AN ABSTRACT OF THE THESIS OF Melissa J. Robelo for the degree of Master of Science in Civil Engineering presented on July 6, 2004. Title: Analysis of Diagonally Cracked Conventionally Reinforced Concrete Girders in the Service Load Range Abstract Approved: Redacted for privacy Christopher C. Performance evaluation of conventionally reinforced concrete (CRC) bridge superstructure elements with diagonal cracks is of interest to the bridge engineering community. Standardized methods to predict service-level stress magnitudes in cracked bridge girders under combined bending and shear forces are not available. An analysis procedure was developed to determine the response of CRC bridge girders with existing diagonal cracks. The method estimates the stirrup stress range without prior knowledge of the previous stirrup strain history. Modified Compression Field Theory (MCFT), a sectional analysis procedure, is used in the current specifications to predict capacity of CRC and prestressed concrete beams and relies on equilibrium and compatibility conditions based on an initial uncracked section. In this research, the method was used to predict service-level performance for cracked sections. Effective response prediction of a previously cracked beam was achieved through alteration of the constitutive relationships to permit softening behavior of the concrete to begin much earlier in the loading history. Validation of the procedure was performed through comparison of analytically predicted stirrup strains with those from full-size laboratory specimens. Empirical data were found to correspond well with the predicted responses. The cracked section analysis procedure was then utilized to estimate the potential for low-cycle fatigue damage of an in-service bridge. A moment and shear interaction surface corresponding to stirrup yielding was estimated and compared with the load effects produced from a data set of over 14,000 permitted trucks to estimate the anticipated life of the bridge based on low-cycle fatigue damage. ©Copyright by Melissa J. Robelo July 6, 2004 All Rights Reserved Analysis of Diagonally Cracked Conventionally Reinforced Concrete Girders in the Service Load Range by Melissa J. Robelo A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Master of Science Presented July 6, 2004 Commencement June 2005 Master of Science thesis of Melissa J. Robelo presented on July 6, 2004. APPROVED: Redacted for privacy Major Professor, representing Civil Engineering Redacted for privacy Head of the Department of Civil, Construction, and Environmental Engineering Redacted for privacy Dean of the Graduate School I understand that my thesis will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my thesis to any reader upon request. Redacted for privacy Melissa J. Robelo, Author ACKNOWLEDGEMENTS I would like to express my sincere appreciation to my Major Professor, Dr. Christopher Higgins, whose encouragement and support have made this thesis possible. I can honestly say that it has been a joy working with Dr. Higgins. I also would like to thank my Minor Professor, Dr. Solomon Yim, and committee members Dr. Thomas Miller, and Dr. Cherri Pancake. A special thank you is given to Dr. Pancake for being a constant inspiration and supporting me in so many ways through the years. I would also like to express my gratitude to my fellow graduate students on the ODOT project. In particular I would like to thank Tanarat Potisuk, Aeyoung Lee, and Theresa Daniels for all their help and support along the way. Finally, I am particularly grateful to my mom, Lucia Robelo, not only for her constant moral support and encouragement, but for being someone that I have always looked up to and been inspired by. TABLE OF CONTENTS Introduction .1 Background .3 AnalysisApproach ......................................................................................................... 16 Amplification Factor for Yielding of the Stirrups ...................................................... 23 ReducedCracking Stress ............................................................................................ 26 Results ............................................................................................................................ 29 -t Prediction ........................................................................................................... 29 M-V Interaction Yield Surfaces ................................................................................. 41 Conclusions .................................................................................................................... 52 References ...................................................................................................................... 54 Appendix ........................................................................................................................ 56 LIST OF FIGURES Figure Eg 1. Principal stress trajectories [MacGregor, 1997] .......................................................... 3 2. Mohr'scircleofstress .................................................................................................. 4 3. Truss analogy model for reinforced concrete members in shear [Collins and Vecchio, 1988] ............................................................................................................................ 5 4. Average strains of a cracked element [Collins and Mitchell, 1991] ............................ 6 5. Average stresses of a cracked element [Collins and Vecchio, 1988] ........................... 6 6. Panel tests performed by Collins and Vecchio [Collins and Mitchell, 1991] .............. 6 7. MCFT compressive stress-strain relationship [Collins and Mitchell, 1991] ............... 8 8. Concrete tension before and after cracking [Collins and Vecchio, 1988] ................... 8 9. Crack spacing parameters [Collins and Mitchell, 1991] ............................................ 12 10. Detailed analysis [Collins and Mitchell, 1991] .......................................................... 14 11. Approximate analysis [Collins and Mitchell, 1991] .................................................. 14 12. Comparison of detailed and approximate procedures [Collins and Vecchio, 1988] 15 13. Specimen T12 used for analysis correlation .............................................................. 17 14. Specimen IT 12 used for analysis correlation ............................................................. 18 15. Specimen T24 used for analysis correlation .............................................................. 18 16. T 12 specimen laboratory setup and crack map .......................................................... 19 17. IT 12 specimen laboratory setup and crack map ......................................................... 20 18. T24 specimen laboratory setup and crack map .......................................................... 21 19. M-V capacity for specimen T12 ................................................................................ 22 20. M-V capacity for specimen ff12 ............................................................................... 22 LIST OF FIGURES (Continued) Figure 21. M-V capacity for specimen T24 ................................................................................ 23 22. Average versus maximum transverse strain ............................................................... 24 23. Average versus maximum transverse strain at high M/V .......................................... 