stress. B6HlOS30: The 0", 15", 30, and 60" skew angles have approximately same stresses by the finite- element and the frame analyses, whereas the stress for the 45" skew is 20% higher than the frame stress. 5-13

58 B6H5S15: This is the configuration that is most critical from the perspective of the difference in the stresses from the two analysis approaches. Stresses are consistently higher than 65% or more than the corresponding frame stresses, with the 60" skew producing a stress 85% greater. B6H5S22.5: As observed in some of the earlier cases the stress does not change by much with the increase in the skew from 0" to 30". The stress for these angles is 27% greater than the corresponding frame stress on an average. The 45", and 60" skew angles show much more deviation of 45 to 54% from the frame stresses. B6H5S30: This is the least critical case in the configurations with B = 6, and H = 5. Stresses are not significantly different from the frame stresses for skew angles 0" to 45". The stress in the 60" skew is I 25% higher than the frame stress. Maximum Deflection As indicated in Figures 5-53 through 5-64, the maximum deflection in almost all models is less than computed from the plane frame analyses. This means an overestimation of the deflections by the two-dimensional frame analysis. It is only in the configurations B6HlOS15 and B6HlOS22.5 that the frame analysis underestimates the maximum deflection. Lateral Stresses The second refinement yields significantly high values for the lateral stresses. The lateral stresses in the mid-span region are higher than those in the support region. The stresses are anywhere between 10 to 30% of the corresponding longitudinal stresses. 5-14

59 5.6 MAGNIFICATION FACTORS The results of the study clearly establish the inadequacy of the conventional two-dimensional analysis in the design of the precast-concrete, short-span, skewed bridges with integral wall abutments and haunches. The correlation diagrams in Figures 5-45 through 5-52 help the designer to convert the moment from the two-dimensional frame analysis to the actual moment that accounts for the effect of the skew angle and the haunch. To use the design magnification diagrams, the designer first computes the design moment using a conventional two-dimensional frame analysis. Then, for the bridge dimensions and the skew angle, the designer refers to the appropriate design magnification diagram. The diagram provides the ratio of the finite-element to frame stress (finite-elemendframe) for the specific bridge configuration. This ratio serves as the design factor in the design. The moment from the two-dimensional frame analysis when multiplied by the design factor, converts to the actual moment in the bridge. The following examples illustrate the use of the design magnification factors. Examde 1 Let the bridge configuration be as follows: B = 4 ft, H = 10 ft, S = 30 ft, A = 30" Let the maximum negative moment from the two-dimensional frame analysis be 20.0 ft-kip. From Figure 5-45, the design magnification factor for the given configuration and skew angle is 1.3. Thus, the modified moment is 20.0 x 1.3 = 26.0 ft-kip 5-15.

60 Example 2 Let the bridge have the following configuration: B = 5 ft, H = 10 ft, S = 22.5 ft, A = 45" Let the maximum negative moment from the two-dimensional frame analysis be 25.0 ft-kip. In this case, the design magnification factor can be interpolated using the correlation diagrams in Figures 5-45 and Referring to the diagrams, the finite-element to frame ratio for configuration B4HlOS22.5 for the skew angle of 45" is 1.1. The ratio for the configuration B6HlOS22.5 for the same skew is 2.0. The interpolation gives the design factor of The modified design moment is 25.0 x 1.55 = ft-kip

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72 I....

73 m 8 E 3; c) B 3 cu E a VI

74 Figure 5-1. Parametric Solid Model of a Haunched Modular Skewed Bridge Figure 5-2. Preliminary Haunched Finite-Element Model with Unit Load Case

75 (d OC v) v)!?! G ca t a LL Skew Angle (deg.) + B4H1 OS1 5 + B4H1 OS22 + B4H 1 OS30 Figure 5-3. Preliminary Haunched FE/Frame Stress Ratios, B = 4 ft, H = 10 ft, Negative Moment.o I I I a a: v) v) G k! I I Skew Angle (deg.) -+ B4H B4H1 OS22 + B4H1 OS30 Figure 5-4. Preliminary Haunched F'E/Frame Stress Ratios, B = 4 ft, H = 10 ft, Positive Moment

76 ;.I E B4H5S15 -t- B4H5S22 + B4H5S Skew Angle (deg.) Figure 5-5. Preliminary Haunched FE/Frame Stress Ratios, B = 4 ft, H = 5 ft, Negative Moment f.i Q a E li ll B4H5S15 -t- B4H5S22 -m- B4H5S I Skew Angle (deg.) Figure 5-6. Preliminary Haunched FE/Frame Stress Ratios, B = 4 ft, H = 5 ft, Positive Moment

77 c g E Q a, a E LL I 1 I I I I Skew Angle (deg.) -+- B6H1 OS22 Figure 5-7. Preliminary Haunched FE/Frame Stress Ratios, B = 6 ft, H = 10 ft, Negative Moment ooo I I I I I I Skew Angle (deg.) -e--- B6H1 OS1 5 -A- B6H 1 OS22 --a- B6H1 OS30 Figure 5-8. Preliminary Haunched FE/Frame Stress Ratios, B = 6 ft, H = 10 ft, Positive Moment

78 # v) v)? El L e- B6H5S15 -A- -m- B6H5S22 B6H5S ! I I Skew Angle (deg.) Figure 5-9. Preliminary Haunched FE/Frame Stress Ratios, B = 6 ft, H = 5 ft, Negative Moment c v) v) Q t- B6H5S15 -t- B6H5S22 -D- B6H5S Skew Angle (deg.) Figure Preliminary Haunched FE/Frame Stress Ratios, B = 6 ft, H = 5 ft, Positive Moment

79 2.OE E E E-08 W U) 1.2E-08 K G c 1.OE c 8.OE-09 6.OE-09 4.OE-09 2.OE-09 O.OE+OO Skew Angle (deg.) Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 15 ft 5.O E E-08 4.OE-08 E 3.5E-08 v v) 3.OE-08 K -0 c 2.5E-08 0 'c 2 2.OE E-08 1.OE-08 5.OE-09 O.OE+OO Skew Angle (deg.) Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 22.5 ft

