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Post-Yield Stiffness Effects on Moment Redistribution in Continuous Reinforced Concrete Beams By Pedro Silva, Ph.D., P.E. R W einforced concrete (RC) beams of the type shown in Figure 1 are commonly designed using ® moment redistribution principles. RC E continuous beams or plane frames may have any number of spans or boundary R restraints; the work presented in this ar- ticle is for a simply supported, two-span, L L continuous RC beam, but many of the A B C U conclusions can be extrapolated to other situations. In design, these continuous Figure 1: Two-span continuous beam under uniform loads. t h yrig T members are typically assumed to dis- play an elasto-plastic response, which Cop Theµrelationships of MR intensile ductility, φ, as a function of strain. terms deforming plastically at end B when the moment and curvature reach Mn and means that after yielding of the tension C of tensile strain and curvature ductility φy, respectively. After this stage, the in- steel any increase in stiffness due to are outlined in Figure 2. cremental uniform applied load on the strain hardening is neglected. In reality, Formulation of MR as a function of 7.5% will impose t ≤ 20% beam ≤ MR = 1000 inelastic rotations and 1 e beams subjected to large inelastic strain U curvature ductility capacity is presented curvatures at support B. The amount of levels may attain a significant post-yield n in terms of the moment curvature (M-φ) MR that the beam can sustain is computed stiffness, which has a strong effect on the i relationships and the statically indeter- as follows: R moment redistribution of continuous 1 z minate beam shown in Figure 3, which RC beams. is a simplified version for the analysis of the MR = 1 − 1 + 3λ μφ − 1 Equation 2 2 T In this article, the basics of moment a two-span beam shown in Figure 1. The redistribution are discussed as a function Modeling the inelastic response of the beam is uniformly loaded and is pinned g of curvature ductility capacity using finite S and fixed at ends A and B, respectively. beam in terms of Release 2 follows the element subroutines. The author further 1 bilinear1 M-φ relationship presented in a Under an increasing load, the beam will illustrates the principles of moment redis- deform elastically up to yielding and then MR = − 1 The beam φ − 1 Figure 3(a).+ 3λ + r μbegins to deform 3 tribution in the design of a two-span RC plastically at end Β when the moment plastically at end B. m continuous beam, including the potential 7.5% ≤ MR = 1000 reach M , and1 , respec- and curvature t ≤ 20% y φy For the nonlinear part of the analysis, effects of post-yield stiffness. tively. After this stage, the beam develops two released structures may be considered. Structural DeSign For Release 1 the beam is considered per- plastic rotations and curvatures that include Basics of Moment fectly plastic at end B, and in Release 2 the post-yield 1 stiffness and plastic hinge Redistribution the beam can be considered restrained 2 MR = 1 Following similar steps, in Release length. − 1 + 3λ μφ − 1 by a plastic rotational spring with the 2 the amount of MR that the beam can Sections 8.4.1 and 8.4.3 of ACI 318- stiffness, β, idealized in terms of the sustain is computed as follows: 05 state that the level of moment re- distribution (MR) that is permitted in a post-yield stiffness, r; initial stiffness, EI; 1 continuous RC beam is: and plastic hinge length as a function of MR = 1 − 1 + 3λ + r μφ − 1 3 Equation 3 beam span length, λL. 7.5% MR=1000εt 20% Equation 1 Modeling the inelastic response of the Equations 2 and 3 can be used to com- where εt is the level of strain in the ex- beam in terms of Release 1 follows the pute the amount of MR that a beam can treme tension reinforcement. As such, elasto-plastic idealization presented in sustain as a function of the plastic hinge this strain must be at least 0.0075 be- Figure 3(a). The beam is assumed to begin length, post-yield stiffness and curvature fore MR is permitted. The permissible levels of MR defined by Equation 1 are 25 25 design issues for structural engineers conservative, and results derived from Moment Redistribution (%) Moment Redistribution (%) this study show that strain levels will in 20 20 many cases fall significantly below 0.005, which violates the ACI 318-05 limit for 15 15 a tension-controlled design. Stipulated by Equation 1, the amounts of MR that 10 ACI 318 10 can be allowed in the design of continu- permissible Proposed 5 MR as a 5 MR as a ous RC beams are only expressed as a function of t function of µ φ function of tensile strains. Because of its 0 0 generality, the work presented in this ar- 0.005 0.010 0.015 0.020 0.025 1 2 3 4 5 6 7 8 9 10 ticle will evaluate MR in RC structures Steel Strain, t Curvature Ductility, µφ as a function of curvature ductility ca- (a) (b) pacity. Previously the author has derived an expression to obtain the curvature Figure 2: Moment redistribution (a) Function of εt, (b) Function of µφ. STRUCTURE magazine 18 January 2010 w From the moment-curvature analysis, the φy φp curvature ductility capacity of the section is nearly µφ≈6.5. From Figure 2(b) this ductility Elasto- L capacity translates into a MR capacity of Mu Plastic 15.7%. Comparatively, for Release 1 the MR A Structural Model B that the section can develop is 12.1%. On the Mn dw other hand, for Release 2 the two-span beam My / can now develop a much greater MR=61.2%. ® rEI θp This simple example clearly shows that the E actual MR that the beam