SOFTWARE FOR AASHTO LRFD COMBINED SHEAR AND TORSION COMPUTATIONS USING MODIFIED COMPRESSION FIELD THEORY AND 3D TRUSS ANALOGY

Transcription

1 Report No. K-TRAN: KSU-08-5 FINAL REPORT October 2011 SOFTWARE FOR AASHTO LRFD COMBINED SHEAR AND TORSION COMPUTATIONS USING MODIFIED COMPRESSION FIELD THEORY AND 3D TRUSS ANALOGY Abdul Hali Hali Hayder A. Rasheed, Ph.D., P.E. Asad Esaeily, Ph.D., P.E. Kansas State University A cooperative transportation research progra between Kansas Departent of Transportation, Kansas State University Transportation Center, and The University of Kansas

2 1 Report No. K-TRAN: KSU Governent Accession No. 3 Recipient Catalog No. 4 Title and Subtitle SOFTWARE FOR AASHTO LRFD COMBINED SHEAR AND TORSION 5 Report Date October 2011 COMPUTATIONS USING MODIFIED COMPRESSION FIELD THEORY AND 3D TRUSS ANALOGY 6 Perforing Organization Code 7 Author(s) Abdul Hali Hali; Hayder A. Rasheed, Ph D., P.E.; and Asad Esaeily, Ph.D, P.E. 9 Perforing Organization Nae and Address Departent of Civil Engineering Kansas State University Transportation Center 2118 Fiedler Hall Manhattan, Kansas Sponsoring Agency Nae and Address Kansas Departent of Transportation Bureau of Materials and Research 700 SW Harrison Street Topeka, Kansas Perforing Organization Report No. 10 Work Unit No. (TRAIS) 11 Contract or Grant No. C Type of Report and Period Covered Final Report January 2008 Septeber Sponsoring Agency Code RE Suppleentary Notes For ore inforation write to address in block Abstract The shear provisions of the AASHTO LRFD Bridge Design Specifications (2008), as well as the siplified AASHTO procedure for prestressed and non-prestressed reinforced concrete ebers were investigated and copared to their equivalent ACI provisions. Response-2000 is an analytical tool developed for shear force-bending oent interaction based on the Modified Copression Field Theory (MCFT). This tool was first validated against the existing experiental data and then used to generate results for cases where no experiental data was available. Several reinforced and prestressed concrete beas, either siply supported or continuous were exained to evaluate the AASHTO and ACI shear design provisions for shear-critical beas. In addition, the AASHTO LRFD provisions for cobined shear and torsion were investigated and their accuracy was validated against the available experiental data. These provisions were also copared to their equivalent ACI code requireents. The latest design procedures in both codes can be extended to derive exact shear-torsion interaction equations that can directly be copared to the experiental results by considering all φ factors as one. In this coprehensive study, different over-reinforced, oderately-reinforced, and under-reinforced sections with highstrength and noral-strength concrete for both solid and hollow sections were analyzed. The ain objectives of this study were to evaluate the shear and the shear-torsion procedures proposed by AASHTO LRFD (2008) and ACI , validate the code procedures against the experiental results by apping the experiental liit points on the code-based exact ultiate interaction diagras, and also develop a MathCAD progra as a design tool for sections subjected to shear or cobined shear and torsion effects. 17 Key Words Coputer Progra, LRFD, Load and Resistance Factor Design, Bridge Truss Torsion 19 Security Classification (of this report) Unclassified For DOT F (8-72) 20 Security Classification (of this page) Unclassified 18 Distribution Stateent No restrictions. This docuent is available to the public through the National Technical Inforation Service, Springfield, Virginia No. of pages 22 Price 105

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4 SOFTWARE FOR AASHTO LRFD COMBINED SHEAR AND TORSION COMPUTATIONS USING MODIFIED COMPRESSION FIELD THEORY AND 3D TRUSS ANALOGY Final Report Prepared by Abdul Hali Hali Hayder A. Rasheed, Ph.D., P.E. Asad Esaeily, Ph.D, P.E. A Report on Research Sponsored By THE KANSAS DEPARTMENT OF TRANSPORTATION TOPEKA, KANSAS and KANSAS STATE UNIVERSITY MANHATTAN, KANSAS OCTOBER 2011 Copyright 2011, Kansas Departent of Transportation

5 PREFACE The Kansas Departent of Transportation s (KDOT) Kansas Transportation Research and New- Developents (K-TRAN) Research Progra funded this research project. It is an ongoing, cooperative and coprehensive research progra addressing transportation needs of the state of Kansas utilizing acadeic and research resources fro KDOT, Kansas State University and the University of Kansas. Transportation professionals in KDOT and the universities jointly develop the projects included in the research progra. NOTICE The authors and the state of Kansas do not endorse products or anufacturers. Trade and anufacturers naes appear herein solely because they are considered essential to the object of this report. This inforation is available in alternative accessible forats. To obtain an alternative forat, contact the Office of Transportation Inforation, Kansas Departent of Transportation, 700 SW Harrison, Topeka, Kansas or phone (785) (Voice) (TDD). DISCLAIMER The contents of this report reflect the views of the authors who are responsible for the facts and accuracy of the data presented herein. The contents do not necessarily reflect the views or the policies of the state of Kansas. This report does not constitute a standard, specification or regulation. ii

6 Acknowledgents This research was ade possible by funding fro Kansas Departent of Transportation (KDOT) through its K-TRAN Progra. Thanks are extended to Ken Hurst, Loren Risch, and Jeff Ruby, all with KDOT, for their interest in the project and their continuous support and feedback that ade it possible to arrive at iportant findings putting the state-of-the-art design procedures into perspective. i

7 Abstract The shear provisions of the AASHTO LRFD Bridge Design Specifications (2008), as well as the siplified AASHTO procedure for prestressed and non-prestressed reinforced concrete ebers were investigated and copared to their equivalent ACI provisions. Response-2000 is an analytical tool developed for shear force-bending oent interaction based on the Modified Copression Field Theory (MCFT). This tool was first validated against the existing experiental data and then used to generate results for cases where no experiental data was available. Several reinforced and prestressed concrete beas, either siply supported or continuous were exained to evaluate the AASHTO and ACI shear design provisions for shearcritical beas. In addition, the AASHTO LRFD provisions for cobined shear and torsion were investigated and their accuracy was validated against the available experiental data. These provisions were also copared to their equivalent ACI code requireents. The latest design procedures in both codes can be extended to derive exact shear-torsion interaction equations that can directly be copared to the experiental results by considering all φ factors as one. In this coprehensive study, different over-reinforced, oderately-reinforced, and under-reinforced sections with high-strength and noral-strength concrete for both solid and hollow sections were analyzed. The ain objectives of this study were to evaluate the shear and the shear-torsion procedures proposed by AASHTO LRFD (2008) and ACI , validate the code procedures against the experiental results by apping the experiental liit points on the code-based exact ultiate interaction diagras, and also develop a MathCAD progra as a design tool for sections subjected to shear or cobined shear and torsion effects. ii

8 Table of Contents Acknowledgents... i Abstract... ii List of Tables... v List of Figures... v Chapter 1: Introduction Overview Objectives Scope...2 Chapter 2: Literature Review General Experiental Studies on Reinforced Concrete Beas Subjected to Shear Only Experiental Studies on Reinforced Concrete Beas Subjected to Cobined Shear and Torsion Procedure for Shear Design of a Concrete Section AASHTO LRFD General Procedure for Shear Design Siplified Procedure for Shear Design of Prestressed and Non-prestressed Concrete Beas ACI Code Procedure for Shear Design of Prestressed and Non-prestressed Reinforced Concrete Beas Design Procedure for Sections under Cobined Shear and Torsion AASHTO LRFD Design Procedure for Sections Subjected to Cobined Shear and Torsion ACI Design Procedure for Sections Subjected to Cobined Shear and Torsion Chapter 3: Forulation Evaluation of Response Review of Experiental Data Exained and Validity of Response-2000 to Deterine the Shear Strength of a Concrete Section Plotting Exact AASHTO LRFD Interaction Diagras for Cobined Shear and Torsion...39 iii

9 3.2.1 Exact Shear-Torsion Interaction Diagras Based on AASHTO LRFD (2008) Provisions Exact Shear-Torsion Interaction Diagras Based on ACI Provisions Chapter 4: Developent of AASHTO Based MathCAD Tool Flow Chart for Math CAD File...43 Chapter 5: Results and Discussion Analysis for Shear Only Analysis for Shear and Torsion...56 Chapter 6: Conclusions and Recoendations Mebers Subjected to Shear Only Mebers Subjected to Cobined Shear and Torsion Recoendations...65 References Appendix A Appendix B Appendix C iv

10 TABLE 2.1 List of Tables Details of the cross-section and suary of the experiental results for the selected panels TABLE 2.2 Properties of reinforcing bars TABLE 2.3 Cross-sectional properties of the bea studied TABLE 3.1 Experiental and Response-2000 shear and oent results at shear-critical section of the bea List of Figures FIGURE 2.1 Traditional shear test set-up for concrete beas... 3 FIGURE 2.2 The ratio of experiental to predicted shear strengths vs. transverse reinforceent for the panels FIGURE 2.3(a) FIGURE 2.4(a) FIGURE 2.5 FIGURE 2.6 Cross-section of the non-prestressed siply supported reinforced concrete bea (b) Cross-section with the crack control (skin) reinforceent Cross-section of the continuous non-prestressed reinforced concrete bea (b) Cross-section with the crack control (skin) reinforceent Profile and cross-section at id-span of the siply supported, Double-T (8DT18) prestressed concrete eber Profile and sections at id-span and at end of continuous Bulb-T (BT-72) eber FIGURE 2.7 Typical bea section tested by Klus FIGURE 2.8 Typical bea sec-tion for RC2 series tested by Rahal and Collins FIGURE 2.9(a) NU2 & HU2; (b) For all other speciens; (c) Hollow section NU3 & HU3. 17 FIGURE 3.1 Typical Response-2000 interface FIGURE 3.2 (Vexp/VResp-2000-Depth) Relationship for 34 reinforced concrete section. 38 FIGURE 5.1 FIGURE 5.2 Predicted shear strength along the length of BM100, non-prestressed siply supported reinforced concrete bea Predicted shear strength along the length of SE100A-M-69, continuous nonprestressed reinforced concrete bea v

11 FIGURE 5.3 FIGURE 5.4 FIGURE 5.5 FIGURE 5.6 FIGURE 5.7 Predicted shear strength for Bulb-T (BT-72) continuous prestressed concrete eber Predicted shear strength along the length of Double-T (8DT18) siply supported prestressed reinforced concrete bea Predicted shear strength along the length of BM100-D siply supported nonprestressed reinforced concrete bea with longitudinal crack control reinforceent Predicted shear strength along the length of SE100B-M-69 continuous nonprestressed reinforced concrete eber with longitudinal crack control reinforceent Shear-torsion interaction diagras along with experiental data for speciens tested by Klus (1968) FIGURE 5.8 Shear-torsion interaction diagras for RC2 series FIGURE 5.9 Shear-torsion interaction diagras for High-Strength over-reinforced speciens HO-1, and HO FIGURE 5.10 Shear-torsion interaction diagra for NO-1 and NO FIGURE 5.11 Shear-Torsion Interaction diagra for High-Strength box section HU vi

12 Chapter 1: Introduction 1.1 Overview In this study the shear or cobined shear and torsion provisions of AASHTO LRFD (2008) Bridge Design Specifications, siplified AASHTO procedure for prestressed and nonprestressed ebers, and ACI for reinforced concrete ebers are coparatively studied. Shear-critical beas were selected to evaluate the shear provisions for the entioned codes. Because of the absence of experiental data for various beas considered for the analysis and loaded with shear, Response-2000, which is an analytical tool for shear force-bending oent interaction based on the Modified Copression Field Theory (MCFT), was checked against the experiental data for cases where such experiental data existed. Consequently, the shear capacity of siply supported beas was slightly under-estiated by Response-2000, while that of continuous beas was accurately quantified. To evaluate the corresponding shear provisions for AASHTO LRFD and ACI Code; a siply supported double-t bea with harped prestressed strands, continuous bulb-t bea with straight and harped prestressed strands, as well as siply supported and continuous rectangular deep beas with and without longitudinal crack control reinforceent were selected for further analysis. The shear capacity using the aforeentioned shear provisions has been calculated at various sections along the bea span and the results are plotted in Chapter 5 of this report. In addition, the AASHTO LRFD provisions for cobined shear and torsion have been investigated and their accuracy has been validated against available experiental data. The provisions on cobined shear and torsion have also been copared to the pertinent ACI code requireents for the behavior of reinforced concrete beas subjected to cobined shear and torsion. The latest design procedures in both codes lend theselves to the developent of exact shear-torsion interaction equations that can be directly copared to experiental results by considering all φ factors to be equal to one. In this coprehensive coparison, different sections with high-strength and noral-strength concrete as well as over-reinforced, oderatelyreinforced, and under-reinforced sections with both solid and hollow cross sections were analyzed. The exact interaction diagras drawn are also included in Chapter 5 of this report. 1