80 1.OE-07 9.OE-08 8.OE-08 7.OE-08 E '5; 6.OE-08 c -2 w 5.OE OE-08 a, 0 3.OE-08 2.OE-08 1.OE-08 O.OE+OO Skew Angle (deg.) 60 Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 30 ft 2.OE E E-08 n 1.4E-08 E '5; 1.2E-08 c -2 c 1.OE-08 0 w- 2 8.OE-09 a, f3 6.OE-09 4.OE-09 2.OE-09 O.OE+OO Skew Angle (deg.) Figure Preliminary Haunched FE and Frame ' Deflections, B = 4 ft, H = 5 ft, CS = 15 ft

81 1.OE-07 9.OE-08 8.O E-08 7.OE-08 v V) 6.OE-08 K -0 c 5.OE-08 0 w- 2 4.OE-08 3.OE-08 2.OE-08 1.OE-08 O.OE+OO Skew Angle (deg.) Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 22.5 ft 1.OE-07 9.OE-08 8.O E-08 7.OE-08 E 6.OE-08 c -0 c 5.OE OE-08 0" 3.OE-08 2.OE-08 1.OE-08 O.OE+OO Skew Angle (deg.) Figure Preliminary Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 30 ft

82 2.OE E E E-08 W V) 1.2E-08 K.O w 1.OE-08 0 c 2 8.OE OE-09 4.OE-09 2.OE-09 0.O E+OO Skew Angle (deg.) Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 15 ft 3.OE E-08 h E 2.OE-08 Y u) K -9 c 1.5E-08 0 a, E Q, n 1.OE-08 5.OE E+OO Skew Angle (deg.) Figure Preliminary Haunched FE and Frame. Deflections, B = 6 ft, H = 10 ft, CS = 22.5 ft

83 n v fn 6.OE-08 5.OE-08 E 4.OE-08 C -2 c 3.OE-08 0 a 5 2.OE-08 n O.OE+OO l.0e-08 ' Skew Angle (deg.) Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 30 ft 1.OE-08 I I I I I I i 9.OE-09 8.OE-09 F 7.OE-09 v L fn 6.OE-09 C 5.OE OE-09 0.c 6 3.OE-09 2.O E-09 1.OE-09 O.OE+OO I

84 3.OE E-08 h Y E 2.OE-08 u) c.o CI 1.5E-08 0 Q) E Q) 1.OE-08 n 5.OE-09 O.OE+OO Skew Angle (deg.) Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 22.5 ft 5.OE E-08 4.OE E-08 Y u) 3.OE-08 C -0 c. 2.5E OE-08 d 1.5E-08 1.OE-08 5.OE-09 0.O E+OO Skew Angle (deg.) Figure Preliminary Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 30 ft

85 Figure First Refinement Haunched Finite-Element Model

86 1.400 I I i Q) E LL & B4HlOSl5 --t B4H B4H Skew Angle (deg.) figure first Haunched FElFrame Stress Ratios, ooo I I Skew Angle (deg.) +B4HlOS15 -+ B4H10S22 -m- B4H figure first Haunched Stress Ratios, ame

87 LT : tj I 1 i -4- B4H5S15 -g- B4H5S22 -is-- B4H5S30 I I I Skew Angle (deg.) figure first Haunched RlFrame Stress Ratios, J a LT v) v) 2 5j a, $ 0.400?il ll e- B4H5S15 -A- B4H5S22 +e B4H5S Skew Angle (deg.) figure first Haunched AlFrame Stress Ratios,

88 ~ ~..'-I fn E $ ~ I i ~ I /"" ; I,/ i ~, 1 "&.- **: / J 1 -y" j & 1-1 )"./ ' 1 1 I 1 --A r U g fj a E % u B6H1 OS Skew Angle (deg.) Figure First Haunched FE/Frame Stress Ratios,

89 i?i (r : fj E $ LL I I I I I I Skew Angle (deg.) figure first aunched m ame Stress Ratios, -A- -m- B6H5S22 B6H5S ).- 0 i K Z I _ * E $ L i : I I I I I I Skew Angle (deg.) --e B6H5S15 -t- B6H5S22 B6H5S30 figure First bunched FElFrarne Stress Ratios,

90 2.O E E E E-08 v UJ 1.2E-08 C 2 1.OE-08 i, 2.c 8.OE-09 E 6.OE-09 4.OE-09 2.OE E+OO Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 15 ft h E v 3.5E-08 3.OE E-08 2.O E E-08 1.OE-08 5.OE-09

91 1.OE-07 9.OE-08 8.OE-08 7.OE-08 W v) 6.OE-08 c c 5.OE-08 0 w- A? 4.OE-08 0" 3.OE-08 2.OE-08 1.OE-08 O.OE+OO Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 30 ft 2.OE E E E-08 v v) 1.2E-08 c -2 c 1.OE OE-09 0" 6.OE-09 4.OE-09 2.OE-09 O.OE+OO Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 15 ft

92 1.OE-07 9.OE-08 8.OE-08 T 7.OE-08 v 6.OE-08 -.O c 5.OE-08 0 U 4.OE-08 3.OE-08 2.O E-08 1.OE-08 O.OE+OO Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 22.5 ft 1.OE-07 9.OE-08 8.O E OE-08 v V) 6.OE-08 C -0 c 5.OE OE-08 d 3.OE-08 2.OE-08 1.OE-08 O.OE+OO Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 30 ft

93 2.OE E E E-08 v V) 1.2E-08 c -2 c 1.OE OE-09 d 6.OE-09 4.OE-09 2.OE-09 O.OE+OO Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 15 ft 3.OE E-08 h Y E 2.OE-08 u) c c 1.5E-08 0 a, % 1.OE-08 n O.OE+OO 5*0E-09 ' Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 22.5 ft

94 6.OE-08 5.OE-08 h v E 4.OE-08 v) c.o c 3.OE-08 u Q, 5 2.OE-08 n 1.OE-08 O.OE+OO Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 30 ft 1.OE-08 9.OE-09 8.O E-09 7.OE-09 v 6.OE-09-0 c 5.OE OE-09 d 3.OE-09 2.OE-09 1.OE-09 O.OE+OO -1 I I 4 I Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 15 ft I

95 3.OE-08 : 2.5E-08 h v E 2.OE-08 t fn C c 1.5E-08 i-*yissr 0 al 1.OE-08 n 5.OE Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 22.5 ft ~12;; 1 4.5E-08!j.OE-O8 L J 4.OE E-08 v 3.OE-08.z c 2.5E OE-08 d 1.5E-08 1.OE-08 5.OE-09 O.OE+OO! I I 1 I I I Skew Angle (deg.) Figure First Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 30 ft