25 24. Cracking strain, 8cr determination of specimen 2T12 north ................................... 27 25. Cracking strain, 26. Cracking strain, 6cr ,determination of specimen 27. V-se 28. at north side, M/V=4 2T12 North ..................................................................... 33 at south side, M/V=4 for specimen 21T12 ........................................................ 34 at south side, MN4 for specimen 10T24 ....................................................... 35 Location of strain gage at south side, MIV=4 for specimen 37. V-es 38. at south side, M/V=4 for specimen 2T12 ......................................................... 32 Location of strain gage at south side, M/V=4 for specimen 21T12 ........................... 34 35. V- 36. at south side, M1V4 for specimen 3T12 ......................................................... 31 Strain gage location relative to crack at north side, M/Vz4 for specimen 2T12 ....... 33 33. V-t 34. south .................................. 28 Strain gage location relative to crack at south side, M/V4 for specimen 2T12 ....... 32 31. V-e, 32. 21T12 Strain gage location relative to crack at south side, MIV=4 for specimen 3T12 ....... 31 29. V-t 30. ,determination of specimen 2T12 south ................................... 27 1 0T24 ........................... 35 at north side, M/V=6 for specimen 2T12 ......................................................... 36 Strain gage location relative to crack at north side, MIV=6 for specimen 2T12 ....... 36 LIST OF FIGURES (Continued) Figure Page 39. V-es at south side, MIV=6 for specimen 2T12 ......................................................... 37 40. Strain gage location relative to crack at south side, MIV=6 for specimen 2T12 ....... 37 41. V-se at south side, MIV=6 for specimen 21T12 .................................................................... 38 42. Location of strain gage at south side, MJV=6 for specimen 21T12 ........................... 38 43. V-st at north side, MIV=6 for specimen 21T12 ........................................................ 39 44. Location of strain gage at north side, MIV=6 for specimen 21T 12............................ 39 45. V-se at north side, MIV=6 for specimen 1 0T24 ....................................................... 40 46. Location of strain gage at north side, MIV=6 for specimen 1 0T24 ........................... 40 47. M-V yield surface and predicted capacities for specimen Ti 2 .................................. 42 48. M-V yield surface and predicted capacities for specimen IT 12 ................................ 43 49. M-V yield surface and predicted capacities for specimen T24 .................................. 44 50. Sections A and B from McKenzie River Bridge........................................................ 45 Si. McKenzie River Bridge at section A ......................................................................... 46 52. McKenzie River Bridge at section B ......................................................................... 46 53. M-V yield curve for McKenzie River Bridge girder section A ................................. 47 54. M-V yield curve for McKenzie River Bridge girder section B ................................. 48 55. McKenzie River Bridge at section A using AASHTO LRFD LL distribution factors. ......................................................................................................... 50 56. McKenzie River Bridge at section B using AASHTO LRFD LL distribution factors. ......................................................................................................... 50 LIST OF TABLES Table Shear to produce yield as % of AASHTO-99 MCFT for specimen T12 ................... 42 2. Shear to produce yield as % of AASHTO-99 MCFT for specimen .................. 1T12 43 3. Shear to produce yield as % of AASHTO-99 MCFT for specimen T24 ................... 44 4. Shear to produce yield as % of AASHTO-99 MCFT for McKenzie River Bridge girdersection A .......................................................................................................... 47 5. Shear to produce yield as % of AASHTO-99 MCFT for McKenzie River Bridge girdersection B .......................................................................................................... 48 Analysis of Diagonally Cracked Conventionally Reinforced Concrete Girders in the Service Load Range Introduction The AASHTO 2003 LRFD Specification contains a relatively new procedure for design and analysis of reinforced concrete beams and beam-columns for shear. The method, developed by Collins and Vecchio at the University of Toronto is called Modified Compression Field Theory (MCFT). MCFT contains relationships that reasonably predict the behavior of cracked reinforced concrete elements in combined shear and bending [Collins and Vecchio, 1986]. In particular, MCFT relationships tend to agree well with the softening behavior after first cracking and reasonably account for the post-cracking strength contribution of reinforced concrete. MCFT is used in the current specifications to predict capacity of conventionally reinforced concrete (CRC) and prestressed concrete girders based on an initial uncracked condition. However, MCFT does not predict the service level performance of CRC sections starting from a cracked condition. Analysis of previously cracked sections prior to reloading requires that MCFT be adjusted to account for the concrete material reloading at a more softened state than in the initially uncracked condition. Through analysis of laboratory data of multiple reinforced concrete beam specimens, in which strain gages were used to record stirrup strains during the entire loading history, a factor was derived to modify the value of the cracking stress, thus allowing the softening behavior of the concrete to begin at an earlier loading stage. Using this adjustment factor 2 for the cracking stress, analysis using MCFT relationships was conducted to produce a moment-shear (M-V) interaction curve at the onset of yielding for the stirrups. This M-V interaction curve was then compared to the ultimate capacity M-V interaction curve produced by Response 2000TM [Bentz, 2000] and the AASHTO-99 MCFT curve. The fraction of ultimate at which yielding occurs was calculated. This relationship may be used, based upon laboratory data, to determine if a reinforced concrete deck girder (RCDG) bridge has experienced loading above the M-V yield surface. The number of loadings above the yield threshold may be calculated based on historical data to produce an estimate of the number of high magnitude events for low-cycle fatigue evaluation. This in turn, may give a much better estimate of the remaining life for a cracked RCDG bridge girder. To demonstrate the use of the M-V yield surface developed from MCFT and the adjustment factor for previously cracked sections, the McKenzie River Bridge was analyzed at the section with the lowest fi, where ,8 is the number of standard deviations away from the mean capacity [Daniels, 2004]. 