13 1.2 Objectives The following are the specific objectives of this study: Evaluate shear and shear-torsion procedures proposed by AASHTO LRFD (2008) and ACI side by side. Develop a MathCAD progra to design sections subjected to shear or shear and torsion. Validate the procedure with experiental results by drawing exact interaction diagras and apping liit experiental points on the. 1.3 Scope Chapter 2 presents the experiental studies on shear or shear and torsion. In addition, the design procedure for shear and cobined shear and torsion using the AASHTO LRFD (2008) Bridge Design Specifications, and ACI are discussed in detail. Chapter 3 addresses the validity of Response-2000 for shear against available experiental data. Furtherore, the procedure to draw exact interaction diagras using the AASHTO LRFD and ACI Code for beas under cobined shear and torsion is discussed. Chapter 4 presents the flow chart for the developed MathCAD design tool for shear or shear and torsion. Chapter 5 presents the results and discussion with all the necessary plots for shear or shear and torsion. Chapter 6 presents the conclusions reached and provides suggestions or recoendations for future research. 2

14 Chapter 2: Literature Review 2.1 General Beas subjected to cobined shear and bending, or cobined shear, bending, and torsion are frequently encountered in practice. Often ties one or two of the cases ay control the design process while the other effect is considered secondary. In this study, structural concrete beas subjected to shear or cobined shear and torsion are considered while the effects of bending oent are neglected. This chapter is devoted to the review of the experiental studies and the design procedures for the structural reinforced concrete beas with negligible bending effects. 2.2 Experiental Studies on Reinforced Concrete Beas Subjected to Shear Only Even though the behavior of structural concrete beas subjected to shear has been studied for ore than 100 years, there isn t enough agreeent aong researchers about how the concrete contributes to shear resistance of a reinforced or prestressed concrete bea. This is ainly because of the any different echaniss involved in shear transfer process of structural concrete ebers such as aggregate interlock or interface shear transfer across cracks, shear transfer in copression (uncracked) zone, dowel action, and residual tensile stresses noral to cracks. However, there is a general agreeent aong researchers that aggregate interlock and copression zone are the key coponents of concrete contribution to shear resistance. FIGURE 2.1 Traditional shear test set-up for concrete beas. Figure 2.1 shows the traditional shear test set-up for concrete beas. Fro the figure, it is concluded that the region between the concentrated loads applied at the top of the bea is subjected to pure flexure whereas the shear spans are subjected to constant shear and linearly 3

15 varying bending oent. It is very obvious that the results fro such test could not be used to develop a general theory for shear behavior. Since it is alost ipossible to design an experiental progra where the bea is only subjected to pure shear, this in turn is one of the ain reasons where the true shear behavior of beas has not been understood throughout the decades. After conducting tests on reinforced concrete panels subjected to pure shear, pure axial load, and a cobination of shear and axial load, a coplex theory called Modified Copression Filed Theory (MCFT) was developed in 1980s fro the Copression Field Theory (Vecchio and Collins 1986). The MCFT was able to accurately predict the shear behavior of concrete ebers subjected to shear and axial loads. This theory was based on the fact that significant tensile stresses could exist in the concrete between the cracks even at very high values of average tensile strains. In addition, the value for angle θ of diagonal copressive stresses was considered as variable copared to the fixed value of 45 assued by ACI Code. To siplify the process of predicting the shear strength of a section using the MCFT, the shear stress is assued to reain constant over the depth of the cross-section and the section is considered as a biaxial eleent in case any axial stresses are present. This in turn produces the basis of the sectional design odel for shear where the AASHTO LRFD Bridge Design Specifications have been based on (Bentz et al. 2006). Even though the earlier AASHTO LRFD procedure to predict the shear strength of a section was straightforward, the contribution of concrete to shear strength of a section was a function of β and varying angle θ for which their values were deterined using the tables provided by AASHTO. The factor β indicated the ability of diagonally cracked concrete to transit tension and shear. The odified copression field theory is now even ore siplified when siple equations were developed for β and θ. These equations were then used to predict the shear strengths of different concrete sections and the results copared to that obtained fro MCFT. Consequently the shear strengths predicted by the siplified odified copression field theory and MCFT were copared with experiental results. To ake sure that the shear strengths predicted by the siplified odified copression field theory are consistent with experiental results, a wide range of concrete panels with and 4

16 without transverse reinforceent were tested in pure shear or a cobination of shear and axial load (Bentz et al. 2006). These panels were ade of concrete with various concrete copressive strengths, f c, different longitudinal reinforceent ratios, ρ, and variety of aggregate sizes. It was found that the results for both siplified odified copression field theory and MCFT were alost exactly siilar and both atched properly to the experiental results. In addition, the results were also copared with the ACI Code where it was pretty uch inconsistent in particular for panels with no transverse reinforceent. V exp /V predicted Vexp/V(MCFT) Vexp/V(Siplified.MCF T) ρ z f y /f' c FIGURE 2.2 The ratio of experiental to predicted shear strengths vs. transverse reinforceent for the panels. Figure 2.2 shows that the ACI ethod to predict the shear strength of a concrete section subjected to pure shear or a cobination of shear and axial load under-estiates the shear capacity of a section. However, the siplified odified copression field theory and MCFT give relatively accurate results. Note that the horizontal line where the ratio of experiental to predicted shear strengths equal to one represent a case where the predicted and the experiental results are exactly equal to each other. On the other hand, points above and below that line siply eans that the shear strength of a particular section is either under or over-estiated. 5

17 Because the points corresponding to the shear strength predicted by siplified odified copression filed theory and MCFT are closer to the horizontal line with unit value, it is concluded that the MCFT can accurately predict the shear behavior of a section. The details of the speciens corresponding to Figure 2.2 are tabulated below. The data provided below is taken fro Bentz et al. (2006). 6

18 TABLE 2.1 Details of the cross-section and suary of the experiental results for the selected panels. Reinforceent Axial load V exp /V predicted f' c, ρ x, *f yx, **S x, Siplified Panel ksi % ksi in ρ z f y /f' c ***f x /v V exp /f' c MCFT MCFT ACI Yaaguchi et al, a g =0.79 in S S S S S S S S S S S S S S Andre a g =0.35 in; KP a g =0.79 in TP TP1A KP TP KP TP KP TP TP4A KP TP KP *f yx Yield stress of longitudinal reinforceent. **S x Vertical spacing between the bars aligned in the x-direction. ***f x /v Ratio of axial stress to shear stress. As stated earlier, the AASHTO LRFD Bridge Design Specifications for shear design are based on the sectional design odel which in turn is based on MCFT. The current AASHTO LRFD (2008) bridge design specifications uses the siple equations for β and θ. These equations reoved the need to use the table provided by AASHTO LRFD to find the values for 7

19 β and θ. In addition, the equations enable the engineers to set up a spreadsheet for the shear design calculations. To evaluate the AASHTO LRFD (2008) shear design procedure for shear-critical sections, six prestressed and non-prestressed reinforced concrete beas were selected for analysis. Aong the total six beas considered, four of the were rectangular non-prestressed reinforced concrete beas which were tested by Collins and Kucha (1999) and are shown in Figure 2.3 and Figure 2.4. The reaining two beas were prestressed Double-T (8DT18) and Bulb-T (BT-72) with harped or a cobination of harped and straight tendons shown in Figure 2.5 and Figure 2.6. Because the AASTHO LRFD shear design procedure takes into account the crack control reinforceent of a section, two of the non-prestressed beas were selected to have crack control (skin) reinforceent. Furtherore, to check the AASHTO LRFD shear design provisions for different support conditions, three of the beas were purposefully selected as siply supported and the reaining three as continuous beas. It is iportant to note that the experiental data existed for only four of the nonprestressed reinforced concrete beas failed in shear at a certain location. Furtherore, the shear strength of the beas at that particular location was also deterined using the analytical tool, Response-2000, which is in turn based on MCFT. It was observed that the shear strength predicted by Response-2000 varied by an average of ±10% fro the experiental results. Since the intention was to evaluate the AASHTO LRFD shear design provisions for different cobinations of oent and shear, the predicted shear strengths at different sections throughout the bea was calculated using AASHTO LRFD (2008) and copared to the results obtained fro Response The validity of the results fro Response-2000 is discussed in Chapter 3 of this report. Note that Response-2000 was also used to verify the predicted shear strength for the prestressed beas. In addition to the AASHTO LRFD (2008), the shear design provisions for the siplified AASHTO and ACI Code were also evaluated. 8

20 2 V in (ksi) in 36.2 in 3.2 in 3.2 in 11.6 in 11.6 in BM 100 BM100D (a) (b) FIGURE 2.3(a) Cross-section of the non-prestressed siply supported reinforced concrete bea (b) Cross-section with the crack control (skin) reinforceent. 9

21 15.75 V V V V x SE100A-M-69 (a) SE100B-M-69 (b) FIGURE 2.4(a) Cross-section of the continuous non-prestressed reinforced concrete bea (b) Cross-section with the crack control (skin) reinforceent. 10

22 8 DT 18 FIGURE 2.5 Profile and cross-section at id-span of the siply supported, Double-T (8DT18) prestressed concrete eber. 11

23 Sec. at Ends Sec. at Mid Span FIGURE 2.6 eber. Profile and sections at id-span and at end of continuous Bulb-T (BT-72) 2.3 Experiental Studies on Reinforced Concrete Beas Subjected to Cobined Shear and Torsion The behavior of reinforced concrete beas subjected to any cobination of torsional, bending, and shear stresses have been studied by any researchers and various forulas have been proposed to predict the behavior of these beas. Structural ebers subjected to cobined shear force, bending oent, and torsion are fairly coon. However, in soe cases one of these actions (shear, bending, or torsion) ay be considered as to have a secondary effect and ay not be included in the design calculations. Significant research has been conducted by different researchers to deterine the behavior of reinforced concrete beas subjected to any cobination of flexural shear, bending, and torsional stresses. Tests perfored by Gesund et al. (1964) showed that bending stresses can increase the torsional capacity of reinforced concrete sections. Useful interaction equations for concrete beas subjected to cobined shear and torsion have been proposed by Klus (1968). 12

24 Moreover, an interesting experiental progra was developed by Rahal and Collins (1993) to deterine the behavior of reinforced concrete beas under cobined shear and torsion. Using siilar experiental progra, Fouad et al. (2000) tested a wide range of beas covering noral strength and high strength under-reinforced and over-reinforced concrete beas subjected to pure torsion or cobined shear and torsion. Consequently, interesting findings were reported about the contribution of concrete cover to the noinal strength of the beas, odes of failure, and cracking torsion for Noral Strength Concrete (NSC) and High Strength Concrete (HSC). It is obvious that ost of the design codes of practice today consider in any different ways the effects of any of the cobinations of flexural shear, bending, and torsional stresses. In other words, there are a variety of equations proposed by each code to predict the behavior of beas subjected to any possible cobination of the stresses entioned above. In this study, the current AASHTO LRFD (2008) and ACI shear and torsion provisions are evaluated against the available experiental data for beas under cobined shear and torsion only. In addition, Torsion-Shear (T-V) interaction diagras are presented for AASTHO LRFD (2008) and ACI and the corresponding experiental data points are shown on the plots. Even though efforts have been ade in the past to check the AASHTO LRFD and ACI shear and torsion provisions; in ost of those cases such efforts were liited to a certain range of concrete strengths or longitudinal reinforceent ratios ρ. As an exaple; Rahal and Collins (2003) have drawn the interaction diagras using the AASHTO LRFD and ACI shear and torsion provisions for bea series RC2. This series was coposed of four beas and subjected to pure shear or cobined shear and torsion. The properties for the reinforcing bars and crosssections for RC2 and other beas studied by the other are tabulated in TABLE and TABLE. The Torsion-Shear (T-V) interaction diagras for AASHTO LRFD provided by Rahal and Collins have been drawn as linear connecting pure shear to pure torsion points. In fact, this is because of the absence of equations at that tie for the factor β and θ, which were calculated using discrete data fro the tables proposed by AASHTO. The factor β as defined earlier 13

25 indicate the ability of diagonally cracked concrete to transit tension and shear, while θ is the angle of diagonal copressive stresses. TABLE 2.2 Properties of reinforcing bars. Yield Yield Yield Noinal Actual Area stress stress stress Dia(in) (in 2 ) (ksi) (ksi) (ksi) E.Fouad., et.al Klus Rahal and Collins 14