96 Figure Final Refinement Haunched Finite-Element Model

97 2.500 I I,.- 0 iii U v) Gj E % LI -+ B4H B4H 1 OS30 Figure Final Haunched FE/Frame Stress Ratios, B = 4 ft, H = 10 ft, Negative Moment o c la v) v) iij a LL Skew Angle (deg.) -f- B4H B4H B4H 1 OS30 Figure Final Haunched FE/Frame Stress Ratios, B = 4 ft, H = 10 ft, Positive Moment

98 2.500 s a U v) L Zi u B4H5S15 -t- B4H5S22 -c- B4H5S Skew Angle (deg.) Figure Final Haunched FERrame Stress B = 4 ft, H = 5 ft, Negative Moment Ratios, I I I L fj a, LL ~ e- B4H5S15 -A- B4H5S22 -c- B4H5S Skew Angle (deg.) Figure Final Haunched FERrame Stress Ratios, B = 4 ft, H = 5 ft, Positive Moment

99 c g a, t U i3 L B6H F- B6H B6H1 OS Skew Angle (deg.) Figure Final Haunched FE/Frame Stress Ratios, B = 6 ft, H = 10 ft, Negative Moment U g g v) a, E $j LL Skew Angle (deg.) -Q-- B6H Figure Final Haunched FE/Frame Stress Ratios, B = 6 ft, H = 10 ft, Positive Moment

100 = a U v) G L E LL a LL ~ -e- B6H5S15 -A- B6H5S22 -a- B6H5S Skew Angle (deg.) Figure Final Haunched FE/Frame Stress Ratios, B = 6 ft, H = 5 ft, Negative Moment I ti U g cn Q) E * 1 I + I 7 1 T 1 $i LL Skew Angle (deg.) Figure Final Haunched FE/Frame Stress Ratios, B = 6 ft, H = 5 ft, Positive Moment + B6H5S B6H5S22 -m- B6H5S30

101 2.OE E E E-08 v cn 1.2E-08 c.o c 1.OE-08 0 w- 2 8.OE-09 d 6.OE-09 O.OE+OO! I I I I Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 15 ft 5.OE E-08 4.OE E-08 v 3.OE-08-0 c 2.5E-08 0 w- U 2.OE-08 0" 1.5E-08 1.OE-08 5 O.OE+OO 5.0E Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS =22.5 ft

102 1.OE-07 9.OE-08 8.OE-08 7.OE-08 v rn 6.OE-08 c.p 4-5.OE-08 0 v- U 4.OE OE-08 2.OE-08 1.OE-08 O.OE+OO Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 10 ft, CS = 30 ft 2.O E E E-08 E 1.4E-08 Y rn 1.2E-08 c -S *-. 1.OE-08 0 U w- 8.OE OE-09 4.OE-09 2.OE-09 O.OE+OO Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 15 ft

103 1.OE-07 I I I I I I I 9.OE-08 8.OE-08 7.OE-08 v cn 6.OE-08 c.o I 5.OE-08 t, U -4-4.OE OE-08 2.OE-08 1.OE Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 22.5 ft 1.OE-07 9.OE-08 8.OE-08 7.OE-08 v cn 6.OE-08 c.p c 5.OE-08 0 U w- 4.OE OE-08 2.OE-08 1.OE-08 O.OE+OO Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 4 ft, H = 5 ft, CS = 30 ft

104 2.OE E E E-08 v V) 1.2E-08, K.O c 1.OE OE-09 6.OE-09 4.OE-09 2.OE E+OO Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 15 ft 3.OE E-08 h Y E 2.OE-08 v) c -P c 1.5E-08 0 a 5 1.OE-08 n L 05*0E-09.O E+OO Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 22.5 ft

105 h 6.OE-08 5.OE-08 v E 4.OE-08 rn C.O c 3.OE a 3 2.OE-08 n - O.OE+OO l.oe-o Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 10 ft, CS = 30 ft 1.OE-08 9.OE-09 8.OE-09 F 7.OE-09 L Y rn 6.OE-09 C 2 5.OE-09 b c U 4.OE-09 E 3.OE-09 2.OE-09 1.OE-09 O.OE+OO I Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 15 ft

106 3.OE E-08 h v E 2.OE-08 v) c 2 c 1.5E-08 0 Q) G= Q) 1.OE-08 n 5.OE-09 O.OE+OO Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 22.5 ft 5.OE E-08 4.O E E-08 v E 3.OE-08,.Q c 2.5E-08 0 * 2 2.OE-08 d 1.5E-08 1.OE-08 5.OE-09 O.OE+OO Skew Angle (deg.) Figure Final Haunched FE and Frame Deflections, B = 6 ft, H = 5 ft, CS = 30 ft

107 6. STATISTICAL STUDY 6.1 GENERAL The large number of analysis models used in the study and voluminous data generated as a corollary make a case for the statistical study of the analysis results obtained. The plots for the negative stresses did not yield a definitive pattern for the structural behavior and thus the effect of haunches. Also the second refinement gave inconsistent positive moment stresses as compared with the previous refinements. The statistical study is aimed at resolving these anomalies in the finiteelement analyses. Statistics is an extremely useful tool in a research study. It helps in the extrapolation or generalization of the results. Any research study usually involves a number of experiments to perform which generates result data. The experiments entail errors as an integral part that affects the answers sought by the study. This in turn may yield inconclusive answers. The statistical study helps in finding the effect of different parameters used in the analysis on the response measured. One can then make a better sense of the data in hand. This study attempts at evaluating the influence of haunches on the behavior of the skewed bridges. The models analyzed using the finite-element method have the span, width, height, and the skew angle defining the geometry of a particular configuration. By varying these parameters different models are generated. Thus these are the variables in the analysis. The responses measured are the stresses and maximum deflection for the structure. The statistical study would evaluate the effect of the variables span, width, height, and the 6-1.