3 Background The performance of conventionally reinforced concrete (CRC) bridge superstructure elements with diagonal cracks is of interest to bridge engineers. Diagonal cracks may result in the premature failure of the member, if web and longitudinal reinforcement are not sufficiently provided. This type of failure is called a shear failure and may occur suddenly in a non-ductile fashion [Collins and Mitchell, 19911. Cracks form perpendicular to the principal stress trajectories which are dependent on the relative quantities of flexural, shear, and axial load applied to the section (Fig. 1). Fig. 1 - Principal stress trajectories [MacGregor, 1997]. The principal stress angles can be found from Mohr's circle for stress and strain (Fig. 2). Given an applied shear stress, rfl,, and normal stresses in the x and y-directions, o, and o,, respectively, the principal tensile stress, ai' principal compressive stress, o, and the angle of inclination for the principal stresses, O, can be derived. Principal stresses are calculated as [Ugural and Fenster, 1995]: ri! I Fig. 2 - Mohr's circle of stress. amax,min l2 2 ax+ay 2 2 2 [1] J The angle of inclination of the principal stresses is calculated by: ay J [2] Early methods to design reinforced concrete beams in shear were developed by Ritter and Morsch, who used a truss analogy for the internal force carrying mechanisms [Ritter, 1899]. The truss analogy became the basis for the strut-and-tie model and describes the cracked reinforced concrete beam as an equivalent truss where the web is made up of diagonal concrete struts, the stirrups are transverse ties, and the compression zone and flexural steel make up the compression and tension chords (Fig. 3). [i1 Compression chord inclined 2P Transverse ties (ie stirrups) 45° Fig. 3 - Truss analogy model for reinforced concrete members in shear ICollins and Vecchio, 19881. The angle for the compression struts has commonly been assumed to be 45°. However, using this steep angle conflicts with many tests and is overly conservative, especially for beams with small amounts of web reinforcement [Collins and Vecchio, 1988]. As a result, the AC! Building code [ACI 318-83] adopted an empirical correction factor to the truss equations to provide better correlation with experiments [Collins and Vecchio, 1988]. However, the ACI approach is known to provide wide scatter between predicted and empirical capacities for shear. To better characterize the behavior of diagonally cracked reinforced concrete, Collins and Vecchio developed the relationships for MCFT from laboratory experiments as well as solid mechanics principles of equilibrium and material compatibility. The method relates average strains in a cracked section and Mohr's circle for stress and strain (Fig. 4 and Fig. 5). Relationships for stress and strain were derived from panel testing of large-sized reinforced concrete elements loaded in various combinations of shear and normal stresses (Fig. 6). 6 /2 j(i \ Ot\ Fig. 4 - Average strains of a cracked element ICollins and Mitchell, 1991]. V -"i ..fcy vcZY\2OJ Fig. 5 - Average stresses of a cracked element ICollins and Vecchio, 1988]. Fig. 6 - Panel tests performed by Collins and Vecchio ICollins and Mitchell, 19911. 7 The stresses and strains were measured across multiple cracks to determine average values for stress and strain across the section. Based on these tests as well as those of other researchers, constitutive relationships were proposed. The average principal compressive stress in the concrete, based on a parabolic-shaped constitutive relationship, depends on the strain magnitude in the longitudinal direction, as well as the principal tensile strain, The average principal compressive stress, f2, at a point along the height of the section is: r f2f2max21 e,cJ [ 21 [3] j where f2max IC C [4] S which is based on experiments of maximum compressive stress for cracked concrete (Fig. 7). The maximum compressive stress, f's, is based on cylinder tests at 28 days and the corresponding strain. is - / / /i_____P__IIl7r(t : Fig. 7 - MCFT compressive stress-strain relationship [Collins and Mitchell, 1991]. Alternatively, the average principal compressive stress over the whole section being analyzed, as opposed to at each point along the height of the section, was derived from Mohr's circle as: f2=(tan9+cot9) bjd fi [5] where 0 is principal angle of strain, V is the shear force, b,, is width of the web, jd is shear depth of the section, andf1 is principal tensile stress. The recommended relationship between the average tensile stress and the average tensile strain is illustrated in Fig. 8. 'C 'C 'C, Cr C cr Fig. 8 - Concrete tension before and after cracking ICollins and Vecchio, 19881. The average tensile stress-strain relationship is linear until the principal tensile strain reaches the cracking strain, 5cr Therefore, if then: fi=ki [6] where E is the modulus of elasticity of the concrete and 5cr is: fcr [7] 5cr An estimate for the stress at which cracking = 42J7 where ,% psi occurs,fcr, is [Collins and Mitchell, 1991]: [8] is a factor that accounts for the density of the concrete [Collins and Mitchell, 1991]: 2 1.00 normal-weight concrete 10 2 = 0.85 for sand-lightweight concrete 2 = 0.75 for all-lightweight concrete After cracking, i > the average tensile stress-strain relationship is a function of the reinforcing bond characteristics, the principal tensile strain, and the cracking stress. However, average tensile stresses will differ with variations of stress along the section, mostly at crack locations. Tensile concrete stresses at the crack approach zero, while the tensile stress in reinforcement crossing the crack increases. The ability of the member to transmit forces across the crack may limit the shear capacity. As a result, a limit on the principal tensile stress is required to maintain equilibrium [Collins and Mitchell, 1991]. The relationship for principal tensile stress after cracking is: a1a2,. fi 1+500s1 vcitanO+(fvy f) Sb [9] Where ai is a factor that accounts for the type of reinforcement bond. ai = 1.0 for deformed reinforcing bars ai = 0.7 for plain bars, wires, or bonded strands ai = 0 for unbonded reinforcement The parameter a accounts for the type of loading. a = 1.0 for short-term monotonic loading a = 0.7 for sustained and/or repeated loads The limiting value of shear stress that can be transmitted along the crack is a function of the concrete compressive strength,f', crack width, w, and the maximum aggregate size, a. V1 2.16%j7 24w 0.3 1+ a+0.63 psi and in. [10] 11 where the crack width is derived as a function of principal tensile strain, and the maximum diagonal crack spacing. [11] W = The maximum diagonal crack spacing, s1, is a function of the maximum longitudinal crack spacing, Smx, the maximum transverse crack spacing, Smv, and the angle of principal strain, 0: SmO=. [12] Smx Smv The maximum longitudinal and transverse crack spacing indicates the ability of the longitudinal and transverse reinforcement to control cracking as: Smx = 2(CX + 10 ) + 0.25k1 Px [13] where c, is the maximum distance to the reinforcement, s is the spacing of the longitudinal reinforcement, and db is the diameter of the longitudinal reinforcement (Fig. 9). The adjustment factor for bond, k1, is 0.4 for deformed bars or 0.8 for plain bars or bonded strands. The longitudinal reinforcement ratio, Px' is: Px where [14] is the area of the longitudinal reinforcing steel and A is the gross area of the concrete. Similarly, the maximum transverse crack spacing is computed as: Smv 2(c +)+o.25k1 10 [15] 12 where c, is the maximum distance to the transverse steel, s is the stirrup spacing, db is the diameter of the stirrups, and Pv is the transverse reinforcement ratio: Pv = A [16] bws where A. is the area cross-sectional area of the stirrups. SI ;. I -: I_I, Fig. 9 - Crack spacing parameters ICollins