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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




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                                                                                                       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




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                                                                                                       curvature ductility capacity is presented                                       curvatures at support B. The amount of
                                                        levels may attain a significant post-yield


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                                                                                                       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:




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                                                        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




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                                                                                                                                   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.




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                                                                                                                                                             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
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        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;


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        continuous beams that correspond to the two                           (iii) the curve for r=0.00 and λ=0.01 follows




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        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


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        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




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                                              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.




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                                                                                   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.▪



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                                                                 STRUCTURE magazine                        20                       January 2010

				
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