108 skew angle on the longitudinal stresses estimated. Both the positive moment and the negative moment stresses are considered. Other responses namely the lateral stresses and the maximum deflections will not be studied statistically as they are less prominent as compared to the longitudinal stresses. Having decided the scope of the statistical study, the next task is to select the statistical tool to be used. The software SAS, which is a strong statistical analysis tool, is used for the study. SAS performs the statistical analysis of the data and provides such useful data as mean values, standard deviations. But more importantly it tells the effect of each of the variables on the result data. Also it provides valuable insight in to whether there are some variables interacting in affecting the results. This could be immensely helpful to this study in explaining some inconsistencies. A code was written for each of the type of stresses analyzed. It had the variables span, width, height, and the skew angle and the stress value given by the finite-element run for each unique model configuration. SAS returned the output of the analysis in the form of ANOVA table and the various mean values and standard deviations. The statistical study proved helpful in elucidating the results from the finite-element analysis with a statistical perspective. Appendix D summarizes the code and output of the statistical study. Following are the results from the statistical study of the analysis results. 6.2 POSITIVE MOMENT STRESSES Among the main parameters, width has the greatest effect on the positive moment stress followed by the span, and skew angle, which has a moderate effect. The other parameter 6-2.

109 height does not have a statistically significant effect on the stress The width of 4 ft gives the higher stress values than the width of 6 ft. The two values of height, 5 ft, and 10 ft, do not have statistically different values. The span of 30 ft gives largest average stress followed by the 22.5 ft, and 15 ft spans. The skew angles 45" to 60" are not different statistically in terms of their effect on the positive moment stresses. Similarly the skew angles 0" to 30" are not different in influencing the stresses. But stress changes significantly as we go from 30" to 45". There seems to be no major interaction, taking place to affect the stresses, except may be the interaction between the width and the span, which is moderate at best. This means what combination of the span and the width the bridge has will affect the positive moment stress to a moderate extent. 6.3 NEGATIVE MOMENT STRESSES 0 Among the main parameters, the span, the width, and the skew angle seem to have a major effect on the negative moment stress. Height has only a moderate effect Average stress for the 4-ft width is greater than the average stress for the 6-ft width. Height of 10 ft yields a higher stress value than that of 5 ft. As expected, stresses are highest for the largest span of 30 ft and lowest for the smallest span of 15 ft. 0 Highest stress values are obtained for the largest skew of 60". Stress decreases with the decreasing skew, with the smallest skew of 0" giving the least value. 6-3

110 7. DISCUSSION 7.1 GENERAL In general, for larger aspect ratios (longer spans), the stress ratios are less influenced by the change of the skew angle (see Figures 4-2 through 4-13). On the other hand, for smaller aspect ratios (shorter spans), the stress ratios are highly influenced by the skew angle. This means that for longer spans, the general behavior is less influenced by the skew angle and the torsional effects. For wider bridges, the positive and the negative stress ratios are changing more rapidly with the increasing skew angle (Figures 4-7 and 4-9), due to the greater effect of the laterally unsymmetrical vertical loading. Comparison between the results of Figure 4-2 (H = 3.05 m = 10 ft) and Figure 4-4 (H = 1.52 m = 5 ft) and also between Figure 4-6 (H = 3.05 m = 10 ft) and Figure 4-8 (H = 1.52 m = 5 ft) shows that, as expected, the negative moments are slightly higher for a taller bridge. These results also show that the lfference is more significant for the shorter spans of 4.57 m and 6.86 m (15 ft and 22.5 ft), and almost insignificant for a 9.14 m (30 ft) span. Also as expected, comparison between the results of Figure 4-3 (H = 3.05 m = 10 ft) and Figure 4-5 (H = 1.52 m = 5 ft) and also between Figure 4-7 (H = 3.05 m = 10 ft) and Figure 4-9 (H = 1.52 m = 5 ft) reveals that the positive moments are slightly lower for a taller bridge. For some aspect ratios and skew angles seen in Figures 4-2 through 4-9, the plane frame method overestimates the positive moments and underestimates the negative moments. If plastic redistribution of the moments is considered in the design process, the plane frame method creates a 7-1,

111 situation where the plasticity is actually higher than intended, causing extensive cracking. The gradual reduction in the negative stress ratios for skew angles greater than 45" can be attributed to the actual span reduction in the short diagonal direction. In general, Figures 4-2 through 4-5 reveal that for most short-span, skewed bridges, the simple frame analysis provides deflections higher than the respective three-dimensional finiteelement analysis. Thus, for most bridges, the deflection calculated by the frame method is more conservative under service level loads. In summary, the major discrepancy between the two calculation methods is in the flexural (longitudinal and lateral) and torsional stresses developing in the bridge's deck and abutment walls. 7.2 BEHAVIOR The load-carrying capacity of the bridge structure is developed by resistance provided by the deck and its integral connection with the abutment walls. The parametric analyses showed that the major resistance effects develop due to a laterally unsymmetrical vertical load at the edge of the deck's mid-span. This load produces torsional phenomena and flexural stress amplification (see longitudinal and lateral stresses in Appendix C). The highest lateral stresses are concentrated along the side edges of the deck, on the top and bottom of the cross section, extending almost over the entire span. These stresses represent the combined torsional effects of the edge load and the geometry of the skew. The direct practical conclusion drawn from this fact is that the lateral reinforcement of the deck requires full continuity (anchorage) at the edges of the deck. This means that the deck's lateral reinforcement cannot be 7-2

112 simply stopped at the edges, as in common practice, but requires a proper detail of continuity between the top and the bottom lateral reinforcement. Generally, in models with larger aspect ratios (longer spans), depending on the particular combination of aspect ratio and skew angle, the lateral stresses at the top areas of the abutment walls are of the same range as the lateral stresses at the deck. The direct practical conclusion that can be drawn from this fact is that the top areas of the abutment walls require lateral reinforcement as much as the deck. In the absence of a more accurate prediction based on the particular geometry, design engineers utilizing the plane frame method will need to comply with a generalized specification. In bridge structures possessing higher skew angles, the positive moments extend close to the abutment walls and their stresses are of the same range as the stresses due to the negative moment at the other lateral side of the deck. This concludes that the entire amount of the deck s bottom reinforcement should be fully anchored at the abutment walls. Also, as a generalized specification for the deck areas close to the abutment walls, the amount of bottom reinforcement considered should be similar to the top reinforcement. 7-3