and Mitchell, 1991]. The resulting shear force is expressed in terms of the concrete and steel contribution for any angle of principal strain, 0 as: V=(Avfv +fib)jdcot0 S The principal compressive strain is derived from Eqn. [1] as: [17] 13 52'cI1/1_ f2 f2max) [18] Longitudinal and transverse strains are derived based on Wagner's expression for the angle of inclination for the diagonal compression and Mohr' s circle [Wagner, 1929]. tan2O=2 [19] S From Mohr's circle: [20] From these two equations, the longitudinal and transverse strain can be found. The longitudinal strain is described by: tan2 0+ [21] l+tanO and the transverse strain is described by: 6= 2 tan2 0+ei [22] 1+tan2G Stirrup stress is computed assuming a bi-linear, elastic perfectly-plastic stressstrain relationship. Therefore, the stress in the stirrups,f, is linear until yielding occurs. After reaching the yield strain,j f:=EStfy , the stress remains constant at yield: [23] The MCFT can be used in two different ways. Either a detailed dual-sectional analysis or an approximate analysis may be conducted [Collins and Vecchio, 1988]. The detailed analysis (Fig. 10) involves dividing the specimen into sub layers, for which each sublayer must satisfy static equilibrium. For this type of analysis, the principal angle of stress, 14 which in MCFT is assumed to be equal to the principal angle of strain, is found at each sublayer. As a result, the principal angle of strain varies throughout the section height. The advantage of the detailed sectional analysis is that the results better predict the capacity and behavior, maintaining the compatibility and equilibrium conditions, but also requires more computational resources. LET- - Th> cross section shear stresses tongitudina strains principal compressive Stress traiectorses Fig. 10 - Detailed analysis [Coffins and Mitchell, 19911. The approximate method of sectional analysis using MCFT sets the angle of principal strain to be constant throughout the height of the cross-section. Thus every sublayer utilizes the same angle of principal strain in this approximate analysis (Fig. 11). di :T hUH fl cross section shear stresses longitudinal strains sectional torces Fig. 11 - Approximate analysis [Collins and Mitchell, 19911. Collins and Vecchio (1988) demonstrated that the approximate method using a constant angle of principal strain provides reasonable results compared with the detailed sectional analysis. Example moment and shear interaction diagrams comparing the approximate with the detailed analysis are shown in Fig. 12. 15 1000 Parabolic shear strain 800 a z Dual secti 800 400 0 200 0 o 200 400 800 MOMENT (kN-m) 800 i Fig. 12 - Comparison of detailed and approximate procedures [Collins and Vecchio, 19881. Several computer programs have been developed based on MCFT for analysis of reinforced concrete sections. Response 2000TM is the most well-known program and was developed by Evan Bentz under the direction of Michael Collins at the University of Toronto Bentz, 2000]. Previously developed programs include Shear (1990) and Response (1990). Shear calculates the resulting shear force, stress in the stirrups, and the principal stresses based on a value of principal strain and concrete section properties. Response does this but also incorporates a greater number of options for sectional analysis. The applied loading is provided as input and the program iterates for the resulting average stresses, average strains, and corresponding average principal angle. It is important to note that these are averages over the entire section as compared to results from Response 2000 TM Response 2000TM uses a detailed analysis approach while Response uses the approximate sectional analysis. However, as described previously, the approximate analysis approach has been shown to correspond well with the detailed analysis approach. 16 Analysis Approach The procedure outlined by Collins and Mitchell (1991) for the programs Shear and Response was used to create a spreadsheet for implementation of MCFT. The spreadsheet results are referred to here as MR. This allowed for a greater flexibility in iterating for solutions using the MCFT relationships. The procedure outlined for Shear was used to calculate the shear force as well as the average stresses and strains across the section by varying the principal tensile strain. The Shear program's output included average stresses and strains including the transverse strain, e, the longitudinal strain, s, the stress in the stirrups, f, principal compressive stress, f2, principal compressive strain, 62, principal tensile stress, f1, and any stresses and strains in prestressing tendons if included. Axial force equilibrium was obtained by iteratively changing the angle of the principal strain. Moment equilibrium was added according to suggestions in Appendix A of Collins and Mitchell (1991). A plane-section analysis method was conducted by numerically integrating concrete sublayers. Any concrete in tension was neglected and concrete in compression was analyzed using the parabolic model (Eqn. 3). Step-by-step implementation of the methodology is illustrated in the appendix. The output for this spreadsheet was compared with output from Response and with instrumented behaviors for three laboratory specimens: a T-beam with 12 in. stirrup spacing, T12 (Fig. 13 and Fig. 16), an inverted T-beam with 12 in. stirrup spacing, 1T12 (Fig. 14 and Fig. 17), and a T-beam with 24 in. stirrup spacing, T24 (Fig. 15 and Fig. 18). The spreadsheet results were found to correspond well with the Response 2000TM capacity 17 curve, the AASHTO-99 MCFT capacity curve, and the Response capacity curve (Figs. 19 -21). In general, MR provides results above Response and below Response 2000TM. This was likely due to improved convergence with the spreadsheet implementation, whereby the user, to a greater extent than with Response, could control the degree of precision of convergence. The analysis performed at low MIV ratios has been found to provide unreliable results. In addition, the procedure is only valid for monotonic loading and does not allow for unloading and reloading of cracked specimens. I in Clear_6in 6TV', 1.5 in Clear cover I #4 Grade 60 L=25ft-9in #4Grade4O 42in 3.75 / Spacing 12 in. #llGrade6O ° 14 in. 1.5 in Clear cover Fig. 13 Specimen T12 used for analysis correlation. 18 1 .5 in Clear cover #11 Grade 60 #4 Grade 40 Spacing 12 in. 42 in --! I 6in I/° __J 1 in Clear COVer Fig. 14 '#11 Grade 60 L=25ft-9in II 0 0 5jn _ 1.5 in Clear cover Specimen 1T12 used for analysis correlation. 1.5 in Clear cover n Clear cover Grade 60 L25ft-9in 42 in rade 40 24n o.c. 1 Grade 60 L24ft-6in Grade 60 L=25ft-9in lear cover ]4 In Fig. 15 Specimen T24 used for analysis correlation. 3(I114 P12 i0©12 in L,, North End P/2 4@6n I 13 5 in. 44 ri 44 I--- 'I 44 '1 fl 21 Wi - '1' -I- .- x=8liri. 18 1I][J y42in x:945,i. i=3O.5in. y1 l2in 144 in 4 in. dssp. sensor I <) Added stralo gage 0.5 in. disp. sensor 0 Strain gage (embedded) 5 in. string pot (on both sides of beam) 0 TIt sensor Note All equipment is on west side of beam unless otherwise noted. 0.5 in. disp. sensors located from nearest corner of stern. Fig. 16 T12 specimen laboratory setup and crack map. 144 in. / /7/ /7 Faikire mode: Shear Compression Peak Load: 378 kips. Widest aad; 0.109 ii. North End P/2 7 in 14 1.1+ 135,n I J 1O©12 in. 4 44ri + I 44wi 1-r-1---r--r-j-r r 44in. 1 -t 2 -i r I I I P/2 I 1O©12 in. 14 In k + 7 ii in. r r I I cfiFrr3r L1 48in. x62.5in. x93.5ri. yl9 75in y2Oin. y22.25. x:78ifl. y 16 5in IJEI ; x=88.5in iLl // // x57.75in y=l5in. / // x=49.75in. yrOri. 