26 TABLE 2.3 Cross-sectional properties of the bea studied. Concrete Diensions Longitudinal Reinforceent f'c Specien* Width Height Cover Top Botto Stirrups b w (in) h (in) (in) (ksi) Type-1** Type-2 Type-1** Type-2 Dia (in) Spacing,s,(in) NU d16 3d16 2d16 3d NU d16 3d16 2d16 3d NU d16 3d16 2d16 3d NO d18 3d18 2d18 3d NO d18 3d18 2d18 3d HU3 (Box) d16-2d HU d18 3d18 3d18 3d HU d18 3d18 3d18 3d HU d18 3d18 3d18 3d HO d25 2d25 2d25 2d HO d25 2d25 2d25 2d *** d18,1d22-2d18,1d d18,1d22-2d18,1d E.Fouad., et.al Klus Rahal and Collins d18,1d22-2d18,1d d18,1d22-2d18,1d d18,1d22-2d18,1d d18,1d22-2d18,1d d18,1d22-2d18,1d d18,1d22-2d18,1d d18,1d22-2d18,1d d18,1d22-2d18,1d RC d25-5d25 5d RC d26-5d25 5d RC d27-5d25 5d RC d28-5d25 5d * HU=High strength Under reinforced; HO=High strength Over reinforced; NU=Noral strength Under reinforced; NO=Noral strength Over reinforced. ** Top layer of reinforceent at the top and lower layer of the botto reinforceent. *** The cover was not given; it was assued to be During this study, exact Torsion-Shear (T-V) interaction diagras were drawn using the AASHTO LRFD (2008) shear and torsion provisions. The word exact is used to indicate that the shear and torsion relationships are not assued as linear. This is due to the fact that the proposed tables for β and θ have been replaced by the siple equations provided in the current AASHTO LRFD Bridge Design Specifications for shear and torsion. For coprehensive evaluation of the AASHTO LRFD and ACI shear and torsion equations for design, a wide range of speciens ade of high-strength and noral strength concrete loaded with shear, torsion, or a cobination of both were investigated in this study. The 15

27 cases studied included under-reinforced, oderately-reinforced, and over-reinforced sections. Aong the total 30 speciens studied, 22 were ade of noral strength concrete while the reaining eight were speciens with high-strength concrete. Two hollow under-reinforced speciens, one ade of high-strength and the other ade of noral strength concrete were considered as well. The procedure for drawing the exact interaction diagras are described in detail in Chapter 3 of this report. Figures given below show soe of the cross-sections for the speciens considered at 3.94 c/c (0.71 ) Both top 1 (0.87 ) and Botto 9.65 FIGURE 2.7 Typical bea section tested by Klus. FIGURE 2.8 Typical bea section for RC2 series tested by Rahal and Collins. 16

28 FIGURE 2.9(a) NU2 & HU2; (b) For all other speciens; (c) Hollow section NU3 & HU Procedure for Shear Design of a Concrete Section The AASHTO LRFD Bridge Design Specifications (2008) proposes three ethods to design a prestressed or non-prestressed concrete section for shear. It is iportant to understand that all requireents set by AASHTO to qualify a particular ethod have to be et prior to the application of that ethod. In this report only two ethods to design a section for shear i.e., the general procedure and the siplified procedure for prestressed and non-prestressed ebers are discussed in detail. In addition, the current ACI provisions for shear design of a concrete section are briefly described. 17

29 2.4.1 AASHTO LRFD General Procedure for Shear Design The AASHTO LRFD general procedure to design or deterine the shear strength of a section is based on the Modified Copression Field Theory (MCFT). As stated earlier, this theory has proved to be very accurate in predicting the shear capacity of a prestressed or nonprestressed concrete section. It is iportant to note that the current AASTHO LRFD provisions for the general ethod are based on the siplified MCFT. The noinal shear strength of a section for all three ethods is equal to V n = V c + V s + V P Equation where: V n = noinal shear strength V c = noinal shear strength provided by concrete V s = noinal shear strength provided by shear reinforceent V p = coponent in the direction of the applied shear of the effective prestressing force V c is a function of a factor β which shows the ability of diagonally cracked concrete to transit tension and shear. The factor β is inversely proportional to the strain in longitudinal tension reinforceent,ε s, of the section. For sections containing at least the iniu aount of transverse reinforceent, the value of β is deterined as β = 4.8 (1+750ε s ) Equation When sections do not contain at least the iniu aount of shear reinforceent, the value of β is deterined as follow β = (1+750ε s ) (39+s xe ) Equation

30 The above equations are valid only if the concrete strength f c is in psi and s xe in inches. If the concrete strength f c is in MPa and s xe in, then 4.8 in Equation and 2.4.3, 3 becoes 0.4 while 51 and 39 in Equation becoe 1300 and 1000 respectively. s xe is called the crack spacing paraeter which can be estiated as s xe = s x 1.38 a g Equation s x is the vertical distance between horizontal layers of longitudinal crack control (skin) reinforceent) and a g is the axiu aggregate size in inches and has to equal zero when f c 10 ksi. Note that if the concrete strength is in MPa and s xe in, the 1.38 and 0.63 in Equation should be replaced by 35 and 16, respectively. The noinal shear strength provided by the concrete V c for the general procedure is equal to β f c b v d v when the concrete strength is in MPa. However, V c = β f c b v d v in case 1 f c is in ksi. The coefficient is and is used to convert the V 1000 c fro psi to ksi. The noinal shear strength provided by the shear reinforceent can be estiated as V s = A vf y d v cotθ s Equation where: A v =area of shear reinforceent within a distance s (inches 2 ) f y = yield stress of the shear (transverse) reinforceent in ksi or psi depending on the case. d v =effective shear depth (inches) and is equal to (d v = A psf ps d p +A s f y d s ). Note that A ps f ps +A s f y d v Max(0.9d, 0.72h) b v = effective web width (inches) s =spacing of stirrups (inches) θ = angle of inclination of diagonal copressive stresses ( ) as deterined below θ = 29(degree) ε s Equation

31 equation The above equation is independent of which units are used for f c or s xe. The strain in longitudinal tension reinforceent ε s is calculated using the following ε s = Mu dv +0.5N u+ V u V p A ps f po) E s A s +E p A ps Equation M u = sactored oent, not to be taken less than V u V p d v (kip-inches) N u = factored axial force, taken as positive if tensile and negative if copressive (kip) V u = factored shear force (kip) A ps = area of prestressing steel on the flexural tension side of the eber (inches 2 ) f po = 0.7 ties the specified tensile strength of prestressing steel, f pu (ksi) E s = odulus of elasticity of the nonprestressed steel on the flexural tension side of the section E p = odulus of elasticity of the prestressing steel on the flexural tension side of the section A s = area of non-prestressed steel on the flexural tension side of the section (inches 2 ) To ake sure that the concrete section is large enough to support the applied shear, it is required that V c + V s should not exceed 0.25f c b v d v. Otherwise, enlarge the section Miniu Transverse Reinforceent If the applied factored shear V u is greater than the value of 0.5φ V c + V p ; shear reinforceent is required. The aount of iniu transverse reinforceent can be estiated as A v f b v s Equation c f y Maxiu Spacing of Transverse Reinforceent According to the AASHTO LRFD Bridge Design Specifications, the spacing of the transverse reinforceent shall not exceed the axiu peritted spacing, s az deterined as 20

32 If v u (ksi) < f c, then s ax = 0.8d v 24 inches If v u (ksi) 0.125f c, then s ax = 0.4d v 12.0 inches. Where v u is calculated as v u = V u φv p b v d v Equation Siplified Procedure for Shear Design of Prestressed and Nonprestressed Concrete Beas The noinal shear strength provided by the concrete V c for perstressed and nonprestressed beas not subject to significant axial tension and containing at least the iniu aount of transverse reinforceent (specified in Section of this report) can be deterined as the iniu of V ci or V cw. V ci = f c b v d v + V d + V im cre M ax f c b v d v Equation where: V ci = noinal shear resistance provided by concrete when inclined cracking results fro cobined shear and oent (kip) V d = shear force at section due to unfactored dead load and include both concentrated and distributed dead loads V i = factored shear force at section due to externally applied loads occurring siultaneously with M ax (kip) M cre = oent causing flexural cracking at section due to externally applied loads (kipinches) M ax = axiu factored oent at section due to externally applied loads (kip-in) 21

33 M cre = S c f r + f cpe M dnc S nc Equation where: S c = section odulus for the extree fiber of the coposite section where tensile stress is caused by externally applied loads (inches 3 ) f r = rupture odulus (ksi) f cpe = copressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) at extree fiber of section where tensile stress is caused by externally applied loads (ksi) M dnc =total unfactored dead load oent acting on the onolithic or noncoposite section (kip-inches.) The web shear cracking capacity of the section can be estiated as V cw = f c f pc b v d v + V p Equation where: V cw = noinal shear resistance provided by concrete when inclined cracking results fro excessive principal tensions in web (kip) f pc = copressive stress in concrete (after allowance for all prestress losses) at centroid of cross-section resisting externally applied loads or at junction of web and flange when the centroid lies within the flange (ksi). In a coposite eber, f pc is the resultant copressive stress at the centroid of the coposite section, or at junction of web and flange, due to both prestress and oents resisted by precast eber acting alone. After calculating the flexural shear cracking and web shear cracking capacities of the section, i.e., V ci and V cw ; the iniu of the two values is selected as the noinal shear strength provided by concrete. 22

34 The noinal shear strength provided by the shear reinforceent is calculated exactly the sae as in Equation with the only difference that cotθ is calculated as following If V ci < V cw ; cotθ = 1 If V ci > V cw ; cotθ = f pc 1. 8 f c Equation To ake sure that the concrete section is large enough to support the applied shear, it is required that V c + V s should not exceed 0.25f c b v d v. Otherwise, enlarge the section. This is condition is exactly siilar to the AASHTO general procedure explained above. Note that the aount of iniu transverse reinforceent and the axiu spacing for stirrups is calculated the sae as in Sections and of this report. More iportantly, the aount of longitudinal reinforceent should also be checked at all sections considered. This is true for both general and siplified procedures described above. AASHTO LRFD (2008) proposes the following equation to check the capacity of longitudinal reinforceent: A ps f ps + A s f y M u d v φ f N u φ c + V u φ v V p 0. 5V s cotθ Equation where: φ f φ v φ c = resistance factors taken fro Article of AASHTO LRFD (2008) as appropriate for oent, shear and axial resistance. For the general procedure, the value for θ in degree is calculated using Equation However, the value for cotθ is directly calculated fro Equation for the siplified procedure for prestressed and non-prestressed beas. 23

35 2.4.3 ACI Code Procedure for Shear Design of Prestressed and Nonprestressed Reinforced Concrete Beas ACI Code presents a set of equations to predict the noinal shear strength of a reinforced concrete section. Experients have shown that the ACI provisions for shear underestiate the shear capacity of a given section and are uneconoical. However, it was recognized that ACI equations for shear over-estiates the shear capacity for large lightly reinforced concrete beas without transverse reinforceent Shioya et al.(1989). As stated earlier, the noinal shear strength of a concrete section is the suation of the noinal shear strengths provided by the concrete V c and the transverse reinforceent V s. The value of V c for a non-prestressed concrete section subjected only to shear and flexure can be estiated as V c = 2λ f c b wd Equation Whereas the shear strength provided by the concrete for prestressed ebers can be estiated using the following equations V ci = 0. 6λ f c b w d p + V d + V im cre M ax 1. 7λ f c b w d Equation or V cw = 3. 5λ f c f pc b w d p + V p Equation where d p need not be taken less than 0.80h for both equations. The value of oent causing flexural cracking due to externally applied loads, M cre at a certain section in (lb.in) is M cre = I y t 6λ f c + f pe f d Equation

36 where: f pe = copressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) at extree fiber of section where tensile stress is caused by externally applied loads (psi). After calculating the values for V ci and V cw, the noinal shear strength provided by the concrete V c is assued as the iniu of V ci or V cw. It is iportant to note that the inclination angle θ for the diagonal copressive stress is assued as 45 in the shear provisions of the ACI Code. Hence to deterine V s which is the noinal shear strength provided by the shear reinforceent, Equation is odified to V s = A vf yt d v s Equation Miniu Transverse Reinforceent According to section of the ACI Code, a iniu area of shear reinforceent A v,in shall be provided in all reinforced concrete flexural ebers (prestressed and nonprestressed) where V u exceeds 0.5φV c, except in ebers satisfying the cases specified by the code. A v,in = f c b w s f yt Equation But shall not be less than 50b w s. Also the concrete strength f f c should be in psi. yt According to section of ACI Code, for prestressed ebers with an effective prestress force not less than 40 percent of the tensile strength of the flexural reinforceent, A v,in shall not be less than the saller value of (Equation ) and (Equation ). A v,in = A psf pu s d Equation f yt d b w The above explanation can be written explicitly as 25