113 8. CONCLUSIONS AND RECOMMENDATIONS 8.1 CONCLUSIONS This report presents the performance assessment of precast-concrete, short-span, skewed bridges, with integral abutment walls. The study is based on the integration of analytical finiteelement assessment in conjunction with experimental field testing. The integrated characterization provided a more realistic conceptualization of the structural response. The commonly accepted design practice for these bridges is based on simplified twodimensional rigid portal frames, neglecting the degrading effects of the skew angle and laterally unsymmetrical vertical loading. To evaluate the adequacy and limitations of this practice and get an objective performance assessment for more reliable evaluation, three-dimensional finite-element models were developed and analyzed. This application demonstrates that the commonly accepted design practice produces under-designed bridges for certain aspect ratios. The lack of resistance of similarly designed bridges is apparently the most probable cause for the extensive cracking and local deterioration symptoms revealed in some short-span, skewed bridges. The practical benefit accruing from the present study is the resulting simple correlation dagrams that enable direct comparison and quantification between a plane frame design method and the corresponding three-dimensional finite-element analysis. In effect, these diagrams provide design magnification factors that enable the usage of the plane frame design method and compensate for the three-dimensional stress amplification effects of the skew angle and laterally unsymmetrical vertical loadmg. 8-1

114 Based on the results of this study on the performance assessment of short-span, skewed bridges, the following conclusions are drawn: Many short-span, skewed bridges are actually subjected to higher stress levels than predicted by the conventional plane frame design method. For certain aspect ratios, the stress amplification is even higher than the safety factors, classifying these bridges as critical or even unsafe. The allowable load limits of existing short-span, skewed bridges, diagnosed as underdesigned, should be lowered, based on the correlation diagrams resulting from this study. Alternatively, these bridges may be upgraded based on a proper evaluation study of the repair method. Future designs, based on the plane frame method, should account for the three-dimensional stress amplification due to the skew angle and laterally unsymmetrical vertical loading. Plane frame design results may be multiplied by the magnification factors extracted from the corresponding correlation diagrams of the present study. Addition of haunches is not totally capable to compensate the inaccurate plane frame design method for the more accurate and amplified stresses calculated with the three-dimensional finite-element design method. The haunch enhances the stiffness of the bridge deck to abutment wall joint. This is corroborated by the fact that the maximum longitudinal stress for the negative bending moment occurs in the haunch. The maximum stress for the negative moment is found at the obtuse comer that contains the longitudinal edge carrying the load. This holds for the bridge without the haunches as well. 8-2

115 Thus, the haunch does not cause a significant change with reference to the location of the critical stress for the negative moment. (7) The maximum stress for the positive moment is located at the area of load application. (8) Both the positive and the negative moment stresses are considerably higher than the stresses given by two-dimensional moment distribution analysis. Stresses are higher by as much as 2 to 2.5 in some bridge configurations. Also, the stresses increase with the increase in the skew angle. (9) Maximum deflection occurs at the area of load application and is less than given by the frame analysis. (10) The statistical analysis carried out on the longitudinal stresses suggests that the width and the span of the bridge have an interactive effect on the negative moment stress. Thus, provision of the haunch is not adequate enough to compensate for the advance analysis method that accounts for the effect of the skew angle. (11) The statistical analysis also proves the significant effect of the width on the positive moment stress. Thus, the statistical analysis confirms the finding of the study that the two-dmensional frame analysis cannot be rationalized by providing the haunches. 8.2RECO NDATIONS One of the foremost objectives of this study was to develop design modification factors that can be used in the design of precast-concrete, short-span, skewed bridges with integral abutments. The design factors modify the moments from the two-dimensional frame analysis to the actual 8-3

116 moment in the bridge so as to encompass the effects of the skew. Thus, there is no need to resort to a more expensive analysis tool such as three-dimensional finite-element software. The design factors from this study are for demonstration purposes only. Further study should be carried out to include the aforementioned parameters so as to obtain more inclusive and definitive design aid. The following design parameters will require additional research: (1) Further study of the finite-element model is recommended for various types of haunches and for the formulation of optimal and more effective haunch geometry for future bridges. (2) The effect of the grouted joints along the side edges of the deck between adjacent modular units requires further practical assessment and optimization of its contribution to the load resistance of the bridge system. Development of an effective continuous type of joint for the side edges of the deck requires both analytical and experimental investigation. (3) Another comprehensive study is recommended to develop repair methods for upgrading the existing under-designed, short-span, skewed bridges. (4) Due to the continuously increasing traffic volume, further investigation is recommended for loads beyond the design limit, to capture the modification of the behavior and the varying contribution of the resistance components. Predictability of the response, beyond the actual design limit is a necessary step for an ever-evolving and active infrastructure design procedure. Furthermore, investigation of the probable failure modes and mechanisms is an essential step in the development and design of prospective structural systems. (5) The obtained results are for the typical haunch size of 17 inch by 17 inch. The influence of additional haunch configurations used in practice should be investigated. (6) Considering the inconsistent variation in the response of longitudinal stresses with the change in the skew angle, intermediate skew angles should be included in future studies.

117 (7) Considering the complex nature of the stresses, further investigation is needed to include the self- weight of the bridge in the loading. 8-5

118 REFERENCES AASHTO (1 92). Standard specijkations for highway "ridges. American Association of ltate Highway and Transportation Officials, Washington, D.C. Dagher, H., Elgaaly, M., Kankam, J., and Comstock, L. (1991). "Skew slab bridges with integral slab abutments - Design guide - Final report." Vol. I, Tech. Paper 90-3, Tech. Sew. Div., Dept. of Civil Eng., Univ. of Maine. Dagher, H.J., Elgaaly, M., and Kankam, J. (1991). "Analytical investigation of slab bridges with integral wall abutments." Transp. Res. Rec. 1319, Transp. Res. Board, Washington D.C., pp Kankam, J.A., and Dagher, H.J. (1995). "Nonlinear FE analyses of concrete skewed slab bridges." J. Struct. Eng., Sept., pp Mahmoudzadeh, M. (1984). "Modification of slab design standards for effects of skew." USDOT, FHWA, Calif. DOT, Sacramento, California. Niday, D.A. (1997). "A Finite-Element Analysis and Field Test of Skewed, Three-Sided Concrete Box Culverts." M.S. Thesis, Dept. of Civil & Env. Eng. & Eng. Mech., Univ. of Dayton, Ohio, October. R-1.