12 ft. 4 in disp sensor 0.5 in. disp. sensor 0 Strain gage (embedded) 12 ft () Added strain gage wi. string pot (on both sides of beam) Failure mode: Shear Tension/Anchorage Peak Load: 358 kips lit sensor Note: All equipment is on west side of beam unless otherwise noted. 0.5 Ni. disp. sensors located from nearest corner of stem tJ 0 Fig. 17 IT1 2 specimen Jaboratory setup and crack map. P/2 ++ P/2 4O6m North End 6in__________________________ in. xr66.51I y20 75 _J[_J ; ,//i//// l2in x-85n yr12 5m xrlO7.75in =24 y=30 75 X661n ysi 6tn x52 5ri y1 3 5 I 144r I - ------- ///4/ / 144in __________________________ - 4in.ClipGage 4 in. disp sensor (> Added strain gage Failure mode; shear-compression 0.5 in. asp. sensor 0 Strain ge (embedded) 5 in. stnng pot (on both sides ci beam) Pe* load: 241 ks Widesi cradi. 0.09Th. Tit sensor Note: M equipment is on west side ci beau unless otherwise noted Eidemal strain gages located on east side of beam I') Fig. 18 T24 specimen laboratory setup and crack map. 22 M-V Capacity Surface 2T 12 stirrup spacing= 12 220 200 180 160 140 120 100 80 - Ultimate Capacity from Response 2000TM Ultimate Capacity from AASHTO-99 MCFT 60 *-* T12 Failure M/V S. Ultimate Capacity from MR 40 A-A Ultimate Capacity from Response 20 0 250 500 750 1,000 1,250 1,500 1,750 2,000 2,250 2,500 2,750 3,000 M (k-ft) Fig. 19 - M-V capacity for specimen T12. M-V Capacity Surface 21T12 stirrup spacingl2" L 2 60 40 cn 20 00 Ultimate Capacity from Response 2000w -+ Ultimate Capacity from AASHTO-99 MCFT 80 ** lTl2 Failure M/V 60 S. Ultimate Capacity from MR 40 A-A Ultimate Capacity from Response 20 0 250 500 750 1,000 1,250 1,500 1,750 2,000 2,250 2,500 2,750 3,000 M (k-ft) Fig. 20 - M-V capacity for specimen 1T12. 23 M-V Capacity Surface 10T24 stirrup spacing24" 160 140 120 iE.II 0. . 80 > 60 - Ultimate Capacity from Response 2000w e-. Ultimate Capacity from AASHTO-99 MCFT 40 *-* T12 Failure MN 20 £4 Ultimate Capacity from MR Ultimate Capacity from Response 0 0 250 500 750 1,000 1,250 1,500 1,750 2,000 2,250 2,500 2,750 M (k-ft) Fig. 21 - M-V capacity for specimen T24. Amplijication Factor for Yielding of the Stirrups Predicting the moments and shears that cause yielding of stirrups is required to determine the potential for low-cycle fatigue on the stirrups. Steel is fatigued when subjected to repeated yielding. Therefore, it is important to be able to predict the moments and shears that may cause yielding of stirrups in a reinforced concrete girder. To estimate the onset of yielding in the transverse reinforcement, an amplification factor was used to relate the average transverse strains over the section to the maximum stirrup strain at a diagonal crack location (Fig. 22). 24 erage Transverse Strain Maximum Transverse Strain Fig. 22 - Average versus maximum transverse strain. The MCFT analysis uses the average strains over a section height, which was taken to be the section of concrete between the compression and flexural reinforcement, referred to here as DV. The transverse strain distribution would be better represented by a parabolic shape over the actual shear depth, geometry and calculus. h. The amplification factor is derived from simple The relationship between the average strain over the whole section and the maximum strain of a corresponding parabolic distribution was found to be: 8parabola_max = where h, 3 DV 2h 8average [24] assumed to extend from the flexural tensile reinforcement to the edge of the deck/stem interface for a T-section is: h = d tdeck [25] and h for an inverted T-section, which was assumed to be the distance between the flexural tensile reinforcement and the edge of the compression zone, a, is: h=da [26] 25 The depth of the compression zone, a, is computed based on an equivalent rectangular stress block as: a= fA [27] O.85f' b where j is the stress in the reinforcement, and A is the cross-sectional area of the reinforcement. A potential limitation of using this strain amplification approach arises in high moment to shear (MIV) ratios. At high M/V ratios, the strain in the flexural steel at the bottom of the section may increase significantly permitting larger transverse strains at the level of the reinforcement. This may distort the strain distribution. The new shape, containing an offset at the bottom where the flexural strains have increased, causes the real strain distribution to approach the average strain distribution. Therefore, for high MN ratios, the amplification factor may be less reliable (Fig. 23). High Strain Due to Flexure Fig. 23 - Average versus maximum strain at high M/V. 26 Reduced Cracking Stress The stress at first cracking used by Response was calculated using Eqn. 8. This is the stress at initial cracking, when concrete material that was previously uncracked begins to crack and thus soften. However, this value of cracking stress no longer functions for the onset of material softening for a concrete member that already contains diagonal cracks. Reloading of a cracked specimen induces material softening behavior once crack surfaces have decompressed. The beam load increases without an increase in strain until a new, reduced, value Ofjr is reached. In order to account for this phenomenon, laboratory data from specimens that are typical of Oregon's 1950's vintage reinforced concrete deck girder (RCDG) bridges were analyzed. The load versus strain data collected from strain gages placed on stirrups during the tests to failure were used to assess concrete softening behavior. For the laboratory specimens, the value for se,. (the onset of material softening behavior on the reloading branch) tended to be in the range between 33 percent and 50 percent of the original er,. value with 33 percent being the more typical for strain gages near diagonal cracks. Therefore, cracking stress of 1/3 the initial value was used to analyze the cracked reinforced concrete sections for reloading response prediction (Figs. 24-26). 27 Shear vs. Stirrup Strain @ 4' from Support Specimen 2T12, stirrup spacing=12" North Side 200 Strain Gage First ()acking @ V=100 kips Softened R)st-Q'acking @ V=33 kips 175 150 : 125 gioo 200 0 400 600 800 1,000 1,200 1,400 1,600 1,800 Stirrup Strain (nc) Fig. 24 - Cracking strain, cr, determination of specimen 2T12 north. Shear vs. Stirrup Strain @ 4 from Support Specimen 2T1 2, stirrup spacing=1 2" South Side 200 175 Strain Gage - First Cracking @ V80 kips - Softened Rst-Cracking @ V=40 kips 150 125 100 50 25 (1 0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 Stirrup Strain (sic) Fig. 25 - Cracking strain, Ecr, determination of specimen 2T12 south, 28 Shear vs. Stirrup Strain @ 4 from Support Specimen 21T12, stirrup spacingl2 South Side 200 175 150 125 Strain Gage - First cracking © V108 kips - Softened Fkst-Cracking © V33 kips I 200 400 600 800 1,000 1,200 1,400 1,600 1,800 Stirrup Strain (nc) Fig. 26 - Cracking strain, Ecr, determination of specimen 21T12 south. Results V-es Prediction Laboratory data were employed to verify the validity of using reduced cracking stress for analyzing the effects of reloading a cracked reinforced concrete section. Three laboratory specimens were evaluated that were characteristic of vintage cracked 1950's RCDG Oregon bridges. Two T-beams with stirrup spacings of 12 in. and 24 in. as well as one inverted T-beam (IT) with stirrups spaced at 12 in were investigated. The