37 A v,in = Min Max f b w s c f yt, 50b ws f yt 80f yt d Equation , A psf pu s d b w Maxiu Spacing of Transverse Reinforceent According to section of the ACI Code, spacing of shear reinforceent placed perpendicular to axis of eber shall not exceed d/2 for non-prestressed ebers or 0.75h for prestressed ebers, nor 24 inches. The axiu spacing shall be reduced by one-half if V s exceeds 4 f c b w d. Furtherore, if the value for V s exceed 8 f c b w d, the concrete at the section ay crush. To avoid crushing of the concrete, a larger section should be selected. 2.5 Design Procedure for Sections under Cobined Shear and Torsion Section of the AASTHO LRFD Bridge Design Specifications (2008) provides pertinent equations to design a concrete section under cobined shear and torsion. The procedure is ainly based on the general ethod for shear discussed earlier. No details have been provided in the code about how to design a section for cobined shear and torsion if the siplified approach is used for the shear part. Hence, only the design procedure which is in the code is discussed here. At the end, the ACI procedure to design a section under cobined shear and torsion is explained AASHTO LRFD Design Procedure for Sections Subjected to Cobined Shear and Torsion As stated earlier, the AASHTO LRFD general procedure is used to design a section under cobined shear and torsion. The section is priarily designed for bending. The geoetry and the external loads applied on the section are then used to check the shear-torsion strength of the section. Since design is an iterative process, the cross-sectional properties and the reinforceent both longitudinal and transverse are provided different values until the desired shear-torsion strength is achieved. Below are the necessary steps to design a section for shear and torsion: 26

38 1. Deterine the external loads applied on the section considered. To do this, the bea has to be analyzed for the external loads using the load cobination that provide the axiu load effects. The section is then designed for bending and the crosssectional diensions and the aount of longitudinal reinforceent are roughly deterined. 2. Having the external load effects (axial force, shear, and bending oent) at the section, the strain in the longitudinal tension reinforceent ε s is calculated using Equation provided above. It is required to substitute V u in Equation with the equivalent shear V u,eq. For solid sections: V u,eq = V u P ht u 2A 0 2 Equation For box sections: V u,eq = V u + T ud s 2A 0 Equation To deterine the noinal shear strength of a section provided by concrete,v c, the value of ε s fro step 2 is substituted into Equation to deterine the value for β. If the concrete strength f c is provided in ksi, V c = β f c b v d v. Otherwise V c = β f c b v d v if f c is given in MPa units. 4. Substitute the value of ε s obtained fro step 2 into Equation to deterine the odified angle of inclination of diagonal copressive stresses θ (in degrees). 27

39 5. Is shear reinforceent required? No shear reinforceent is required if V u < 0.5φ(V c + V p ). 6. If V u > 0.5φ V c + V p, solve Equation for A v s after substituting the value for θ obtained in step 4. Note that V s = V u φ V c V p. 7. Calculate the torsional cracking oent for the section considered using the given equation: T cr = f c A cp 2 P c 1 + f pc f c Equation where: T u = factored torsional oent (kip-inches). T cr = torsional cracking oent (kip-inches). A cp = total area enclosed by outside perieter of concrete cross-section (inches 2 ). P c = the length of the outside perieter of the concrete section (inches). f pc = copressive stress in concrete after prestress losses have occurred either at the centroid of the cross-section resisting transient loads or at the junction of the web and flange where the centroid lies in the flange (ksi). φ = 0.9 (specified in Article of the AASHTO LRFD (2008). 8. Should torsion be considered? If the external factored torsional oent T u applied on the section is such that T u > 0.25φT cr, torsion ust be considered. Otherwise, ignore the torsion. 28

40 T n = 2A 0A t f yt cotθ s Equation where: A 0 = area enclosed by the shear flow path, including any area of holes therein (inches 2 ). It is peritted to take A 0 as 85% of the area enclosed by the centerline of stirrups. A t = area of one leg of closed transverse torsion reinforceent in solid ebers (inches 2 ). θ = angle of crack as deterined in accordance with Equation using the odified strain ε s calculated in step Solve Equation for 2A t s 5. and su it with the output of step A v+t s = A v + 2A t s s Equation The aount of transverse reinforceent obtained fro step 8 should be equal to or greater than the aount given by the equation below A v,in f c b v s f yt Equation According to the AASHTO LRFD, the spacing of transverse reinforceent shall not exceed the axiu peritted spacing, s ax, deterined as: If v u (ksi) < f c, then s ax = 0.8d v 24 inches If v u (ksi) 0.125f c, then s ax = 0.4d v 12.0 inches 29

41 Note that v u given in Equation is odified for torsion using V u,eq provided by Equations and Is the cross-section large enough? If V c + V s < 0.25f c b v d v, the section is large enough, otherwise enlarge the section. 13. As a last step, the longitudinal reinforceent in solid sections shall be proportioned to satisfy A ps f ps + A s f y M u N u φd v φ + cotθ V 2 u φ V p o. 5V s Equation P 2 ht u 2A 0 φ while for box sections the longitudinal reinforceent for torsion, in addition to that required for flexure, shall not be less than A l = T np h 2A 0 f y Equation ACI Design Procedure for Sections Subjected to Cobined Shear and Torsion To design a prestressed or non-prestressed eber under cobined shear and torsion loading using the ACI provisions, the following steps can be followed: 1. Should torsion be considered? If the applied torsion on a section (prestressed or non-prestressed) is greater than the corresponding value given by Equation 2.5.9, the section has to be designed accordingly. Otherwise, torsion is not a concern and could be ignored. For non-prestressed ebers: 30

42 T th = φλ f c A2 cp P cp Equation 2.5.9a For prestressed ebers: T th = φλ f c A2 cp P cp 1 + f pc 4λ f c Equation 2.5.9b P cp is the outside perieter of concrete cross-section and is equal to P c defined earlier. φ is the resistance factor which is equal to Note that T th is the threshold torsion. 2. Equilibriu or copatibility torsion? According to section of ACI Code, if the applied factored torsion, T u in a eber is required to aintain equilibriu and is greater than the value given by Equation depending on whether the eber is prestressed or non-prestressed, the eber shall be designed to carry T u. However, in a statically indeterinate structure where significant reduction in T u ay occur upon cracking, the axiu T u is peritted to be reduced to the values given by Equation For non-prestressed ebers: T u = φ4λ f c A2 cp P cp Equation a For prestressed ebers: T u = φ4λ f c A2 cp P cp 1 + f pc 4λ f c Equation b 31

43 4. Is the section large enough to resist the applied torsion? To avoid crushing of the surface concrete due to inclined copressive stresses, the section shall have enough crosssectional area. The surface concrete in hollow ebers ay crush soon on the side where the flexural shear and torsional shear stresses are added. For solid sections: V u b w d 2 + T up h 1.7A 2 oh 2 φ V c b w d + 8 f c Equation a For hollow sections: V u + T up h φ V c + 8 f b w d 1.7A 2 oh b w d c Equation b Note that the above equations can be used both for prestressed and non-prestressed ebers. For prestressed ebers, the depth d in the above equations is taken as the distance fro extree copression fiber to centroid of the prestresses and non-prestressed longitudinal tension reinforceent but need not be taken less than 0.80h. 5. The stirrups area required for the torsion is calculated using Equation This area is then added to the stirrups area required by shear calculated based on Equation The angle θ in Equation is assued as 45 for non-prestressed and 37.5 for prestressed ebers. 6. The iniu area of transverse reinforceent required for both torsion and shear shall not be less than 32

44 A v +2A t s f c b w s f yt Equation Note that the spacing for transverse torsion reinforceent shall not exceed the saller of P h 8 or 12 inches. 7. The longitudinal reinforceent required for torsion can be calculated using the following equation A l = A t s P h f yt f y cot 2 θ Equation The required longitudinal reinforceent for torsion should not be less than the iniu reinforceent proposed by ACI and given below A l,in = 5 f ca cp f y A t s P h f yt f y Equation

45 Chapter 3: Forulation The purpose of this chapter is to evaluate the analytical tool used to deterine the shear capacity of a concrete section and develop exact interaction diagras for concrete ebers subjected to cobined shear and torsion. In Chapter 2 of this report necessary inforation about Modified Copression Field Theory (MCFT) and its application to deterine the shear or cobined shear and torsion capacity of a section were provided. Research perfored by Bentz et al.(2006) show that the MCFT and its siplified version give alost exactly the sae results and confors well to the experiental results. In this chapter, output fro an analytical tool called Response-2000 which is based on odified copression field theory is evaluated. In addition, exact interaction diagra for the general procedure of AASHO LRFD are drawn. 3.1 Evaluation of Response-2000 Response-2000 was developed by Bentz and Collins (2000). This Windows progra is based on MCFT which can analyze oent-shear, shear-axial load, and oent-axial load responses of a concrete section. Response-2000 is designed to obtain the response of a section using the initial input data. The input data depends on the desired response of a section i.e., oent-shear, shear-axial load, oent-axial load. However, cobined shear and torsion is not covered by this software. Knowing the fact that Response-2000 is based on MCFT, the output values ay shift slightly copared to AASHTO LRFD (2008) general procedure for shear which is based on siplified MCFT. 34

46 FIGURE 3.1 Typical Response-2000 interface Review of Experiental Data Exained and Validity of Response-2000 to Deterine the Shear Strength of a Concrete Section. The purpose of this section is to show how close Response-2000 can approxiate the shear capacity of a eber at a particular section. To study the shear behavior of concrete ebers, often ties siply supported rectangular reinforced concrete beas without shear reinforceent are tested in research laboratories. These beas often have a depth of 15 inches or less and loaded by point loads over short shear spans (NCHRP-549). Unfortunately these tests can not represent real cases such as continuous bridge girders supporting distributed loads and have shear reinforceent. To address this deficiency in available experiental data and generate experiental data for cases siilar to real-world situations for which no experiental data exists, the output fro Response-2000 was evaluated for 34 beas. The experiental shear strengths for these beas were taken fro Collins and Kucha (1999). 35

47 Aong the 34 beas selected, 22 beas were siply supported (Figure 2.3) with an overall depth, d, ranging between 5 inches to 40 inches These beas had a constant crosssectional width, b w, of 11.8 inches, longitudinal reinforceent ratio, ρ l of 0.5% to1.31%, and varying copressive strength, f c of 5 ksi to 14 ksi. The yield strength of longitudinal and shear reinforceent varied fro 69 ksi to 80 ksi. In addition, two beas had shear reinforceent of #3 bars spaced 26 inches apart while the reaining 20 beas didn t have any shear reinforceent. Twelve beas fro the total 34 beas selected for the analysis were continuous (Figure 2.4) with an overall depth, d, and cross-sectional width,b w each ranging between 20 inches to 40 inches and 6.7 inches to 11.6 inches respectively. The longitudinal reinforceent ratio, ρ l, varied between 1.03 to 1.36% while the concrete copressive strength, f c varied between 7.25 ksi and 13.2 ksi. The yield strength for the longitudinal and shear reinforceents varied between 69 ksi and 86 ksi. Four beas fro the total 12 beas studied had shear reinforceent of D4 with spacing ranging between 10.9 inches to 17.3 inches All of the beas were shear critical in the sense that the eber had enough capacity to support the associated bending oent. The longitudinal reinforceents for the siply supported beas were continued up to the ends. However, the longitudinal reinforceents for continuous beas were cut-off where bending oent had lower values. The critical section in the siply supported bea was assued to be at the iddle of the bea. This is due to the fact that the bending oent is a axiu at the iddle and reduces fro the full shear capacity of the section while the critical section for the continuous bea was located 3.94 ft. fro the right support. The critical section is not where shear is a axiu; rather it is a section along the bea where the bea tends to fail in shear. For continuous beas, the critical section was located where soe of the longitudinal bars on the flexural tension side of the section were not continued further. This in turn helped the strain ε s to increase. Because the provided shear reinforceent was not enough, the cross-section was assued to fail at that location. To ake sure that the bea exactly fails at this location, the shear-oent capacity along the length of the bea was deterined using Response-2000 and the location so called the critical-section provided the lowest oent-shear capacity. The experiental shear and oent 36