119 APPENDIX A DOT SURVEY RESPONSES The following information is included in this Appendix. 0 Blank Survey Form 0 Survey Response List 0 Response Summary 0 Typical Culvert Parameters The attached survey form was sent to the United States DOTs and to the Washington D.C., Puerto Rico and Ontario, Canada DOTs: 53 total surveys were sent. The intent was to identify current design and analysis practices. The response to the survey is summarized in the next few sheets. The speed and thoroughness of the 30+ responses was very helpful and much appreciated for this project. A-1

120 ~~ ~~ -\ The University of Dayton DOT Questionnaire - Survey Page 1 The following questions are intended to survey the current practice used by state DOTS in the design and manufacture of precast concrete box culverts. Please answer the questions below: Department of Transportation (State?): Name of Engineer?: Engineer's Position?: Business Address?: '3 1. Does your office design and/or specify and/or approve three sided concrete box 2. If yes, approximately how many of these structures are... a) in service in your state? b) under construction in your state? c) anticipated to be constructed in the next five years? 3. What ranges of the following parameters of these type structures have you used or would you expect to use? See the following sketches for notation.: a) Span Length: From To b) Height of Abutment Walls: From To If different, opposite wall height: From To A-2

121 -7 The University of Dayton DOT Questionnaire - Survey Page 2 c) Width of segments: From To d) Thickness of the top slab: From To e) Thickness of abutment walls: From To f) Skew angles: From To g) Haunch Details: Width: From To Depth: From To Three Sided BoxCukert Bridge A Sketch 2 Sketch I A-3

122 ~~ ~ The University of Dayton DOT Questionnaire - Survey Page 3 3. What loads are applied to the top slab, do loads vary with span length? Are soil loads applied to the top slab? (Please define practice) 4. What soil loads are applied to the abutment walls? 5. What abutment end restraints are assumed? (Fixed, Pinned, Combination, Other) 6. What method of analysis is used for these structures? (Moment Distribution, Other) 7. Are provisions made for the increased stresses associated with the skew angle in the analysis? (Please provide details) A-4

123 The University of Dayton DOT Questionnaire - Survey Page 4 8. Define the reinforcement steel layout used for a typical structure (Please provide sketch as appropriate) 9. Have you observed cracking in these three sided box culvert bridge structures? (Please elaborate upon the details of such cracking, if any) A-5

124 -- SURVEY RESPONSE LIST State Returned form Response State Returned Form Response Alabama N Nevada Y survey/plan Alaska Y survey NewHampsh N Arizona Y not used New Jersey Y survey Arkansas Y not used New Mexico Y survey/plan California Y survey NewYork Y survey Colorado N North Caro. Y survey/plans Connecticut Y not used NorthDakota N Delaware Y mail returned Ohio Y survey/plans Florida Y survey Oklahoma Y not used Georgia Y survey Ontario, CN Y survey Hawaii N Oregon N 3 Idaho d: Illinois Indiana Y Y Y survey survey survey Pennsylvania Y PuertoRico Y Rhode Island Y survey not used survey/plans Iowa Y not used South Caro. Y not used Kansas Kentucky Y N not used South Dakota Y Tennessee N not used Louisiana N Texas Y survey Maine Maryland N Y survey Utah Vermont N N Massachus. Michigan N N Virginia Wash.D.C. N N Minnesota Y not used Washington Y survey Mississippi Missouri Y N not used West Virginia Y Wisconsin N not used Montana Y not used Wyoming N P Nebraska Y not used TOTAL = 34/53 (64+%) A-6

125 . -_ RESPONSE SUMMARY 1 Number of these type bridges in State... State In service Under Construction Planned Alaska about California approx 6 approx 2 appror 30 Florida Georgia Idaho Illinois Indiana Maryland Nevada New Mexico New York North Carolina Ohio RhodeIsland Texas Washington A-7

126 - Dimensional Data: 1 State Suans Heiph ts Widths Skews Thick Haunch California 5' - 22' 4.5' - 11' 4' - 6' 0 deg 6" - 18" 4" - % spn Florida 4' -14' 4' - 14' 10' 0-50 deg 9'' - 12" Georgia 8' - 40' 8' - 20' 0-10 deg ' Idaho 15' - 20' 5' - 10' 4' - 5' 0-10 deg 7" - 10'' Illinois 14' - 42' 5' - 14' 4' - 8' 0-45 deg 8'' - 20'' Maryland 12' - 16' 8' - 16' deg 12" - 25" (6 Nevada 24 ' 8l-2" 5' 15 deg 10" - 14" 12" New Mexico 24' 10 '-6 " 61-91' 0 deg 10" - 14" 18" 3 New York 10-40' North Caro. 24' 38' 4' - 10' 4' 9-1/2' 4' - 8' 2' - 8' 0-45 deg 0-20 deg 8-18" 12-18" 8'' - 12'' 16-18" Ohio 14' - 34' 4' - 10' 4' - 8' 0-30 deg 10" - 16" 1' - 6' Pennsylvania 8' - 20' 8' - 20' 60 deg 12" mode Island 15' - 45' 10' - 20' 6' 0-35 deg 18" - 36" 1' -6' Texas 20' 8' 8' 12" - 18" Washington 15' - 40' 5' - 15' 5' - 10' 0-45 deg 8'' - 18" 15" - 20" Range: 5' - 45' 4' - 20' 2' - 10' 0-61 deg 6" 36" 0"- %span Typical: 20' 8' 4' 20 deg 16" 18" A-8

127 I, Analysis Methods and Performance: State Analysis Method Skew Provision Loads End Restraints Alaska Frame program no standard practice AASHTO pinned California Moment Distribution none HS-20 pinned or roller Florida Stiffness Method AASHTO HS-25 pinned Georgia Moment Distribution extra rebar Soil and LL fixed or pinned Idaho Moment Distribution none HS-25 pinned Illinois STAAD III none AASHTO pinned Indiana fabricators do it, same Maryland Moment Distribution none AASHTO soil & LL pinned pinned 0 Nevada Moment Distribution none New Jersey computer program none soil & pvmnt pinned HS-25 New Mexico Moment Distribution none AASHTO pinned New York Stiffness Method increased LL ' Ohio Rigid Frame none AASHTO HS-20 pinned fixed Pennsylvania Stiffness Method None DL, LL, soil fixed or pinned Rhode Island Moment Distribution none AASHTO fixed Washington Moment Distribution none HS-25 pinned Note: No reports were made of damage beyond normal surface cracking. i A-9

