6 THEORETICAL STUDIES ON MOMENT REDISTRIBUTION 447

Transcription

1 HPTER 6 THEORETI STUDIES ON OENT REDISTRIBUTION ONTENTS 6 THEORETI STUDIES ON OENT REDISTRIBUTION Introduction iterature Review oment Redistribution oncept Plastic Hinge pproach oment Redistribution in National Standards Fundamental oncept of Fleural Rigidity () pproach Simplified Fleural Rigidity () pproach Journal Paper: oment Redistribution In ontinuous Plated R Fleural embers. Part Fleural Rigidity pproach Further Discussions on Simplified pproach Derivation of athematical Equations for Beams with Different Hogging and Sagging Stiffnesses One End ontinuous Beam Subjected to Point oad Both Ends ontinuous Beam Subjected to Uniformly Distributed oads omparison Between Eperimental and Theoretical Results Test Series S (Specimens With Eternally Bonded Plates) Test Series NS and NB (Specimens With NS Strips) Parametric Studies Based on Simplified pproach Journal Paper: oment redistribution parametric study of FRP, GFRP and steel surface plated R beams and slabs inear Fleural Rigidity () pproach Journal Paper: oment redistribution in FRP and steel plated reinforced concrete beams Further Discussions on inear pproach Derivation of Equivalent One End ontinuous Beam Subjected to Point oad Both Ends ontinuous Beam Subjected to Uniformly Distributed oads

2 6.5.. omparison Between Eperimental and Theoretical Results Test Series S (Specimens With Eternally Bonded Plates) Test Series NS and NB (Specimens With NS Strips) Summary References Notations

3 Theoretical Studies on oment Redistribution 6. INTRODUTION The eisting design guidelines (fib ; oncrete Society ) tend to neglect any moment redistribution occurring in plated structures. However, the eperimental studies presented in hapter 5 and s performed by other researchers, such as ukhopadhyaya et al. (998), El-Refaie et al. () and shour et al. (4), clearly showed that significant amounts of moment redistribution can be obtained in both eternally bonded (EB) and near surface mounted (NS) plated beams. Therefore, new approaches are required to analyse the moment redistribution behaviour of continuous plated members that takes into account premature debonding failure prior to concrete crushing. The member ductility design of reinforced concrete (R) continuous beams or frames often uses the plastic hinge concept (Darvall and endis 985; Barnard 964) and the neutral ais depth factor (ku), which is common to most national standards (i.e. S6), to quantify both collapse and the associated ability to redistribute moment within a continuous beam prior to collapse. These approaches work well in unplated reinforced concrete structures as the material ductility of the steel tension reinforcing bars, that is their strain capacity, can be assumed to be very large which ensures that compressive crushing of the concrete, at an often specified strain εc, always controls failure of the beam (Oehlers et al. 4a). However for plated beams where premature debonding often occurs, this method of determining moment redistribution is found to be unsuitable. To date, very limited research has been carried out on moment redistribution of plated structures. few researchers, such as ukhopadhyaya et al. (998) and El-Refaie et al. (), have developed different indees to measure the ductility of beams with eternal reinforcement, but none can be used to quantify moment redistribution of continuous plated structures. Through the eperimental investigations performed in hapter 5, it has been shown that moment redistribution is affected by the etent of cracking along the beam. That is moment redistribution is dependent on the variation in stiffness along the beam. Therefore in this research, the fleural rigidity () approach was developed for evaluating the moment redistribution behaviour of plated and unplated reinforced concrete members. This approach takes into account the variation in stiffness along the beam, while assuming that there is zero rotation at the hinges. To allow for the variation in stiffness, two methods are proposed: () the simplified approach, where the stiffness of the hogging (hog) and the sagging (sag) are different, while within each region is assumed to be constant; and () the linear approach, where the stiffness within the hogging and sagging regions of the beam is assumed to vary linearly