48 capacity and the capacity deterined using Response-2000 at shear-critical sections are tabulated in Table3.1. TABLE 3.1 the bea. Experiental and Response-2000 shear and oent results at shear-critical section of Siply Supported and Continuous* Beas Noral strength concrete High strength concrete w/o crack control reinf. With crack control reinf. w/o crack control reinf. With crack control reinf. Bea Type Bea Depth Exp.Shear Force(kips) Exp.Moent (kip.ft) (inch) (kip.ft) B BN BN BN BN B100L B100B BM100(w/stirrups) SE100A SE50A B100D BND BND BND BM100D (w/stirrups) SE100B SE50B B100H B100HE BH BH BH BRL SE100A SE100A-M-69 (w/stirrups) SE50A SE50A-M-69 (w/stirrups) BHD BHD BHD SE100B SE100B-M-69 (w/stirrups) SE50B SE50B-M-69 (w/stirrups) * Data for continuous beas are highlighted in the table above. Response 2000 (kips) Resp-2000 Moent V exp /V resp2000 To generate data using Response-2000, the experiental shear and oent at shear critical sections and the necessary properties of the section such as f c, f y, b w, overall depth, h, 37

49 and reinforceent configuration were used as the initial input values. Fro Figure 2.5 and Figure 2.6. It is known that the shear is constant along the beas and is equal to V, which is the external applied load. To find the exact shear and oent applied at the critical section, the shear and oent fro self-weight of the beas were also added. Table 3.1 presents the total shear and oent (including self-weight) at the critical section. Refer to Collins and Kucha (1999) for further details about the cross-sectional properties of the beas Siple Support Continuous Vexp/Vresp2000= V exp /V resp Depth, FIGURE 3.2 section. (Vexp/VResp-2000-Depth) Relationship for 34 reinforced concrete Figure 3.2 shows the ratio of experiental shear and shear obtained fro Response V exp It is observed that the ratio of is close to 1.0 for continuous beas while the values are V Res2000 considerably higher for siply supported beas. The line drawn at the iddle shows the boundary where the experiental shear strength is equal to that obtained fro Response The data points lower than the line show cases where Response-2000 over-estiates the shear capacity at the critical section roughly by 15% while the values above the line show cases where 38

50 Response-2000 under-estiates the shear strength of the sections. Overall, it is concluded that Response-2000 can be used to predict the shear capacity of sections for real-world cases where no experiental data exists. The graphs in Chapter 5 for the purpose of coparison between the AASTHO LRFD general procedure for shear, siplified procedure for prestressed and nonprestressed ebers, ACI include both the shear capacity predicted by Response-2000 and the 85% of that capacity. 3.2 Plotting Exact AASHTO LRFD Interaction Diagras for Cobined Shear and Torsion Shear-torsion interaction diagra for a section provides the ultiate capacity of a section under various cobinations of shear and torsion. Depending on the equations used for the cobined shear and torsion response of a section, the interaction diagra could either be linear, a quarter of a circle, an ellipse, or coposed of several broken lines. In the following section, the procedure to plot exact shear-torsion interaction diagras using the corresponding provisions of AASHTO LRFD (2008) is presented. To deterine the noinal torsional capacity of a section (Equation 2.4.4), section of the ACI Code perits to give θ values fro 30 to 45 while it is always assued 45 for shear. For the purpose of coparison, the ACI shear-torsion interaction diagras for θ equal to 30 and 45 are also plotted Exact Shear-Torsion Interaction Diagras Based on AASHTO LRFD (2008) Provisions Knowing that the transverse reinforceent required for shear and torsion for a section shall be added together, this fact provides the basic equation to plot T V interaction diagras. Fro Equation and 2.4.4, the aount of transverse reinforceent required to resist shear and torsion can be found as A t f yt s = T n 2A 0 cotθ + V n V c 2d v cotθ Equation

51 The noinal shear strength provided by the concrete V c can be substituted with β f c b v d v when f c is given in ksi. However V c is equal to β f c b v d v when the concrete strength is given in MPa. The factor β in Equation is given in ters of longitudinal strain ε s. Depending on the case, the value for ε s in Equation shall be odified. Furtherore, assuing the section is subjected to cobined shear and torsion, the value for shear in Equation should also be odified using the equivalent shear given in Equation and for solid and box sections respectively. The odified expression for ε s is then substituted into Equation as a result of which an expression for β would be obtained in ters of V and T. In addition, the odified expression for strain ε s is also substituted into Equation to deterine an expression for θ. If the section is subjected to cobined shear, torsion, and bending oent; the bending oent could either be written in ters of shear or a fixed value shall be provided. Consequently V c and θ are substituted into above Equation Knowing the reinforceent and cross-sectional properties of the section, Equation would yield an equation containing V and T as the only variables. For a certain range of values for V provided it does not exceed the pure shear capacity of the section, the corresponding torsion is easily deterined using Excel Goal Seek function or any other coputer progra. To deterine the axiu torsion that a section can resist corresponding to the shear values provided, the shear stress in Equation is set equal to the axiu allowable value of 0.25f c and the shear V u odified using Equation or For a given value of shear, the related value for torsion is then deterined by solving Equation On the other hand, Equation is used to deterine torsion that causes the longitudinal reinforceent to yield. To solve Equation 2.5.7, the sae shear values as in the previous stages are substituted into the equation. Meanwhile the expression given as Equation for V s is also substituted. Note that the equation ay further be odified depending on the case considered i.e., A ps, f ps and V p for non-prestressed ebers and other ters not satisfying for a certain case shall be set to zero. It is extreely iportant to reeber that V shall not be odified because it is already odified in Equation Finally the equation is solved for T using Excel Goal Seek function. 40

52 For a particular value of shear, the corresponding iniu value for torsion is selected fro the three analyses explained. Note that all resistance factors are assued as 1.0 because the strength of a section that has already been designed is evaluated. Six T V interaction diagras representing 20 beas are included in Chapter 5 of this report Exact Shear-Torsion Interaction Diagras Based on ACI Provisions The procedure to draw T V interaction diagras using the corresponding ACI provisions is siple copared to AASHTO LRFD (2008). The ain equations used to plot the interaction diagras are the equations based on the fact that the shear and torsion transverse reinforceent are added together and that the shear stress in concrete should not exceed beyond the axiu allowable liit of 10 f c. A t f yt s = T n + V n V c 2A 0 cotθ 2d Equation Having V c = 2 f c b w d when the concrete strength is given in psi and θ equal to 30 and 45 ; the above equation is solved for T by providing different values for V. Making sure that V does not exceed the pure shear capacity of the section. Equation a or b is solved for T depending on whether the section is solid or hollow to deterine the axiu torsion that a section can support corresponding to a certain value of shear. The axiu torsion eans that the concrete at section ay crush if slightly larger torsion is applied on the section. Note that the resistance factor is set equal to 1.0. The saller values for T is selected for a particular value of shear and the sae process is followed for other points on T V interaction diagras. The ACI interaction diagras both for θ equal to 30 and 45 are included in Chapter 5 of this report. 41

53 Chapter 4: Developent of AASHTO Based MathCAD Tool A MathCAD design tool was developed to design sections subjected to cobined shear and torsion using the corresponding AASHTO LRFD provisions. However, sections under shear and torsion where torsion is negligible can also be designed using the developed design tool. The progra is developed for kip-inches units and the initial input values shall be entered in the highlighted yellow fields. In addition, the address of each equation used is also provided in the AASHTO LRFD (2008) code. This ay help to locate the equation in the code. Brief description where ever needed has been provided in the progra to help understand different variables used. It is essential to enter the required initial input with proper units as written in the progra. Below is the flow chart for the MathCAD design tool to show how the progra functions. Furtherore, an exaple solved using the developed file has been added in Appendix C. 42

54 4.1 Flow Chart for Math CAD File 1 Input Values* Calculate V p, E c, and f po Is ConTyp= No Noral? Yes φ f = 0.7 φ f = 0.9 Calculate d v1, d e Eq. C , Select d v = ax (0.9d e, 0.72h, d v1 ) Calculate A2 cp P c Calculate v u Eq Is SecType = "Solid"? This is used to calc. A2 cp P c Yes No Is A2 cp P c 2A 0 b v? Yes No Calculate 2A 0 b v *h, h nc, A cp, P c, A 0, P h, b v, b w, d s, d p, SecType, ConType, f c, f y, f pc, f pu, f 1t1, A s, E s, A ps, E p, f ct specified, α p, a g, α, K 1, w c, A ct, s CrackControl, V u, T u, M u, N u, HasMin, A v A 43

55 A Is ConType= Noral? Yes f pc A 2 cp T cr = f c 1 + P c f c No Is f ct_specified = Yes? Yes T cr = 0.125xMin 4.7f ct, f c A2 cp P c 1 + f pc 0.125xin(4.7f ct,f c ) No Is ConType= AllLightweigh? Yes f pc A 2 cp T cr = 0.125x0.75 f c 1 + P c 0.125x0.75 f c No f pc A 2 cp T cr = 0.125x0.85 f c 1 + P c 0.125x0.85 f c Ignore Torsion (Design for shear only) Yes Is T u 0.25φT cr? Eq No Consider torsion in design V u eq = V 2 u + 0.9P 2 ht u 2A 0 Yes Is SecType= Solid? B V u + T ud s 2A 0 No 44

56 B Is M u V u V p d v? Article ( ) No M u1 = (V u V p )d v Yes M u1 = M u Claculate ε s = M u1 dv +0.5N u+ V u V p A ps f po E s A s +E p A ps (Eq ) Is ε s > 0 No Is ε s < Yes Yes No No Is ε s < ε s = M u1 dv +0.5N u+ V u V p A ps f po E s A s +E p A ps +E c A ct Article ( ) Yes ε s Calculate s x = in (d v, s CrackControl ) Calculate s xe = s x 1.38 a g (Eq ) C 45

57 C Is s xe < 12 No Is s xe > 80 No s xe (Eq ) Yes Yes s xe = 12 s xe = 80 Is HasMin= Yes? No β = ε s s xe (Eq ) Yes Calculate β = 4.8 (1+750ε s ) (Eq ) Calculate θ = ε s (Eq ) V c = β f c b v d v (Eq ) Is V u > 0.5φ(V c + V p ) No Shear reinf. NOT Required Yes Shear reinf. Required D 46

58 D Calculate V n = in ( V u, φ 0.25f c b vd v + V p ) Calculate V s = V n V c V p Is V s 0? Yes No s req = A vf y d v (cotθ + cotα)sinα V s Provide s in (Eq ) s = in (s in, s req ) Is V u φv p b v d v < 0.125f c No s ax = in (0.4d v, 12 in. ) Yes s ax = in (0.8d v, 24 in. ) s actual = in (s, s ax ) E 47

59 E Is V u φ V p < 0.25f c b v d v? No Enlarge the Section Yes SecEnough Calculate: avss = A v s actual Is IgnoreTorsion= Yes? No avst1 = T uφ 2A 0 f y cotθ Yes avst1 = 0.0 avs = avss + 2(avst1) s ShearTorsion = A v avs F 48

60 F Calculate: tpval = M u N u + φd v φ cotθ V u V φ p 0.5V s P ht u 2A 0 φ 2 No Is SecType= Solid? No Is IgnoreTorsion= Yes? Yes Yes Is A ps f ps + A s f y tpval? Yes A 1 = 0 No A 1 = tpval A psf ps f y A s A 1 = T up h 2A 0 f y A s_total = A s + A 1 End 49

61 Chapter 5: Results and Discussion 5.1 Analysis for Shear Only In Figure 3.1, the predicted shear strength at different sections along the span for BM100 using AASHTO LRFD general procedure, siplified AASHTO procedure for prestressed and non-prestressed concrete ebers, ACI , and Response-2000 are plotted. For ACI , the noinal shear strength provided by concrete was calculated both using ACI Equation (11-3) and ACI Equation (11-5). Knowing that Response-2000 underestiated the shear strength by 24% for noral strength concrete siply supported beas without crack control reinforceent (Figure 11), it can be concluded that the results obtained using the general AASHTO procedure are reasonably accurate. On the other hand, both siplified AASHTO and ACI see to slightly overestiate the shear capacity. As shown in the figure, both ACI Equation (11-3) and (11-5) used to predict V c led to alost the sae overall shear capacity of sections. However, using ACI Eq (11-3) the shear strength at different sections along the bea is constant because the bea is prisatic and has the sae spacing 16 inches for transverse reinforceent throughout the span while the shear strength using ACI Equation (11-5) follows decreasing trend because of the increasing oent towards center of the bea. The shear strength for the general AASHTO procedure and Response-2000 varies along the bea span because of the varying longitudinal strain ε x which is one-half of the strain in non-prestressed longitudinal tension reinforceent given in Equation