128 APPENDIX B STRUCTURED FINITE-ELEMENT MODEL /FILNAME, bool! * File name : " AxBxHxSxLx. db" /TITLE, Skewed Bridge /UNITS, SI /SHOW /GRAPHICS, FULL /PBC,ALL,,1 "AFUN, DEG! Pa = N/m2! Show graphics! PowerGraphics OFF: Full average of stresses! Show boundary conditions! Degrees!* >>>---> INPUT PARAMETERS <---<<< A= 0! <---<e Skew angle: Alpha [degree] (A in file name) *** BFT=4! <---<<< Width [ft] (B in file name)*** HFT= 10! <---<<< Height [ft] (H in file name)*** CSFT= 15! <---<<< Clear span [ft] (S in file name)***!* CONSTANT PARAMETERS TWFT=12 TSFT=12 HHI=17 HLI=17! Thickness of wall [in]! Thickness of slab [in1! Haunch Height [in1! Haunch Length [in]!* CALCULATED PARAMETERS & SI-METRIC CONVERSION HH=HHI*2.54*0.01! Haunch Height in meter HL=HLI*2.54*0.01! Haunch Length in meter INM=2.54*0.01! inch to meter FTM=12 * INM! foot to meter B=BFT* FTM! Width [ml BY=B*COS (A)! Width component in direction Y [ml BX=B*SIN (A)! Width component in direction X [ml CS=CSFT*FTM! Clear span [ml TW=TWFT* INM! Thickness of wall [ml TWX=TW/COS (A)! Thickness of wall in direction X [ml TS=TSFT*INM! Thickness of slab [ml H=HFT* FTM! Height [ml CH=H-TS! Clear height [ml L=CS+2 *TWX! Full length [ml!* / PREP7 I* ET,l,SOLID95! 20-node elements!* MP, EX, 1,2.8E10! MP,EX,MAT,CO - E [Pa] * SQRT(5000 psi) * 6895 Pa/psi MP,NUXY,1,0.2! Poisson's Ratio: AASHTO Standard 1983, 8.7.3, p60 MP,DENS,1,22778! Density: 145 pcf = 2320 kg/m3 = N/m3!* LEFT WALL!* NODES N,1,0,0,0 N, 2, TWX/4,0,0 N,3,TWX/2,0,0 B-1

129 N,4, (3*TWX/4), 0 N, 5, TWX, 0, 0 NGEN,21,5,1,5,1,0,0, (CH-HH)/20,! NGEN,ITIME,INC,NODEl,NODE2,NINC,DX,DY,DZ,SPACE!* NGEN,9,5,101,105,1,0,O,HH/8, NGEN,5,5,141,145,1,0,O,TS/4, N, 166, TWX+ (HL/ 8 ), 0, CH- ( 7 *HH/ 8 ) Nr167,TWX+(HL/8),O,CH-(6*HH/8) Nrl68,TWX+(HL/8),O,CH-(5*HH/8) N, 169, TWX+ (HL/8), 0, CH- (4*HH/8) N,J-70,TWX+(HL/8),0tCH-(3*HH/8) N,171,TWX+(HL/8),0,CH-(2*HH/8) N,172,TWX+(HL/8),O,CH-(HH/8) N,173,TWX+(HL/8),O,CH N,174,TWX+(HL/8),O,CH+(TS/4) N, 175, TWX+ (HL/ 8, 0, CH+ (TSI 2 ) N,176,TWX+(HL/8),O,CH+(3*TS/4) N, 177, TWX+ (HL/8), 0, H NGEN,2,11,167,173,1,HL/8,0,0, NGEN,2,11,173,177,1,HL/8,0,0, NGEN,2,10,179,188,1,HL/8,0,0, NGEN,2,9,190,198,l,HL/8,0,0, NGEN,2,8,200,207,l,HL/8,0,0, NGEN,2,7,209,215,l,HL/8,0,0, NGEN,2,6,217,222,1,HL/8,0,0, NGEN,2,5,224,228,l,HL/8,0,0,!* RIGHT WALL NGEN,2,486,1,105,1,TWX+CS,O,O, NGEN,9,5,587,591,1,0,O,HH/8, NGEN, 5,5,627,631,1,0,O,TS/4, I* N,424,TWX+CS-(7*HL/8),O,CH-(HH/8) N,425,TWX+CS-(7*HL/8),O,CH N,426,TWX+CS-(7*HL/8),O,CH+(TS/4) N, 427, TWX+CS- (7*HL/8), 0, CH+ (TS/2) N, 428, TWX+CS- (7*HL/8), 0, CH+ (3*TS/4) N, 429, TWX+CS- (7*HL/8), 0, H N,430,TWX+CS-(6*HL/8),O,CH-(HH/4) NGEN,2,7,424,429,l,HL/8,0,0, N,437,TWX+CS-(5*HL/8),O,CH-(3*HH/8) NGEN,2,8,430,436,l,HL/8,0,0, N,445,TWX+CS-(4*HL/8),O,CH-(4*HH/8) NGEN,2,9,437,444,1,HL/8,0,0, N, 454, TWX+CS- (3*HL/8), 0, CH- (5*HH/8) NGEN,2,10,445,453,1,HL/8,0,0, N,464,TWX+CS-(2*HL/8),O,CH-(6*HH/8) NGEN,2,11,454,463,l,HL/8,0,0, N,475,TWX+CS-(HL/8),O,CH-(7*HH/8) NGEN,2,12,464,474,1,HL/8,0,0, I!* SLAB NGEN,39,5,229,233,1, (CS-2*HL)/38,0,0,!*!* LATERAL GENERATION NGEN,21,1000,1,651,1,BX/20,BY/20,0,!* ELEMENTS!* LEFT WALL E,1,2,1002,1001,~6,7,1007,1006 EGEN, 32,5, -1 1 EGEN,ITIME,NINC,IEL1,IEL2,IEINC,MINC,TINC,RINC,CINC B-2