softened cracking stress was taken as one-third of the original cracking stress, as discussed previously. The results of this analysis corresponded well with the reloading branches of the laboratory specimens demonstrating that the reduced cracking stress provided a suitable modification to model the reloading behavior of cracked concrete specimens (Fig. 27 46). The model for reloading based on MCFT, adjusted with a reduced cracking stress, can be seen as the dashed curves on each plot. In some cases, the curve has been shown both at the origin and offset to a distance where softening behavior is more readily evident. In addition, the amplification factor from Eqn. 24 was applied to the model based on MCFT (spreadsheet MR) and compared with the laboratory data. In general, the results showed that the amplified models corresponded more reasonably to the initial loading branch, while the unamplified model provided a better correlation with the softened reloading branches after cracking. A possible explanation is that the stirrups, on initial loading, are fully bonded. Therefore, the strains vary locally throughout the height of the 30 stirrup, reaching maximum magnitude at the crack location. This corresponds well with the purpose of the amplification factor, which is to modify the average transverse strain to account for the maximum transverse strain over the section. On the reloading branch after cracking, the concrete softening results in diminished bond to the steel. With less bond to cause local variations in strain, the strain over the stirrup height approaches the average. Stirrup strains for a fully unbonded stirrup are constant (average) over the stirrup height. Consequently, the strain in the stirrup for the reloading branch is predicted more realistically by the average strain over the section height (the unamplified model) due to reduced bond between the softened concrete and the stirrup. One additional factor affecting the degree of correlation between the data and the MCFT model (spreadsheet MR) is the location of the strain gage relative to the crack(s). The model assumes that the strain is measured at the crack location. Comparing the model predictions with a strain gage located at the crack produces good results. Conversely, the comparison of the model predictions with a strain gage located a distance away from the nearest crack shows results that are not as well correlated. An example of this can be seen in Fig. 33. 31 Shear-Stirrup Strain © M/V4 and S12" 3T12 Precrack South Side 20C 190 180 170 160 150 140 - - Cl) a. G' LL. c 0 120 110 100 90 80 . U) 60 50 40 30 20 10 n 0 300 600 900 1,200 1,500 1,800 2,100 2,400 2,700 3,000 Stirrup Strain (J.Le) * * Strain Gage yield strain=1 760 rvR rvR j.IE c arrplified by 1.5 arrplified by 3*/(2h)_1 .63 VR&rlI3cr_original Rr=l/3Ecr_origjn offset (for derronstration) Fig. 27 - V- at south side, M/V=4 for specimen 3T12. WAW Fig. 28 - Strain gage location relative to crack at south side, M/V=4 for specimen 3T12. 32 Shear-Stirrup Strain @ M/V4 and S=12' 2T1 2 South Side L 1 1 1 (0 0. ai 0 U. I- G) U) 'I I 0 300 600 900 1,200 1,500 1,800 Stirrup Strain (t) * * Strain Gage yield strain=1 760 p fflMR MR arTplif led by 1.5 MR c arrpllf led by 3*DV/(2h)....1 .63 - MR Ir1'3Ecr_originaI Fig. 29 - V- at south side, M/V=4 for specimen 2T12. IIIrI: Fig. 30 - Strain gage location relative to crack at south side, M/V=4 for specimen 2T12. 2,100 33 Shear-Stirrup Strain @ M/V4 and S12" 2112 North Side 2" U 2 25 2 10 95 80 65 (j 3: Wi '-1 0 1I 0 . Cl) 0 250 500 750 1,000 1,250 Stirrup Strain 1,500 1,750 2,000 2,250 (l.L) * * Strain Gage yield strain=1 760 (> MRcarrplified by 1.5 UIffiMR < MRcarrplified by 3*Dv/(2h)163 MR r'13cr_orIginaI MREcrl/3cr_originai offset (for derronstration) Fig. 31 - V- at north side, M/V=4 2T12 North I( I I Fig. 32 - Strain gage location relative to crack at north side, M/V=4 for specimen 2T12. 2,500 34 Shear-Stirrup Strain @ MN=4 and S12" 2 ITI 2 South Side -'U 1 70 1 30 1 50 1 40 1 30 ' 0. 0 U(5 w C/) 501 f: I;' 0 150 300 450 600 750 900 1,050 1,200 Stirrup Strain (.tc) Strain Gage yield strain=1 760 rvRarrplifled by 1.5 rvRc arrplified by 3*DV/(2h)....1 44 - IVR Ccrl /3Ccr_orig inal FvR CCr=l I&cr_orig inal offset (derronstration) Fig. 33 - V-E at south side, M/V=4 for specimen 21T12. I -------I I I I I C:.., '. Fig. 34 - Location of strain gage at south side, M/V=4 for specimen 21T12. 1,350 35 Shear-Stirrup Strain @ MN=4 and S24" I 0T24 South Side 1 DI 14 13 12 11' j 10' 0 a 0' 0 U- J 3 2 1 0 300 600 900 1,200 1,500 Stirrup Strain * * mm 1,800 2,100 2,400 (j.t) Strain Gage yield strain rvREan-pliIied by 1.5 rvR c an-plified by 3*DV/(2h).1 .63 - IVREcrlI3rcr_originai FvR Cr=lI3Ecr_originaI offset (for derronstration) Fig. 35 - V-E at south side, M/V=4 for specimen 10T24. wAfl\ Fig. 36 - Location of strain gage at south side, M/V4 for specimen 10T24. 2,700 36 Shear-Stirrup Strain © M/V6 and S12" 2T1 2 North Side 22C 200 180 160 -'I--. 140 . d.- 120 I- 0 u. 100 I- 80 Cl) 60 40 20 C 0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 Stirrup Strain () * * K) Strain Gage yield strainl 760 E K) MR amplified by 1.5 MRcarrplified by 3*/(2h).....163 MRrl/3cr_origina MRrl/3cr_originai (for derronstration) Fig. 37 - V-E at north side, MIV=6 for specimen 2T12. I Fig. 38 - Strain gage location relative to crack at north side, M/V6 for specimen 2T12. 2,000 37 Shear-Stirrup Strain © M/V6 and S12" 2T1 2 South Side 'U I 1 30 I 1 30 1 50 1 U) a. I 30 20 10 I )0 1 1 w 0 UI- 5 0 30 U) 30 50 30 20 10 0 300 600 900 1,200 1,500 1,800 2,100 Stirrup Strain () * * mm yieId=1760 O 0 IVR c arrplified by 1.5 X X R c arrplif led by 3*DV/(2h)1 .63 Strain Gage R Ccr'3Cr_OriginaI -M Fig. 39 - V- grl/3Ecr_originaI offset (for derronstration) at south side, MIV=6 for specimen 2T12. I Fig. 40 - Strain gage location relative to crack at south side, MIV=6 for specimen 2T12. Shear-Stirrup Strain @ MN6 and S12" 2 ITI 2 South Side U'-) 1 90 I 80 I 1 I 1 C', 0. 0 U. I- w .c Cl) 0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 Stirrup Strain () * * Strain Gage yield strain=1 760 pz MRcarrplified by 1.5 FvRcarrphuied by 3*/(2h)...144 - rvR &r=l/&cr_originaI VRrl/3Fr_originaI Fig. 41 - V- offset (for derronstration) at south side, MIV=6 for specimen 21T12. Fig. 42 - Location of strain gage at south side, M/V=6 for specimen 21T12. 2,000 39 Shear-Stirrup Strain @ MN=6 and S12" 21T1 2 North Side J,.J 1 1 1 (ii e 0 1 30 70 50 50 40 30 20 10 DO U- .z U) ic n 0 400 800 1,200 1,600 2,000 2,400 2,800 Stirrup Strain () * * Strain Gage yield strain=1 760 pz fflmR O 0 rVRcarrphlied by 1.5 FvRcarrplif led by 3*[/(2h)144 cr I3r ofig irial RrlI3r_originai offset (for derronstration) Fig. 43 - V- at north side, M/V=6 for specimen 21T12. Fig. 44 - Location of strain gage at north side, MIV=6 for specimen 21T12. 3,200 40 Shear-Stirrup Strain @ MN6 and S=24" I 0T24 North Side U') 1 40 I 30 1 1 U) .x e 0 U- I- 20 V 10 70 60 C, 50 40,.. f 30 20 10 n 0 400 800 1,200 1,600 2,000 2,400 2,800 Stirrup Strain (.tc) Strain Gage yield strain IvRarrplified by 1.5 rvR c ariplified by 3*DV/(2h)....1 .63 - FVR Erlt3Ecr_onginaI 'Rrl/3Ccr_originai Fig. 45 - V-E offset (for derronstration) at north side, MJV=6 for specimen 10T24. Fig. 46 - Location of strain gage at north side, MIV6 for specimen 10T24. 