can develop is sig- Moment Release 1: Perfectly Plastic nificantly higher than what is allowed by ACI Bilinear R = rEI L p 318. Figure 6 (page 20) shows the moment EI dw profiles for three cases. One curve shows the profiles considering the elastic design condition, φy U φu / θp another corresponds to r=0, and the third rep- Curvature, φ Release 2: Bilinear resents r=0.035. It is not apparent from these ht yrig T curves the salient differences between a design (a) M- Idealization p Co(b) Uniform Load, w that considers r=0 and the actual response of Figure 3: Basics of moment redistribution. the beam with r=0.035. C continued on next page ductility capacity. Obviously, these principles Some other trends of the MR levels presented of MR capacity only apply to the beam geom- in Figure 4 are as follows: (i) as the post-yield 12 in. e U etry presented in Figure 1. stiffness ratio increases, so does MR; (ii) as the n The permissible levels of MR in two-span plastic hinge length increases, so does MR; i continuous beams that correspond to the two (iii) the curve for r=0.00 and λ=0.01 follows R releases are depicted graphically in Figure 4. below the permissible MR curve computed The post-yield stiffness (r) and plastic hinge based on Equation 1, and depicted in Figure z #4 Stirrups T a length (λL) have a marked effect on the MR 2. The next section presents the effects that @ 6 in o.c. 21.6 in. 24 in. capacity of two-span continuous beams. It these trends have on the actual performance g S is envisioned that this same observation will of beams designed using MR principles. also apply to other continuous structures. a Design and #5 Top & Bottom m Performance Evaluation 40.0 As discussed, the levels of MR that can be MR Permissible Moment Redistribution (%) achieved in continuous beams depend strictly r = 0.00 & = 0.01 (a) Beam Cross-Section 30.0 on the plasticr rotation capacity of members Release 1 = 0.00 & = 0.02 Curvature Ductility, µφ r = In this = 0.04 at plastic hinges. 0.00 & section, a design ex- 0 1 2 3 4 5 6 7 8 20.0 r 0.00 & = to investigate the ample has been=established 0.08 200 r = 0.05 stiffness has effects that post-yield & = 0.01 on MR. r = parameters Reflecting the 0.05 & = 0.02 of Table 1, 160 Moment (kips-ft) 10.0 Release 2 required a beam with the cross-section design r = 0.05 & = 0.04 dimensions and=reinforcement layout shown r 0.05 & = 0.08 120 0.0 in Figure 5(a), which consists of 6-#5 (Grade 0.0 2.0 4.0 6.0 8.0 10.0 60) top and bottom bars. The moment- 80 Release 1: r = 0.00 Curvature Ductility, µφ curvature analysis for this section is presented 40 Release 2: r = 0.035 in Figure 5(b). The solid curve is the moment- curvature section analysis that is used to 0 MR Permissible evaluate the performance of the two-span 0.0000 0.0005 0.0010 0.0015 r = 0.00 & = 0.01 continuous RC beam under Release 2 with Curvature (1/in.) r = 0.00 & = 0.02 (b) Moment - Curvature Relations Release 1 r=0.035. The dashed curve is for the same r = 0.00 & = 0.04 evaluation under Release 1 with r=0. It is im- Figure 5: Cross Section Dimensions and r = 0.00 & = 0.08 portant to emphasize that in current practice, Capacity Analysis. r = 0.05 & = 0.01 r=0 is generally assumed for design. r = 0.05 & = 0.02 Release 2 r = 0.05 & = 0.04 Span length = 20 feet Steel bars required = 6-#5 (Grade 60) r = 0.05 & = 0.08 Uniform dead load = 900 plf Uniform Live Load = 1400 plf Dead load factor = 1.2 Live load factor = 1.6 .0 10.0 Figure 4: Moment redistribution versus ductility. Ultimate factored load = 3,320 plf MR per ACI (6-#5) = 15.7% for µφ Release 1: With r=0, MR = 12.1% Release 2: With r=0.035, MR = 61.2% Table 1: Design Parameters. STRUCTURE magazine 19 January 2010 200 1.4 Elastic Design Mu Bending Moment (kips-ft) 1.2 r = 0.00 Ratio of Demand/Capacity Elasto-Plastic r = 0.00 Design (r = 0) Mn 100 M'y 1.0 r = 0.035 0.8 ® 0 0.6 E 0.4 r = 0.035 M'y r = 0.035 0.2 R -100 0.0 0.0 0.2 0.4 0.6 0.8 1.0 U Location from end A to B (x/L) Plastic Rotations Tension Strains ht yrig T Cop Figure 6: Moment profiles for end spans A-B. Figure 7: Ratio of Demand versus Capacity. C Future Investigations Figure 7 shows the ratio of the plastic rotation This article presented some of the basics of Pedro Silva, Ph.D., P.E. and tensile strain demand versus capacity. moment redistribution principles and ap- e U (silvap@gwu.edu), is an For r=0, demand exceeds the section plastic plied them to a two-span continuous RC Associate Professor in the Civil n rotation and tensile strain capacity by a ratio beam. Results show that post-yield stiffness & Environmental Engineering i R of 1.02 and 1.3, respectively. For r=0.035, has a marked effect, an important observation Department at The George z there is a drastic decrease in the demand versus that should be investigated in further detail Washington University in capacity ratio to 0.10 and 0.25, indicating for structures that have a higher order of in- T Washington DC. His research that the degree of conservatism is on the order of 10. These ratios show that post-yield a determinacy. Issues of moment redistribution g for continuous beams with a number of spans interests include analysis and design of structures subject to S stiffness has a marked effect on the moment greater than two and plane frames will be un- seismic and blast loading. a redistribution of continuous RC beams. dertaken in the future.▪ m ADVERTISEMENT - For Advertiser Information, visit www.STRUCTUREmag.org STRUCTURE magazine 20 January 2010

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