4 Theoretical Studies on oment Redistribution In this chapter a literature review on the eisting methods for moment redistribution analysis of statically indeterminate beams is firstly presented, followed by a description of the fundmental concept of the fleural rigidity approach developed in this research. The simplified approach is described and verified with the eperimental results of EB plated specimens (hapter 5) in the journal paper included in the Section Further discussions on the simplified approach are given in Section 6.4., which includes the derivation of the mathematical epressions of the simplified approach and the application of the approach to the NS specimens in hapter 5. Parametric studies on varying plating positions and materials were also performed on EB plated beams using the simplified approach and are presented in the journal paper in Section Finally, the linear approach is described and verified in the journal paper in Section 6.5., with further discussions on the derivation of the mathematical equations and the comparison between the eperimental and results given in Section ITERTURE REVIEW oment redistribution is an important and beneficial behaviour in statically indeterminate structures as it allows transfer of moments from the most stressed to less stressed areas, hence giving a more economical and efficient design. The total moment redistribution in a statically indeterminate system consist of two parts (EB-FIP 998): () related to the change of stiffness in the span and over the support due to different cracking; () governed by ductility of reinforcement when passed the yielding moments in the hinge that occurs first. In the following section, the concept of moment redistribution is firstly revised, then the plastic hinge approach presently used to determine the moment redistribution of reinforced concrete structures is reviewed, and finally, the neutral ais depth factor used by various R codes and standards to determine the amount of moment redistribution is discussed. 6.. OENT REDISTRIBUTION ONEPT onsider the encastre or built in unplated reinforced concrete beam of length in Figure 6.c, which is equivalent to an internal span of a continuous beam. For convenience, it is assumed that the same longitudinal reinforcing bars are in the top and bottom of the beam. Hence, the hogging (hog) and sagging (sag) regions have the same moment/curvature (/χ) relationships as shown in Figure 6.a, where: the idealised perfectly elastic portion has a fleural rigidity of ()elas up to a moment capacity

5 Theoretical Studies on oment Redistribution of u at a curvature χy; after which there is a perfectly plastic ductile plateau in which the secant stiffness ()sec reduces up to a curvature of χu at which failure occurs when the secant stiffness is at its minimum ()min. The beam in Figure 6.c is subjected to a uniformly distributed load w, so that whilst the fleural rigidity of the whole beam remains at, the moment at the supports hog is twice that at mid-span sag. Hence for this specific beam, there is no moment redistribution whilst the maimum hogging moment hog is equal to twice the maimum mid-span moment sag. onversely, when hog sag, then there is moment redistribution. Therefore in this contet, moment redistribution is defined as occurring when the distribution of moment within a beam is not given by elastic analyses that assume is constant within the beam. (a) u elastic sag hog ductile plateau hog (b) hog u ( static ) u w /8 ( static ).5 u w /8 elastic u / sag () elas () sec () min χ y χ χ u (c) continuous beam sag u / sag u B non-elastic w (kn/m) hogging joint Figure 6. oment redistribution concept s the uniformly distributed load w is gradually applied to the beam in Figure 6.c, the beam is initially elastic so that hog sag and there is no moment redistribution. When the support moment first reaches its moment capacity u as shown as the point hog in Figure 6.a, then the mid-span moment reaches a value of u/ which is shown as sag. t this stage, the static moment is (static).5u w /8 as shown in Figure 6.b and the distribution of moment is given by line which is labelled elastic. Up to this point, the beam behaviour remains linear elastic. s the load is increased, the beam deflects further resulting in an increase in sag above u/ in Figure 6.b. However, the moment at the support remains at u. The only way that the increase in deflection or deformation, due to the increased load, can be accommodated is for the curvature at the supports to be increased from hog to hog as shown in Figure 6.a and the hogging curvature will keep increasing until the sagging curvature sag reaches sag in Figure 6.a, that is the mid-span moment has reached its capacity u

6 Theoretical Studies on oment Redistribution whilst the behaviour of the hogging region is no longer elastic. The static moment has now reached (static) u w /8 in Figure 6.b, which is the maimum static moment. Hence, the maimum load w that can be applied as all the joints, that is the positions of maimum moments in the hogging and the sagging regions, have reached their moment capacities and a collapse mechanism has formed. The distribution of moment within the beam is now given by line B which has been labelled non-elastic as shown in Figure 6.b. It can be seen in the eample shown in Figure 6., that it is the hogging joints that are required to maintain the moment whilst their curvature is increasing. Hence in this eample, it is the hogging joints that have to redistribute moment and it is their ductility that governs the amount of moment redistribution. If for eample it was necessary for hog in Figure 6.a to eceed the curvature capacity of the section χu, to achieve the static moment (static) in Figure 6.b, then sag in Figure 6.a cannot achieve u and the continuous beam would fail before reaching its theoretical plastic capacity. It can be seen in this eample that the sagging moment joint has only to reach its moment capacity, u in Figure 6.a at point sag, that is its curvature has only to reach χy. Hence its ductility, that is its capacity to etend along the plateau in Figure 6.a, is of no consequence. Unless of course the beam is required to absorb energy such as under seismic loads, in which case it may be a requirement that point sag is also etended into the plastic zone to allow the beam to deflect further and absorb energy without an increase in load. 6.. PSTI HINGE PPROH To determine whether a beam is ductile enough to redistribute moment is an etremely comple problem and there is much good ongoing research (EB-FIP 998; Bigaj 999; El-Refaie et al., ; ukhopadhyaya et al. 998) to develop a comprehensive and simple solution. These researches generally involve the development of different indees to measure the ductility of beams, but none of which can be used to quantify moment redistribution of continuous plated structures. The problem is to understand how the beam can deform to accommodate the non-elastic distribution of moment (line B in Figure 6.b and also shown in Figure 6.b) and then to determine whether the deformation capacity of the beam can accommodate this required deformation. The presently used method for analysing the redistribution of moment in statically indeterminate structures is known as the plastic hinge approach. This method assumes that there is a discontinuity of the slope at the supports as shown in line in Figure 6.e