3,200 41 M- V Interaction Yield Surfaces Low-cycle fatigue of reinforced concrete girders may occur when load effects (M and V) combine to produce yielding of stirrups. The moment-shear (M-V) interaction yield surface is needed to perform a low-cycle fatigue evaluation. Using the MCFT with the adjustment factor for cracking stress, as well as an amplification factor to achieve the maximum strain in the section based on the average strains, an M-V interaction curve at yielding of the stirrups was developed. The yield M-V interaction diagrams were developed for the three laboratory specimens and two critical sections of the McKenzie River Bridge, located in Lane County, Oregon. The ratio between yielding and capacity values were calculated with respect to the AASHTO-99 MCFT capacity curve. Yielding was conservatively based on amplified strains corresponding to well bonded stirrups. The amplified yield curve exhibited lower moments and shears to produce the yield condition than that based on average strains. Through analysis of the M-V interaction yield surfaces, it was found that for higher M/V ratios, there is a point at which the stirrups do not yield before capacity is reached for the section. As a result, sections at high M/V ratios do not appear to be prone to low-cycle fatigue. Laboratory Specimens The three laboratory specimens analyzed were the T12, IT 12, and T24. On average, the shear and corresponding moment to cause yielding in the stirrup was approximately 67% of the ultimate capacity as predicted by the AASHTO-99 MCFT for the T12 specimen for MJV ratios of 3 to 14, approximately 69% for 1T12 for MJV ratios of 3 to 10, and for T24 was about 63% for MN ratios 42 from 3 through 20. These results are displayed in Tables 1 to 3, and graphically displayed in Figs. 47 to 49. M-V Yield Surface T12 stirrup spacingl2" I 1 I 1 I 0. > 0 0 * * Ultimate Capacity from Response 2000TM Ultimate Capacity from AASHTO-99 MCFT T12 Failure MN A A MR cr=1/3cronginaI 30 MR Er=l'&cr onginal AMPLIFIED 15 0 0 300 600 900 1,200 1,500 1,800 2,100 2,400 2,700 M (k-ft) Fig. 47 - M-V yield surface and predicted capacities for specimen T12. Table 1 - Shear to produce yield as % of AASHTO-99 MCFT for specimen T12. MN 1 2 3 4 6 8 10 12 14 Moment at V,,, (k-ft) V, kips % V by AASHTO-99 MCFT 44%* 80 55%* 200 100 400 133.3 73% 500 125 70% 625 104 61% 800 100 61% 1000 100 64% 1200 100 68% 1350 96 69% *Low MN values not well predicted by method. 80 43 M-V Yield Surface IT 12 stirrup spacing=12" lu 1 95 1 1 1 1 35 .1 20 > 90 /' 05 60 0 0 Ultimate Capacity from Response 2000 Ultimate Capacity from AASHTO-99 MCFT * * 1T12 Failure M/V A A MR Fr=lI3F.tronginaI 45 30 MR cr=l/3Ccr onginal AMPLIFIED 15 0 0 300 600 900 1,200 1,500 1,800 2,100 2,400 2,700 M (kip-ft) Fig. 48 - M-V yield surface and predicted capacities for specimen 1T12. Table 2 - Shear to produce yield as % of AASHTO-99 MCFT for specimen 1T12. MAT Moment at V, (k-ft) Vs,, kips % V, by AASHTO-99 MCFT 1 120 2 300 400 500 700 900 120 150 133.3 125 116.7 112.5 110 65%* 81%* 3 4 6 8 10 1100 *Low MN values not well predicted by method. 72% 69% 68% 68% 70% 44 M-V Yield Surface T24 stirrup spacing=24" .1 rc 1 40 ____________ 1 30 * * 1 1 1 Il . > 80 70 60 50 40 0 30 0 * * 20 Ultimate Capacity from Response 2000TM Ultimate Capacity from AASHTO-99 MCFT T24 Failure M/V MR Fcr1/3Ecronginal MR FcrlI3Ecr original AMPLIFIED 10 0 0 300 600 1,500 1,800 2,100 2,400 M (k-ft) MR with original cracking strain (r original) reach ultimate before yield. Therefore, ultimates are plotted for rorigIna. 900 1,200 2,700 Fig. 49 - M-V yield surface and predicted capacities for specimen T24. Table 3 Shear to produce yield as % of AASHTO-99 MCFT for specimen T24. M/V 3 4 6 8 10 12 14 20 Moment at Vs,, 250 350 500 625 740 875 1000 1400 (k-ft) Vs,, kips 83.3 87.5 83.3 78.1 74 72.9 71.4 70 % V, by AASHTO-99 MCFT 60% 64% 63% 62% 61% 62% 63% 69% 45 McKenzie River Bridge The two sections of the McKenzie River Bridge analyzed were an IT and T beam, sections A and B (Fig. 50) with stirrups spaced at 9 in. and 12 in., respectively (Fig. 51 and Fig. 52). The McKenzie River Bridge section tended to yield at higher percentages of the AASHTO-99 MCFT ultimate capacity than the laboratory specimens. For M!V between 1 and 4, the McKenzie River Bridge section A yielded on average approximately 81% of AASHTO-99 MCFT capacity while for the same range of MIV ratios, section B on average yielded about 86% of ultimate capacity predicted by AASHTO-99 MCFT. This data can be seen in Tables 4 and 5, and is plotted in Figs. 53 and 54. A 42ft 5Oft B 112.5ft ' 5Oft 5Oft Fig. 50 Sections A and B from McKenzie River Bridge. 46 3 #11 #4 50.0 9.00 in 4- #11 Fig. 51 - McKenzie River Bridge at section A. 108.0 4- #11 r 03 t #4 12.00 in 3- #11 4i4 Fig. 52 - McKenzie River Bridge at section B. 47 M-V Yield Surface Mckenzie Bridge, Section A stirrup spacing=9" uU 180 160 140 1 > 0 o 0 Ultimate Capacity from Response 2000 Ultimate Capacity from AASHTO-99 MC MR ftrl/3Ecr_originap MR Fcrl/3Ecr orlgina! AMPLIFIED 20 0 0 150 450 300 600 900 750 1,050 1,200 M (kip-ft) Fig. 53 - M-V yield curve for McKenzie River Bridge girder section A. Table 4 - Shear to produce yield as % of AASHTO-99 MCFT for McKenzie River Bridge girder section A. M/V Moment at V,,, (k-fl) V,,, kips 1 115 228 115 340 440 113.3 110 2 3 4 114 % V11 by AASHTO-99 MCFT 81% 80% 80% 84% M-VYieId Surface Mckenzie Bridge at Section B stirrup spacing=12" 1 8C 1 6C 1 4C 1 2C i bc . > 8C 6C 4c o o Ultimate Capacity from Response 2( Ultimate Capacity from AASHTO-99 * * MR 0 0 MR 2C 0 100 rl'3Ecr_originaI cr=lI&cr original 200 300 AMPLIFIED 400 500 600 700 800 900 M (kip-ft) Fig. 54 M-V yield curve for McKenzie River Bridge girder section B. Table 5 - Shear to produce yield as % of AASHTO-99 MCFT for McKenzie River Bridge girder section B. MN Moment at Vs,, (k-ft) 1 95 180 2 3 4 260 340 V, kips % V by AASHTO-99 MCFT 95 90 86.7 85 89% 85% 82% 86% As can be seen from comparing the yielding shear level with the capacity, the laboratory specimens yielded between approximately 60% and 73% of the nominal AASHTO-99 MCFT capacity, while the sections analyzed for the McKenzie River Bridge yielded between about 80% and 89% of the nominal AASHTO-99 MCFT capacity. 49 The load effects produced by service level trucks combined with dead load at the sections of interest were determined and compared with the predicted yield shear force and moment. To estimate service level truck loads at the McKenzie River bridge, weigh-inmotion data was used which had been collected for southbound truck traffic on interstate highway I-S during the period of one year, that being 2003. The truck data was collected at the Wilbur weigh station, located just North of Roseburg, OR. In total, 900,000 trucks passed through Wilbur weigh station going southbound in 2003 [Daniels, 2004]. Of those, approximately 14,000 trucks were classified by ODOT as being permit table 3, 4, and 5 vehicles [Daniels, 2004], plotted in Fig. 55 and Fig. 56. These are vehicles that carry loads which exceed the legal limits. Load effects calculated using AASHTO LRFD live load distribution factors combined with AASHTO impact factors [Daniels, 2004] caused some trucks to induce yielding of the stirrups. For the location of section A, approximately 5 of the 14,000 permit table 3, 4, and 5 trucks caused yielding in the stirrups. 50 1 1 1 ti) I65 Cl) E E (5 0 100 200 300 400 500 700 600 800 900 1000 Moment (kip-ft) Fig. 55- McKenzie River Bridge at section A using AASHTO LRFD LL distribution factors. 