130 E,2,3,1003,1002,7,8,1008,1007 EGEN, 32,5, -1 E,3,4,1004,1003,8,9,1009,1008 EGEN, 32,5, -1 E,4,5,1005,1004,9,10,1010,1009 EGEN,32,5,-1!*!* RIGHT WALL EGEN, 2,486, ALL!*!* LEFT HAUNCH E,110,166,105,105,1110,1166,1105,1105 E,167,178,166,166,1167,1178,1166,1166 E,179,189,178,178,1179,1189,1178,1178 E,190,199,189,189,1190,1199,1189,1189 E,200,208,199,199,1200,1208,1199,1199 E,209,216,208,208,1209,1216,1208,1208 E,217,223,216,216,1217,1223,1216,1216 E,224,229,223,223,1224,1229,1223,1223 E,110,166,1166,1110,115,167,1167,1115 E,115,167,1167,1115,120,168,1168,1120 E,120,168,1168,1120,125,169,1169,1125 E,125,169,1169,1125,130,170,1170,1130 E,130,170,1170,1130,135,171,1171,1135 E,135,171,1171,1135,140,172,1172,1140 E,140,172,1172,1140,145,173,1173,1145 E,145,173,1173,1145,150,174,1174,1150 E,150,174,1174,1150,155,175,1175,1155 E,155,175,1175,1155,160,176,1176,1160 E,160,176,1176,1160,165,177,1177,1165 E,167,178,1178,1167,168,179,1179,1168 EGEN, 10,1, -1 E,179,189,1189,1179,180,190,1190,1180 EGEN,9,1,-1 E,190,199,1199,1190,191,200,1200,1191 EGEN,8,1,-1 E,200,208,1208,1200,201,209,1209,1201 EGEN,7,1,-1 E,209,216,1216,1209,210,217,1217,1210 EGEN,6,1,-1 E,217,223,1223,1217,218,224,1224,1218 EGEN, 5,1, -1 E,224,229,1229,1224,225,230,1230,1225 EGEN,4,1,-1 I *!* RIGHT HAUNCH E,592,587,475,475,1592,1587,1475,1475 E,475,592,1592,1475,476,597,1597,1476 E,476,597,1597,1476,477,602,1602,1477 E,477,602,1602,1477,478,607,1607,1478 E,478,607,1607,1478,479,612,1612,1479 E,479,612,1612,1479,480,617,1617,1480 E,480,617,1617,1480,481,622,1622,1481 E,481,622,1622,1481,482,627,1627,1482 E,482,627,1627,1482,483,632,1632,1483 E,483,632,1632,1483,484,637,1637,1484 E,484,637,1637,1484,485,642,1642,1485 E,485,642,1642,1485,486,647,1647,1486 E,476,475,464,464,1476,1475,1464,1464 E,464,476,1476,1464,465,477,1477,1465 EGEN, 10, 1, -1 B-3.

131 E,465,464,454,454,1465,1464,1454,1454 E,454,465,1465,1454,455,466,1466,1455 EGEN, 9,1, -1 E,455,454,445,445,1455,1454,1445,1445 E,445,455,1455,1445,446,456,1456,1446 EGEN,8,1,-1 E,446,445,437,437,1446,1445,1437,1437 E,437,446,1446,1437,438,447,1447,1438 EGEN,7,1,-1 E,438,437,430,430,1438,1437,1430,1430 E,430,438,1438,1430,431,439,1439,1431 EGEN,6,1,-1 E,431,430,424,424,1431,1430,1424,1424 E,424,431,1431,1424,425,432,1432,1425 EGEN,5,1,-1 E,425,424,419,419,1425,1424,1419,1419 E,419,425,1425,1419,420,426,1426,1420 EGEN, 4,1, -1 I *!* SLAB E,229,234,1234,1229,230,235,1235,1230 EGEN,38,5,-1 E,230,235,1235,1230,231,236,1236,1231 EGEN,38,5,-1 E,231,236,1236,1231,232,237,1237,1232 EGEN,38,5,-1 E,232,237,1237,1232,233,238,1238,1233 EGEN,38,5,-1 I *!* LATERAL GENERATION EGEN,20,1000,ALL I* GPLOT! PLOT CONTROLS /PLOPTS,INFO,l / PLOPTS, LEG1,1 / PLOPTS, LEG2,O /PLOPTS,LEG3,1 / PLOPTS, FRAME, 0 /PLOPTS,TITLE,l / PLOPTS, MINM, 1 /PLOPTS,VERS,O / PLOPTS, WINS, 1 / PLOPTS, WP, 0 /TRIAD,ORIG /REPLOT!* VIEWING CONTROLS /VIEW, 1,,-1 /ANG, 1 /REP /ANG, 1,-~o.oooooo,YS,1 /REP /ANG, 1,-1O.OOOooo,YS,1 /REP /ANG, 1,~o.ooOooo,XS,1 /REP /ANG, 1,~o.oooooo,XS,1 /REP '/ANG, 1,~o.oooooo,XS,~ /REP /ANG, 1., 10. oooooo,xs, 1 B-4

132 /REP /FOC, 1,, ,,1 FINISH I* / SOLU!* CONSTRAINTS: DISPLACEMENTS UX = W = UZ = 0 D,3,UX, 0,,20003,1000,W,UZ! D,NODE,Lab,VALUE,VALUE2,NEND,NINC,Lab2,Lab3,Lab4,Lab5,lab6 D,489,UX,0,,20489,10OO,UY,UZ!* >>>--> INPUT "UNIT" ELEMENT PRESSURE LOAD (ELEM = L IN FILE NAME) *** <-- <<< SFE, 525,6, PRES,, ( (38*20)/ (8*(CS-2*HL)*BY)),,, I SFE,ELEM,LKEY,Lab,,VALl,VAL2,V?G3,VAL4 SFE,526,6,PRES,, ((38*20)/(8*(CS-2*HL)*BY)),,, SFE,1069,6,PRES,, ((38*20)/(8*(CS-2*HL)*BY)),, SFE,1070,6,PRES,, ((38*20)/(8*(CS-2*HL)*BY)),, SFE,1613,6,PRES,, ((38*20)/(8*(CS-2*HL)*BY)),,, SFE,1614,6,PRES,, ((38*20)/(8*(CS-2*HL)*BY)),, SFE,2157,6,PRES,,((38*20)/(8*(CS-2*HL)*BY)),, SFE,2158,6,PRES,, ((38*20)/(8*(CS-2*HL)*BY)),,, /REPLOT!* / STAT, SOLU ANTYPE, 0 SOLVE FINISH!* RESULTS / POST1 AVPRIN,0,0.2! AVPRIN,KEY,EFFNU - Effective Poisson's ratio for Eq. Strain B-5.

133 APPENDIX C FINITE-ELEMENT STRESS RESULTS c- 1

134 o r -

135 N N cn 0 a Y Iu x rn

136 0 M m 0

137

138 z 0 H c- 5 Y r 0 I p1 r M N I a QJ 3 QJ A m

139 N CD v) m rl 0 0 N M rl R 0 H N m co N La I II B m

140 N La v) m SI ijl R 4 M rl N m M I1 x 5; m