62 Shear Capacity, kip Siplified-AASHTO General AASHTO ACI-08 (Eq 11-3) Resp-2000 ACI-08 Detailed (Eq 11-5) Length, ft FIGURE 5.1 Predicted shear strength along the length of BM100, non-prestressed siply supported reinforced concrete bea. Figure 5.2 shows the shear strength predictions for SE100A-M-69. Fro the previous evaluation of Response-2000, it was obtained that Response-2000 underestiates the shear strength by 3.9% (average) for high strength concrete continuous beas without crack control reinforceent. In Figure 5.2 it is seen that the shear strength predictions using the general AASHTO procedure closely follow Respnse-2000 predictions for ost of the sections along the span. Note that Response-2000 highly underestiates the shear strength for sections subjected to large oent and relatively less longitudinal reinforceent. Such locations happen to be at 40 inches and 320 inches fro the left. Accordingly, Response-2000 highly overestiates the shear strength for locations with approxiately zero oent and enough longitudinal reinforceent. An exaple for such location would be a section at 360 inches fro left along the bea. As shown in the figure, the siplified AASHTO and ACI where V c is calculated using ACI Eq (11-3) give conservative results while ACI Eq (11-5) is better in this regard. Overall, the general AASHTO procedure gives convincing results for this case. Meanwhile, the shear strength is influenced by the variations in oent and longitudinal tensile reinforceent both for siplified AASHTO and a case where V c is calculated using ACI Eq (11-5). 51

63 Shear Capacity, kip Siplified-AASHTO General AASHTO ACI-08 (Eq 11-3) Resp-2000 ACI-08 Detailed (Eq 11-5) Lenght, ft FIGURE 5.2 Predicted shear strength along the length of SE100A-M-69, continuous nonprestressed reinforced concrete bea. Figure 5.3 shown below shows the predicted shear capacity along the length of BT-72, continuous prestressed reinforced concrete bea. The bea as depicted in Figure 2.6 has a span of 120 ft. and a total nuber of 44, half-inch diaeter, seven wire, 270 ksi low relaxation prestress strands. The bea had a cobination of draped and straight strands such that 12 of the strands were draped and the reaining 32 were straight. In Figure 5.1, the shear strength predictions using the aforeentioned procedures for continuous prestressed high strength concrete girder (BT-72) are shown. Noting the fact that Response-2000 was not validated for prestressed concrete beas, the shear strength results for all the ethods are reasonably close to each other. In contrast to the previous cases, the shear strength for the entire ethods follow decreasing trend as it goes far fro the support. This is due to the fact that the detailed ACI Eq s. (11-10) and (11-12) takes into account the bending oent effects present at the section. 52

64 Siplified AASHTO General AASHTO ACI-08 Resp-2000 Shear, kips length, ft FIGURE 5.3 Predicted shear strength for Bulb-T (BT-72) continuous prestressed concrete eber. In Figure 5.4, the results for (8DT18) siply supported double-t prestressed bea with harped strands are plotted. The bea shown in Figure 2.6 was 40 ft. long and did not have any transverse reinforceent and the whole noinal shear strength for the section was provided by the concrete and the P/S effects. In other words, the results plotted show the noinal shear strength provided by the concrete V c. As stated earlier, Response-2000 gives higher shear strength at section 1.5 ft. fro the support because of sall bending oent and underestiated the shear strength at 16 ft. fro the support where the oent was alost a axiu. For cases other than this, both ACI and siplified AASHTO give consistent results, however the general AASHTO procedure highly overestiated the shear strength or V c in this case. To verify which ethod gives reliable results, ore experiental work has to be ade available. 53

65 Shear capacity, Kips Siplified- AASHTO General AASHTO ACI-08 Resp Length, ft FIGURE 5.4 Predicted shear strength along the length of Double-T (8DT18) siply supported prestressed reinforced concrete bea. Figure 5.5 shown below shows the predicted shear capacity at different sections along the length of the eber, BM100D. This eber is siilar to BM100 except that crack control reinforceent was provided along the eber length as shown in Figure 2.3. The results for BM100D are plotted in Figure 5.5. Fro the previous knowledge about Response-2000, it was found that Response-2000 underestiated the shear strength by 51% (average) for noral strength concrete siply supported beas with crack control reinforceent. As shown in the figure, the siplified AASHTO highly underestiates the shear strength while the general AASHTO procedure gives reasonable results. The results for ACI are alost exactly the sae as for BM100 (without crack control reinforceent). The only difference is that the predicted shear strength by the general AASHTO procedure increases and akes ACI results relatively accurate for BM100D. 54

66 Shear Capaity, Kip Siplified-AASHTO General AASHTO ACI-08 (Eq 11-3) Resp-2000 ACI-08 Detailed (Eq 11-5) Length, ft FIGURE 5.5 Predicted shear strength along the length of BM100-D siply supported nonprestressed reinforced concrete bea with longitudinal crack control reinforceent. Figure 5.6 presents results for SE100B-M-69. Both Response-2000 and the general AASHTO procedure give very close results except for the critical locations as entioned earlier. This is in total conforance with the results showing 3.1% (average) difference obtained fro qualifying Response-2000 against experiental results tabulated in Table 3.1. The shear strength predicted using the general AASHTO procedure and Response-2000 show considerable increase, while it reains unchanged for ACI and siplified AASHTO. In other words, ACI and siplified AASHTO fail to encounter the effect of crack control reinforceent on the noinal shear strength of a section. 55

67 Shear Capacity, kip Length, ft Siplified-AASHTO General AASHTO ACI-08 (Eq 11-3) Resp-2000 ACI-08 Detailed (Eq 11-5) FIGURE 5.6 Predicted shear strength along the length of SE100B-M-69 continuous nonprestressed reinforced concrete eber with longitudinal crack control reinforceent. 5.2 Analysis for Shear and Torsion Figure 5.7 shows the T-V interaction diagras for AASHTO LRFD (2008) and ACI Code. Details of the reinforceent for these beas tested by (Klus 1968) are tabulated in Table 2.2 and Table 2.3. Having the related properties of the section, the torsion obtained fro Equation controlled. This eans that the section will neither fail due to yielding of the longitudinal tension reinforceent nor the concrete crushing. For the pure shear case, the predicted shear capacity is the sae for ACI when θ is equal to 45 and 30. This is due to the fact that the angle θ in Equation is only used in the ter that includes torsion which in turn equals zero for the pure shear case. The equation for noinal shear capacity provided by shear reinforceent, V s, of a section is independent of θ for ACI. The value for θ is inherently assued as

68 AASHTO-LRFD(2008) ACI (45 Degree) ACI (30 Degree) Experiental Data Points 100 T(kip.in) V(kip) FIGURE 5.7 Shear-torsion interaction diagras along with experiental data for speciens tested by Klus (1968). The flat plateau at the top of the graphs is due to the fact that the applied shear force is less than the noinal shear strength provided by concrete, V c. Hence, the total transverse reinforceent is used to resist the applied torsion. In other words, the applied shear does not alleviate fro the full noinal torsional capacity of the section. This is because of the fact that for V < V c, the applied shear V is resisted by the concrete and not the shear reinforceent. This situation will continue until the applied shear, V, is greater than V c. 57

69 AASHTO Series1 LRFD (2008) ACI (45 Degree) ACI (30 Degree) Experiental Data Points T(kip.in) V(kip) FIGURE 5.8 Shear-torsion interaction diagras for RC2 series. In Figure 5.8, the AASHTO LRFD shear-torsion interaction curve for (RC2 series) is flat approxiately up to a shear force of 60.5 kips; while for the curve based on the ACI it is horizontal up to a shear force of 50 kips. This is due to the fact that the value of V c for AASHTO LRFD is calculated to be kips while it is equal to 49 kips for ACI. After the section is subjected to greater shear force and torsion, the curve follows a decreasing trend as shown in the figure. Fro Figure 5.7 and Figure 5.8, it is evident that the experiental data is perfectly atching the AASHTO LRFD curve. On the other hand, the corresponding ACI curves for 30 and 45 are consistent in both figures. In a sense the ACI provisions for cobined shear and torsion are very conservative and uneconoical when θ is equal to 45. However, these provisions see to be slightly less conservative when θ is equal to

70 AASHTO Series1 LRFD (2008) ACI (45 Degree) ACI (30 Degree) Experiental Data Points T(kip.in) V(kip) FIGURE 5.9 Shear-torsion interaction diagras for High-Strength over-reinforced speciens HO-1, and HO-2. The figure shown above shows the T-V interaction diagra for HO-1 and HO-2 speciens based on AASHTO LRFD and ACI As stated earlier, the AASHTO LRFD provisions closely approxiate the torsion-shear strength of HO-1 and HO-2 sections. The ACI procedure underestiates the shear-torsion strength for θ equal to 30 and

71 AASHTO Series1 LRFD (2008) ACI (45 Degree) ACI (30 Degree) Experiental Data Points 250 T(kip.in) V(kip) FIGURE 5.10 Shear-torsion interaction diagra for NO-1 and NO-2. As shown in Figure 5.10, again the AASHTO LRFD provisions closely approxiate the cobined shear and torsion capacity for NO-1 and NO-2 speciens. However, when the cobined shear force and torsion reaches 125 kips and 86 kip-inches respectively, the equation produced fro substituting the shear stress v u with 0.25f c in Equation and substituting V u with V u eq yields a negative nuber under the square root. This eans that the concrete crushes and no torsion would be obtained fro the corresponding equation for the applied shear force greater than 125 kips. To obtain the pure shear capacity of the section, T was set equal to zero and the pure shear capacity of the section was found to be 0.25f c b v d v. The estiated capacity of the speciens where the equation yields negative nuber under the square root is shown as straight line on the AASHTO LRFD curve. 60

72 AASHTO-LRFD(2008) ACI (45 Degree) ACI (30 Degree) Experiental Data Points T(kip.in) V(kip) FIGURE 5.11 Shear-Torsion Interaction diagra for High-Strength box section HU-3. In the above figure, the predicted shear-torsion capacity for the box section HU-3 subjected only to cobined shear force and torsion is shown. According to ref (13) the section is slightly under-reinforced. Using the ACI provisions, the torsion is controlled by Equation b when the angle θ is equal to 30. This iplies that the concrete crushes if shear-torsion greater than that shown in Figure 5.9 are applied on the section. However, the torsion is controlled by Equation when θ is equal to 45 and the shear force is lower than 5 kips. For shear force greater than 5 kips, the axiu torsion that the section can resist is controlled by Equation b. This siply eans that the concrete ay crush before the reinforceent yields if a larger torsion is applied. Since Equation b is independent of the angleθ, both curves for ACI (30 and 45 ) give exact siilar results after the curve for θ equal to 45 bifurcates. T he experiental result for pure torsion is exactly the sae as predicted by ACI when θ is 30. The results for NU-3 which is not included here were also consistent with that shown in Figure 5.9. The only difference was that the experiental pure torsion strength was slightly greater copared to the strength predicted by ACI using

73 T(k-in) AASHTO LRFD (2008) ACI (45 Degree) ACI (30 Degree) Experiental Data Points V(kip) FIGURE 5.12 Shear-Torsion interaction diagra for NU-2. Fro the above figure, it is obvious that the ACI provisions for θ equal to 30 are extreely un-conservative for NU-2 which is an under-reinforced specien ade of noral strength concrete. The AASHTO LRFD curve sees to be conservative for ost of the cases studied. However, when the shear and torsion reaches 21 kips and kip-inches respectively; the longitudinal reinforceent starts yielding. As a value for shear force greater than 21 kips is substituted in Equation knowing that V p, N u, A ps, and M u are zero, a negative nuber under the square root is produced and the equation reains unsolved. To deterine the pure shear capacity of the section, T n in Equation was set equal to zero and the equation was solved for V n. The portion of the curve where the reinforceent yields is shown by dashed lines in FIGURE. The sae responses were observed for NU-1, HU-1, and HU-2 where ACI Code for θ equal to 30 gave extreely un-conservative results. 62

74 Chapter 6: Conclusions and Recoendations 6.1 Mebers Subjected to Shear Only The AASHTO LRFD (2008) general procedure to deterine the shear strength of prestressed and non-prestressed reinforced concrete ebers proved to be ore econoical than the siplified-aashto procedure for prestressed and non-prestressed reinforced concrete ebers, and ACI shear provisions. This is due to the fact that the provisions for the AASHTO LRFD general procedure is based on the Modified Copressions Field Theory which takes into account the longitudinal strain ε s in the longitudinal non-prestressed reinforceent and assue a variable angle for the diagonal copressive stresses in the web of the eber. Furtherore, the theory assues that significant tensile stress ay exist in reinforced concrete ebers after cracking has occurred. After analyzing six prestressed and non-prestressed shear critical reinforced concrete beas, it was found that the required stirrup spacing for the general AASHTO procedure was significantly larger copared to the siplified AASHTO and ACI -08 procedures. Nevertheless, it predicted the shear capacity of the section consistently in coparison with the siplified- AASHTO and ACI In addition, the siplified-aashto procedure underestiated the shear strength of sections copared to ACI for all cases studied except where the iniu shear reinforceent doinated. When analyzing the shear capacity of a reinforced concrete eber, It is extreely iportant to note that a shear-critical section could not be only liited to the location where the shear is axiu, rather a section ay be shear critical if insufficient longitudinal reinforceent is provided. Since in the ACI shear provisions, the concrete contribution to shear resistance, V c, is based on the load at which diagonal cracking is expected to occur, hence it is useful to check whether or not the eber cracks under service loads. This is not true in particular for AASHTO LRFD (2008) because the concrete contribution in AASHTO LRFD (2008) is based on the factor β showing the ability of diagonally cracked concrete to transit tension and shear. 63