1 1 - - - - - - - 10 - AASHTO-LRFD 98): DFv= 0 884 1 U) 0. .) * 30 Dead ad Moment àidShei: -2 Q3 ft-kips,V01= 22.42 '5 w 70 MDL = Cl) DO d=41 2 in, b'430 in' . E E '5 = , = YDL $0 'S 30 * AASHTO-MCFT 99 M-V Interaction Yield Points (amplified) WIM Vehicles (Permit Tables 3 4 5) 100 150 20 10 0 50 400 450 Moment (kip-ft) 200 250 300 350 I 500 550 600 650 700 Fig. 56- McKenzie River Bridge at section B using AASHTO LRFD LL distribution factors. 51 This procedure may be used to determine the yield moments and shears for any bridge. However, the iterative process of finding these moment and shear combinations may be bypassed if conservatively one were to choose a percentage of yielding to use as an initial screening for possible low-cycle fatigue damage accumulation. Additional analyses are needed for other bridge spans and indeterminacies to develop a reliable recommendation, but based on the analyses performed, yielding occurs between 60% and 89% of ultimate. A percentage within this range may be acceptable for roughly estimating low-cycle fatigue potential. 52 Conclusions The Modified Compression Field Theory (MCFT) is currently used in the AASHTO LRFD specification to compute the shear capacity of reinforced concrete and prestressed beams and beam-columns. It uses strain compatibility and equilibrium to predict the capacity of sections under combined flexure and shear. Currently, the method is used to analyze beams assuming an initial uncracked condition. However, analysis of cracked sections is needed to evaluate potential for LCF in bridge girders. MCFT was used to predict the behavior of girders with existing diagonal cracks. The constitutive relationships were modified to realistically predict stirrup strains for girders in the previously cracked condition. Analyses were performed for laboratory specimens and the method was applied to an existing bridge with diagonal cracks. Based on these analyses, the following conclusions are presented: Stirrup strain behavior for reloading of previously cracked girders can be reasonably predicted by MCFT and incorporating an adjustment factor to the cracking stress, fcr Transverse strains in the cross-section were amplified to account for higher stresses in the stirrups at diagonal cracks to provide a better correlation with experimental results. For high MIV ratios, there is a limit at which the stirrups do not yield when the section reaches capacity. There is no risk for low-cycle fatigue at these sections. For very low M/V ratios, the MCFT does not predict the shear and moment capacity very well. 53 . Assessment of low-cycle fatigue for a bridge requires comparison of the load effects from service level trucks and dead load with the yielding MIV interaction curve of the section. The analysis technique described provides a method to predict this threshold. 54 References AASHTO. (2003). "LRFD Bridge Design Specifications." Ed., Washington, D.C. ACI Committee 318-83. (1983). Building Code Requirements for Structural Concrete. American Concrete Institute, Detroit, Michigan. ACI Committee 318-02. (2002). Building Code Requirements for Structural Concrete. American Concrete Institute, Farmington Hills, Michigan. Bentz, E. (2000). "Sectional Analysis of Reinforced Concrete Members." Ph.D. Thesis, Department of Civil Engineering, University of Toronto, Toronto, Canada. Collins, M.P. and Mitchell, D. (1991). Prestressed Concrete Structures. Prentice Hall, Englewood Cliffs, New Jersey. Collins, M.P. and Vecchio, F.J. (1982). "The Response of Reinforced Concrete to InPlane Shear and Normal Stresses." Publication No. 82-03, Department of Civil Engineering, University of Toronto, Mar., 332 pp. Collins, M.P. and Vecchio, F.J. (1986). "The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear." ACT Journal, Vol. 83, No. 2, Mar-Apr., pp 219-23 1. Collins, M.P. and Vecchio, F.J. (1988). "Predicting the Response of Reinforced Concrete Beams Subjected to Shear Using Modified Compression Field Theory." ACI Structural Journal, May-June, pp. 258-268. Daniels, T. (2004). "Reliability Based Bridge Assessment using Modified Compression Field Theory and Oregon Specific Truck Loading." M.S. Thesis, Department of Civil, Construction, and Environmental Engineering, Oregon State University, Corvallis, OR. Felber, A.J. (1990). "Response: A program to determine the load-deformation response of reinforced concrete sections." M.S. Thesis, Department of Civil Engineering, University of Toronto, Toronto, Canada. Kani, G.N.J. (1964). "The Riddle of Shear Failure and Its Solution." ACI Journal, Vol. 61, Apr., pp. 44 1-467. MacGregor, J.G. (1997). Reinforced Concrete Mechanics and Design. 3rd Ed. Prentice Hall, Upper Saddle River, New Jersey. Response. (Version 1.0) [Computer software]. (1990). Shear. (Version 1.0) [Computer software]. (1990). 55 Ritter, W. (1899). "Die Bauweise Hennebique." (Construction Techniques of Hennebique). Schweizerische Bauzeitung, ZUrich, Feb. Ugural, A.C. and Fenster, S.K. (1995). Advanced Strength and Applied Elasticity. 3 Ed. Prentice Hall, Upper Saddle River, New Jersey. Wagner, H. (1929). "Ebene Blechwandträger mit sehr dUnnem Stegblech" (Metal Beams with Very Thing Webs). Zeitschrfi für Flugtechnik und MotorIuftschfJahr, Vol. 20, Nos. 8 to 12. 56 Appendix MCFT Procedure 57 Following is the algorithm for shear, moment, and axial load as outlined by Collins and Mitchell in Prestressed Concrete Structures (1991) for the program Shear (1990). LOOP FOR ITERATING ON Step 1: Choose a value of e at which to perform the calculations. LOOP FOR ITERATING ON 0 Step I: Estimate 0 Step H: Calculate the crack width, w, and the shear along the crack, v1 LOOP FOR ITERATING ON VERTICAL EQUILIBRIUM Step a: Estimate f Step b: Calculate the principal stress f1 Step C: Calculate the shear, V, as a resultant of the shear stress. Step d: Calculate the average compressive stress, f2 Step e: Calculate fimax, the maximum compressive stress Step f: Check that f2 f2. If f2> f2m,x, the concrete crushes and the solution is not possible. Return to Step 1 and choose a 58 smaller 61. Step g: Calculate compressive strain, 62 Step b: Calculate the average longitudinal and transverse strains, e and s. Step i: Calculate the stress in the stirrup utilizing the transverse strain. Step j: Check if the stirrup stress matches the initial guess for the stirrup stress. If the values do not match, return to Step a with an improved estimate of the stress in the stirrup. END LOOP FOR ITERATING ON VERTICAL EQUILIBRIUM Step III: Calculate the axial force, N, on the section resulting from the concrete stresses in the effective shear area, bjd, and the moment caused by N. tan 0 M = Ny where y is the distance from the center of the effective shear area to the moment axis. [All [A2] 59 Step IV: Using a plane-section analysis procedure, find the stress resultants for the section corresponding to the longitudinal strain, and a curvature, b, that corresponds with the applied moment and longitudinal strain. N = f(&,Ø) = Ng,.oss Nbjd = f(s, q) = [A3] Mbjd {A4J Step V: Calculate the axial resultant and moments N=N + [A5] MM+M [A6] Step VI: Change the principal angle of strain 0 until the axial force resultant equals the applied axial load. END LOOP FOR ITERATING ON 0 Step VII: Check that the steel demand (right side of inequality) has not exceeded its strength (left side of inequality). (i i=1 + i=1 (, fi)4 f1bJd+[fi (f bjd _fV)Jtan2O [A7} 60 Step 2: Verify that the applied shear force equals the resultant shear force. If the two are not close enough, revise the estimate of Si. END LOOP FOR ITERATING ON