75 During this study, it was found that both siplified AASHTO and ACI poorly perfored to predict the effects of longitudinal crack control reinforceent on the shear strength of a section, however, AASHTO LRFD (2008) and Response-2000 perfored well in this regard. In addition, Response-2000 has proved itself as a useful tool to accurately predict the shear strength of non-prestressed reinforced concrete sections. 6.2 Mebers Subjected to Cobined Shear and Torsion A research progra was conducted to explore the accuracy and validity of the AASHTO LRFD (2008) provisions for cobined shear and torsion design, validating against 30 experiental data fro different sections. These sections covered a wide range of speciens fro over-reinforced to under-reinforced and ade fro noral tor high strength concrete. Solid or hollow sections were aong the speciens for which the experiental data was used for coparison. AASHTO LRFD (2008) provisions were also copared to the ACI provisions for cobined shear and torsion design. AASHTO LRFD (2008) provisions consistently were ore accurate and the predictions, while conservative in ajority of the cases, were uch closer to the experiental data for close to all of the speciens. This included over-reinforced and underreinforced sections ade of high strength and noral strength concrete. During this study it was found that the AASHTO LRFD (2008) provisions to analyze a section under cobined shear and torsion ay not be able to predict the coplete T-V interaction curve for cases leading to negative ters under the square root in the derivation process. This particularly happens for over-reinforced or under-reinforced sections ade of high strength or noral strength concrete. The analytical reason is the liitation dictated by the AASHTO LRFD Equation related to the aount of longitudinal steel and equations and or related to the axiu sustainable shear stress by concrete which iplicitly affects the level of the cobined shear and torsion. However, it should be noted that the axiu shear stress liit of 0.25f c dictated by the AASHRO LRFD 2008, was accurate in prediction of the behavior of sections experiencing relatively high levels of shear stress. This was especially true for over-reinforced sections. 64

76 On the other hand, the results by the ACI are frequently un-conservative when the angle θ is equal to 30 degrees. This is especially true for the under-reinforced sections. However, when the angle θ is considered to be 45 degrees, the results are conservative for close to all of the speciens. An iportant point for the ACI code is that the angle θ is always considered as 45 degrees for shear even if the angle for torsion is used as 30 degrees. This is a discrepancy in the ACI code, while AASHTO is consistent fro this perspective. Copared to the ACI code, AASHTO LRFD (2008), provides a ore detailed process to assess the shear/torsion capacity of a section. As a result, the capacities evaluated by the AASHTO LRFD (2008) were found to be closer to the experiental data, copared to those predicted by the ACI code. It should be noted that the strain copatibility is not directly considered in the ACI code, while it plays a critical role in derivation of the AASHTO LRFD (2008) design equations. This in turn has added ore value to the AASHTO process in accurate assessent of the shear-torisonal capacity of a section. 6.3 Recoendations The AASHTO LRFD (2008) Bridge Design Specifications and ACI Code need to address the following ites: 1. The strain ε s in the longitudinal tensile reinforceent can be deterined using Eq of the AASHTO LRFD (2008). In the code, it is not explicitly stated whether the whole area for longitudinal reinforceent A s should be used for sections subjected to pure torsion or negligible shear and high torsion, or only the positive longitudinal reinforceent as defined on page 5-73 of the code shall be used to deterine the value of ε s. 2. Coentary C on Page 5-61 of the AASHTO LRFD (2008) is not clear on where to substitute V u with V u eq obtained fro Equations and 7 for sections subjected to cobined shear and torsion. Although it explains why 65

77 Equations and 7 were added, it does not provide enough details where exactly to substitute these equations. 3. The resistance factor φ in the denoinator of Equation should be reoved. 4. In the coentary C on page 5-73 of the AASHTO LRFD (2008), 0.5 cot θ = 1 should replace 0.5 cot θ = 2 5. In the ACI , the axiu liit of 10 f c as the axiu overall induced shear stress is very conservative and ay lead to un-econoical design. 66

78 References AASHTO. (2008). LRFD bridge design specification (interi). Washington, D.C.: Aerican Association of State Highway and Transportation Officials, 5-59 to Adebar, P. & Collins, M. P. (1996). Shear strength of ebers without transverse reinforceent. Canadian Journal of Civil Engineering, 23(1), Aerican Concrete Institute Code Requireents for Structural Concrete (ACI ), and Coentary, An ACI Standard, Reported by ACI Coittee 318, Farington Hills, MI. Badawy, I. E. H., McMullen, E. A., & Jordan, J. I. (1977). Experiental investigation of the collapse of reinforced concrete curved beas. Magazine of Concrete Research, 29(99), Bentz, C. E., Vecchio, J. F., & Collins, P. M. (2006). Siplified odified copression field theory for calculating shear strength of reinforced concrete eleents. ACI Structural Journal, 103(4), Bentz, E. C. & Collins, M. P. Response (2000). Collins, M. P. & Kucha, D. (1999). How safe are our large, lightly reinforced concrete beas, slabs, and footings? ACI Structural Journal, 96(4), Collins, M. P. (1978). Towards a rational theory for RC ebers in shear. Proceedings of the ASCE, 104(ST4), Collins, M. P., Mitchell, D., Adebar, P. & Vecchio, J. F. (1996). A general shear design ethod. ACI Structural Journal, 93(1), Ewida, A. A., McMullen, E. A. (1982). Concrete ebers under cobined torsion and shear. Journal of the Structural Division, Proceedings of the Aerican Society of Civil Engineers, 108(ST4), Fouad, E., Ghonei, M., Issa, M., & Shaheen, H. (2000). Cobined shear and torsion in reinforced noral and high-strength concrete beas (I): experiental study. Journal of Engineering and Applied Science, 47(6), Klus, P. J. (1968). Ultiate strength of reinforced concrete beas in cobined torsion and shear. ACI Journal, 65(17),

79 Nilson, H. A., Darwin, D., & Dolan, W. C. (2004). Design of Concrete Structures. New York: McGraw-Hill, Rahal, N. K. (2006). Evaluation of AASHTO LRFD general procedure for torsion and cobined loading. ACI Structural Journal, 103(5), Rahal, N. K., & Collins, P. M. (1995). Effect of thickness of concrete cover on shear-torsion interaction an experiental investigation. ACI Structural Journal, 92(3), Rahal, N. K., & Collins, P. M. (2003). Experiental evaluation of ACI and AASHTO LRFD design provisions for cobined shear and torsion. ACI Structural Journal, 100(3), Shioya, T., Iguro, M., Nojiri, Y., Akiyaa, H., & Okada, T. (1989). Shear strength of large reinforced concrete beas, fracture echanics: application to concrete. Fracture Mechanics: Application to Concrete, SP 118, (V. C. Li & Z. P. Bezant, Eds.). Aerican Concrete Institute, Farington Hills, Michigan. Transportation Research Board of The National Acadeies. (2005). Siplified shear design of structural concrete ebers, NCHRP Web-Only-Docuent 78, D-7. Transportation Research Board of the National Acadeies. (2005). Siplified shear design of structural concrete ebers, NCHRP Report-549, 47. Vecchio, J. F. & Collins, P. M. (1986). The odified copression field theory for reinforced concrete eleents subjected to shear. ACI Journal Proceedings, 83(2), Wight, K. J., & MacGregor, G. J. (2009). Reinforced concrete, 5th edition. New Jersey: Pearson Prentice Hall, ,

80 Appendix A AASHTO LRFD (2008) SAMPLE CALCULATIONS FOR SHEAR DESIGN Based on Modified Copression Field Theory (MCFT) and Siplified Provisions for P/S and Non-P/S Concrete Beas fc', girder fc', deck/topping Concrete Properties: 7ksi 4ksi Legend: Input Values n(odular ratio) concrete 0.15 kcf Are not used in this file/other files also E c,slab,topping ksi Iportant Notes: E c, bea ksi 1). x is calculated at the section and Prestressing Strands: all reinforceent should be on the A ps in 2 tension side at that particular section. (0.5 in. dia., seven wire, low relaxation) 2).dp? 3).fpe f pu f py f po f se E p 270 ksi 243 ksi 189 ksi ksi ksi Reinforcing Bars As in 2 f y 60 ksi E ksi Precast, Three Span Girder with Distributed Load: Continues for Barrier, Future Wearing Surface, and Live Load Siply Supported for Bea and Slab Dead Loads * Note: In case the topping and girder both have the sae f'c, ake sure that you enter f'c, deck, topping equal to that of f'c,girder (f po = A paraeter for P/S) (f se= Effective prestress after all loses) Final Answers for iportant paraeters for coposite section where the oent is negative equals zero. 4).for calculating Mcr in negative oent region, f'c for topping should be considered.also ore iportantly, if the applied oent is positive, substitute the correct values for "y or c" in flexure forula. 5). The eccentricity, e, used to calculate fpc is the distance between the centroid of P/S and centroid of NON coposite section. 69

81 Over All Geoetry and Sectional Properties: Non Coposite Section Span Length, L Over All Depth of Girder,h Width of Web, b v Area of Cross Section of Girder, A g Moent of Inertia, I g Dis. Fro centroid to ext. botto fiber, y b Dis. Fro centroid to ext. top fiber, y t Sec.odulus, ext.botto fiber, S b Sec.odulus, ext.top fiber, S t Weigh of Bea Over all depth of the coposite section, h c Slab thickness, t s Total Area (transfored)of coposite sect.,a c Moent of Inertia of the coposite sec. I c Coposite section odulus for the extree top fiber of bea, S tg Coposite section odulus for the extree top fiber of slab, S tc = 1/n*(I c /y tc ) Critical in case n=0 Total # of P/S strands Area of P/S tension reinforceent Sectional Forces at Design Section Axial Load N u Coposite Section Dis. Fro centroid of coposite section to extree botto fiber,y bc Dis. Fro centroid of coposite section to extree top fiber of bea,y tg Dis. Fro centroid of coposite section to extree top fiber of slab,y tc Coposite section odulus for the extree botto fiber of bea, S bc Shear Forces (D.L, L.L) Unfactored shear force due to bea weight, V d,girder Unfactored shear force due to deck slab, V d,slab Unfactored shear force due to barrier weight, V d,barrier Unfact. shear force due to future wearing surface, V d,wearing Unfactored shear force due to TOTAL D.L, V d Unfactored shear forces due to live load, V LL FACTORED SHEAR FORCE, V u Moents (D.L,L.L) Unfactored oent due to bea weight, M d,girder Unfactored oent due to deck slab, M d,slab Unfactored oent due barrier, M d,barrier Unfact. oent due to future wearing surface, M d,wearing Unfactored oent due to TOTAL D.L, M d Unfactored oent due to live load, M LL FACTORED MOMENT, M u 120 ft 72 in. 6in. 767 in in in in in in k/ft 80 in. 8in in in in in in 2 0 kips in in. in 3 in kips 64.6 kips 7.8 kips 14.2 kips kips kips kips kips ft kips ft kips ft kips ft kips ft kips ft kips ft *Note: If odulus of Elasticity for topping, slab, is different than odulus of elasticity for girder, deterin n=e c,slab/e c,bea called odular ratio and ultiply it by the area of slab( b eff x slab thickness). Add the given value to Area of Girder for total area of coposite section. *Note: The factored shear and oent is calculated using the following cobinations: V u =0.9(V d,girder +V d,slab +V d,bearing )+1.50(V d,wearing )+1.75(V LL ) V u =1.25(V d,girder +V d,slab +V d,bearing )+1.50(V d,wearing )+1.75(V LL ) Select the MAX shear fro above M u=0.9(m d,girder+m d,slab+m d,bearing)+1.50(m d,wearing)+1.75(m LL) M u =1.25(M d,girder +M d,slab +M d,bearing )+1.50(M d,wearing )+1.75(M LL ) Select the MAX oent fro above It is conservative to select the Max oent rather than the oent corresponding to Max shear. (check the forula for M u for abs.value) 70

82 SOLUTION: Calculation of Effective Depth, dv: de a (depth of copression) dv=de a/2 dv=0.9de dv=0.72h Max dv (controls) in in in in in in. *de=effective depth fro extree copression fiber to the centroid of the tensile force in the tensile reinforceent *Note: the value of "a"depends on location of cross section. a=(as or Aps)*fy/(0.85*f'c*b) because in this particular case it is intended to deterine "a" at critical section 7.1 ft fro support for continuous bea, only non prestrestressed reinforceent at the deck is considered in the calculation. Control : V-Gxy a).evaluation of Web Shear Cracking Strength: Vcw=(0.06* f'c+0.3fpc)bvdv+vp Vp e(eccentricity ) Pse fpc Vcw 35.2 kips in kips ksi kips Mcr=(Ic/ytc)*(0.2* f'c+fpe fd) where f'c for section with neg.oent is that for topping. fd=mdwytc/ic Vi kips Max kips ft fpe =Aps*No.P/S*fse/Ag+Aps*No.P/S*fse*e*c/Ig 0ksi fd, put the right y tc Mcr Vci fpc=pse/ag Pse*e*(ybc yb)/ig+(mdg+mds)*(ybc yb)/ig pse= # of Strands*Astrand*fse b).evaluation of Flexure Shear Cracking Strength: Vci=0.02* f'c bvdv+vd+vimcr/max >=0.06 f'c bvdv Vi=Vu Vd Max=Mu Md ksi kips ft kips *Make sure that you put right f'c, ytc and fpe while calculatin g Mcr *f pc = copressive stress in concrete (after allowance for all pretension losses) at centroid of cross section resisiting externally applied loads. ***f pc = copressive stress in concrete after all prestress losses have occurred either at the centroid of the crosssection resisting live load or at the junction of the web and flange when the centroid lies in the flange. In a coposite section, f pc is the resultant copressive stress at the centroid of the coposite section, or at the junction of the web and flange when the centroid lies within the flange, due to both prestress and to the bending oents resisted by the precast eber acting alone. Capacity Predictions: (V=Vc+Vs) *V p= vertical coponent of effective pretension Siplified force at section.(strands which are not straight or draped or harped) Response Method ACI (Kips) *e= eccentricity of P/S strands fro the centroid of the non coposite section of crosssection. *Ig = (Kips) 2000 (Kips) AASHTO LRFD (Kips) oent of inertia of non coposite section. *fpe = copressive stress in concrete due to effective pretension forces ONLY (after allowance for all pretension losses) at extree fiber of section where tensile stress is caused by externally applied loads. In this particular case, where the bea at this section is under net negative oent, hence the top portion of deck slab is in tension where prestressing doesn't affect because P/S is liited to non coposite section. fpe=0 (Always satisfy for coposite section under negative oent). When calculating fpe using flexure forula, consider Ic or cop.oent of inertia. *fd was calculated using fd=mdwytc/ic. Because this section is under net negative oent Mdw was evaluated conservatively by considering the DL negative oent as that resulting fro the DL acting on a continuous span. *Vci >0.06 f'cbvdv Control : M-Phi Note: Vresp2000 on the graph is Vu 71

83 c).evaluation of Concrete Contribution: V c =Min(V ci,v cw ) Vc kips d).evaluation of Required Transverse Reinforceent: V s =A v *f y *d v *cotθ/s CHECK: Transverse Reinforceent Required cotθ 1 Vs (Req'd) A v,in /S A v /s (Req'd) Assue: kips in 2 /in in 2 /in. A v /s (Provided) in 2 V s (Provided) s 12 in. c/c Area (#5 stirrups) in kips OK OK * cotθ=1 if M u >M cr else cotθ=1+3f pc / f'c *Note: the assued spacing and selected bars are valid for CSA approach also. e).checks: Maxiu Spacing Liit of Transverse Reinforceent: v u /f'c in. c/c S ax Miniu Reinforceent Requireent: A v,in f'cb v S/fy A v,in in 2 Maxiu Noinal Shear Resistance V n 0.25f'cb v d v +V p OK 72

84 Modified CSA Approach or Copression Field Theory: a).evaluation of x x=(m u/dv+0.5n u+ ІV u V pі A ps*f po)/(2*(e sa s+e p*a ps)) # of A ps 12 Aps in 2 x in/in *Note: The paraeters for caculating x is quite dependent on the location of crosssection for the support,such that x is the tensile stress at cross sec caused by external and internal loads. *A ps, A s = P/S and non prestressed reinforceent respectively at tensile zone NOT all If thereis no cross section. *M u = tension reinforcee Absolute value of total factored oent at the cross section. *N u = nt in Factored axial force, taken as positive if tensile and negative if copressive (kips) conjunction b).evaluation of β and θ θ= * x θ 32.4 Degrees β 2.78 * For Calculating β, it is assued that at least iniu aount of shear reinforceent is provided. c).evaluation of Concrete Contribution Vc Vc=0.0316β f'c b vd v kips d).evaluation of Required Transverse Reinforceent V s=a v*f y*d v*cotθ/s Check Transverse Reinforceent Required kips V s (Req'd) A v /s Req'd) in 2 /in. Spacing, S 11.0 in. A v/s (Porvided) in 2 /in. V s (Provided) kips e).checks: Maxiu Spacing Liit of Transverse Reinforceent: v u /f'c in. S ax Miniu Reinforceent Requireent: A v,in f'cb vs/fy A v,in in 2 Maxiu Noinal Shear Resistance V n 0.25f'cb vd v+v p OK 73

85 Appendix B VERIFICATION OF RESPONSE-2000 For Siply Supported and Continuous Beas Siply Supported Beas: B100 SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro right support, L concrete W self wt V expeiriental / 3 / Shear Force at Critical Section, L.L and D.L Control : V-Gxy Note: Vexperiental is the shear where it causes shear failure at critical section of the eber. V crit,d.l 0 V crit,l.l 225 Response2000 Vu 225 Vresp Moent at Critical Section, LL and D.L Vexp 225 M cirt,d.l Vexp/Vresp M cirt,l.l Mu Control : V-Gxy Control : M-Phi B100D SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro right support, L concrete / 3 W self wt / V expeiriental 320 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 320 Response2000 Vu 320 Vresp Moent at Critical Section, LL and D.L Vexp 320 M cirt,d.l Vexp/Vresp M cirt,l.l 864. Mu Control : M-Phi

86 B100H SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro right support, L 2.7 concrete / W self wt / V expeiriental 193 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 193 Response2000 Vu 193 Vresp Moent at Critical Section, LL and D.L Vexp 193 M cirt,d.l Vexp/Vresp M cirt,l.l Mu Control : V-Gxy Control : M-Phi B100HE SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro 2.7 right support, L concrete / W self wt / V expeiriental 217 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 217 Response2000 Vu 217 Vresp Moent at Critical Section, LL and D.L Vexp 217 M cirt,d.l Vexp/Vresp M cirt,l.l Mu Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro right support, L.. B100L SIMPLY SUPPORTEDC CASE: Control : V-Gxy Control : V-Gxy Control : M-Phi Control : M-Phi concrete / 3 W self wt / V expeiriental 223 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 223 Response2000 Vu 223 Vresp Moent at Critical Section, LL and D.L Vexp 223 M cirt,d.l Vexp/Vresp M cirt,l.l Mu

87 Control : M-Phi B100B Control : V-Gxy SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro 2.7 right support, L concrete / W self wt / V expeiriental 204 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 204 Response2000 Vu 204 Vresp Moent at Critical Section, LL and D.L Vexp 204 M cirt,d.l Vexp/Vresp M cirt,l.l Mu BN100 SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro 2.7 right support, L concrete / W self wt / V expeiriental 192 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 192 Response2000 Vu 192 Vresp Moent at Critical Section, LL and D.L Vexp 192 M cirt,d.l Vexp/Vresp M cirt,l.l Mu Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1.. BND100 SIMPLY SUPPORTEDC CASE: Control : V-Gxy Control : V-Gxy Control : M-Phi Control : M-Phi critical section fro 2.7 right support, L concrete / W self wt / V expeiriental 258 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 258 Response2000 Vu 258 Vresp Moent at Critical Section, LL and D.L Vexp 258 M cirt,d.l Vexp/Vresp M cirt,l.l Mu

88 BH100 SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro 2.7 right support, L concrete / W self wt / V expeiriental 193 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 193 Response2000 Vu 193 Vresp Moent at Critical Section, LL and D.L Vexp 193 M cirt,d.l Vexp/Vresp M cirt,l.l Mu BHD100 SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro 2.7 right support, L concrete / W self wt / V expeiriental 278 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 278 Response2000 Vu 278 Vresp Moent at Critical Section, LL and D.L Vexp 278 M cirt,d.l Vexp/Vresp M cirt,l.l Mu Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1.. BRL100 SIMPLY SUPPORTEDC CASE: Control : V-Gxy Control : V-Gxy Control : V-Gxy Control : M-Phi Control : M-Phi Control : M-Phi critical section fro 2.7 right support, L concrete / W self wt / V expeiriental 163 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 163 Response2000 Vu 163 Vresp Moent at Critical Section, LL and D.L Vexp 163 M cirt,d.l Vexp/Vresp M cirt,l.l Mu

89 BM100 SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro 2.7 right support, L concrete / 3 W self wt / V expeiriental 342 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 342 Response2000 Vu 342 Vresp Moent at Critical Section, LL and D.L Vexp 342 M cirt,d.l Vexp/Vresp M cirt,l.l Mu BM100D SIMPLY SUPPORTEDC CASE: Meber Properties: Total Spans Length 5.4 width, w 0.3 height, h 1 critical section fro 2.7 right support, L concrete / 3 W self wt / V expeiriental 461 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l 0 V crit,l.l 461 Response2000 Vu 461 Vresp Moent at Critical Section, LL and D.L Vexp 461 M cirt,d.l Vexp/Vresp M cirt,l.l Mu Control : V-Gxy Control : V-Gxy Control : M-Phi Control : M-Phi

90 Continuous Beas: SE100A 45 CONTINUOUS CASE: Meber Properties: Total Span lengths 9.2 width, w height, h 1 critical section fro right support, L concrete W self wt V expeiriental / 3 / Shear Force at Critical Section, L.L and D.L Control : V-Gxy Note: Vexperiental is the shear where it causes shear failure at critical section of the eber. *Vresp2000=Vresp2000 Actual V critd.l V crit,d.l V crit,l.l 201 Response2000 Vu Vresp Moent at Critical Section, LL and D.L Vexp 201 M cirt,d.l Vexp/Vresp M cirt,l.l Mu Control : M-Phi SE100B 45 CONTINUOUS CASE: Control : V-Gxy Control : M-Phi Meber Properties: Total Span lengths 9.2 width, w height, h 1 critical section fro right support, L concrete / W self wt / V expeiriental 281 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l V crit,l.l 281 Response2000 Vu Vresp Moent at Critical Section, LL and D.L Vexp 281 M cirt,d.l Vexp/Vresp M cirt,l.l Mu SE100A 83 CONTINUOUS CASE: Control : V-Gxy Control : M-Phi Meber Properties: Total Span lengths 9.2 width, w height, h 1 critical section fro right support, L concrete / W self wt / V expeiriental 303 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l V crit,l.l 303 Response2000 Vu Vresp Moent at Critical Section, LL and D.L Vexp 303 M cirt,d.l Vexp/Vresp M cirt,l.l Mu

91 SE100B 83 CONTINUOUS CASE: Meber Properties: Total Span lengths 9.2 width, w height, h 1 critical section fro right support, L concrete W self wt V expeiriental 365 Shear Force at Critical Section, L.L and D.L V crit,d.l V crit,l.l 365 Response2000 Vu Vresp Moent at Critical Section, LL and D.L Vexp 365 M cirt,d.l Vexp/Vresp M cirt,l.l Mu / 3 / Control : V-Gxy Control : M-Phi Note: Vexperiental is the shear where it causes shear failure at critical section of the eber. 1.1 SE100A M 69 CONTINUOUS CASE: Meber Properties: Total Span lengths 9.2 width, w height, h 1 critical section fro right support, L Control : V-Gxy Control : M-Phi concrete / 3 W self wt / 7.8 V expeiriental 516 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l V crit,l.l 516 Response2000 Vu Vresp Moent at Critical Section, LL and D.L Vexp 516 M cirt,d.l Vexp/Vresp M cirt,l.l Mu SE100B M 69 CONTINUOUS CASE: Meber Properties: Total Span lengths 9.2 width, w height, h 1 critical section fro right support, L Control : V-Gxy Control : M-Phi concrete / W self wt / V expeiriental 583 Note: Vexperiental is the shear where it causes shear failure at critical Shear Force at Critical Section, L.L and D.L section of the eber. V crit,d.l V crit,l.l 583 Response2000 Vu Vresp Moent at Critical Section, LL and D.L Vexp 583 M cirt,d.l Vexp/Vresp M cirt,l.l Mu

92 Appendix C EXAMPLES SOLVED USING THE DEVELOPED MATHCAD DESIGN TOOL USING AASHTO LRFD SHEAR AND TORSION PROVISIONS 81

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