7 Theoretical Studies on oment Redistribution (a) plastic hinge w plastic hinge hog (b) sag hog / sag hog ( static ) el static B (c) static deformations: (dy/d) static (d) support moment deformations: (dy/d) support (e) overall deformation: dy/d > : plastic hinge approach Figure 6. Plastic hinge approach In the hinge approach, it is assumed that most of the beam of length remains linear elastic at a fleural rigidity as shown in Figure 6.a, and that there are small hinge regions at the joints of length hinge where moment redistribution requires ductility. The hinge length hinge <<, being of the order of magnitude of the depth of the beam. It is assumed that the hinge, often referred to as the plastic hinge, accommodates the discontinuity of slope at the supports in line shown in Figure 6.e. This discontinuity is caused by the non-elastic moment distribution (line B in Figure 6.b) where sag > hog/. The discontinuity of slope can be determined from the static moment in Figure 6.c and the redundant moment shown in Figure 6.d. The slope at the supports (dy/d)static and (dy/d)support in Figure 6.c&d can be derived by integration of the curvature along the length of the beam. Hence the discontinuity of the slope in line in Figure 6.e is equal to the difference between (dy/d)static and (dy/d)support which is accommodated by the plastic hinge in Figure 6.a. s the length of the hinge is very small, it is often assumed that the curvature within the hinge χu is constant so that the rotation capacity of the hinge θcap is:

8 Theoretical Studies on oment Redistribution θ cap χ u hinge Equation 6. Therefore, in order for moment redistribution to occur, the condition given in Equation 6. needs to be satisfied, where θreq is the rotation required at the plastic hinge which can be determined from the fundamental relationship given in Equation 6. (EB-fib 998). Using the concept of plastic hinge, the available rotation capacity θcap can be determined from Equation 6.. One of the difficulties with applying the plastic hinge theory is the determination of the plastic hinge length. θ req θ cap Equation 6. θ req χd Equation 6. The earliest plastic hinge approaches assumed that the hinge occurs at a point, that is hinge. This posses some conceptual difficulties for beams with horizontal or falling branch moment/curvature relationships, since the rotation of the beams requires hinges of zero length where the curvature was increasing, whilst adjacent to the hinge the curvature was decreasing which requires a sudden step change in the curvature at the boundary of the hinge. cknowledging this problem, Johnson (Barnard and Johnson 965), Barnard (964) and Wood (968) proposed the concept of a finite hinge length. This concept, however, is specifically for reinforced concrete beams only where concrete crushing failure occurs, such that large amounts of rotation are present at the hinges to achieve the required rotation (Equation 6.). Therefore, the plastic hinge approach is unsuitable for plated structures where premature debonding failure occurs prior to concrete crushing. 6.. OENT REDISTRIBUTION IN NTION STNDRDS International standards tend to base the ability of (unplated) reinforced concrete beams and slabs to redistribute moment on the neutral ais parameter ku given by Equation 6.4, where d and dn are the effective depth of the beam and the depth of the neutral ais from the compression face. This ku factor, which measures the ductility of a structure, is based on the plastic hinge approach where the hinge length is assumed to be equal to the depth of the beam i.e. hinged. Therefore from Equation 6., the rotation at the hinge is given by Equation 6.5, where the curvature χ is equal to εc/kud. For a constant concrete crushing strain εc, the rotation is, hence, directly proportional to /ku

9 Theoretical Studies on oment Redistribution d k n u Equation 6.4 d ε ε θ c c cap χ u hinge d kud k Equation 6.5 u Typical eamples of ku from five standards (DIN45, EB-FIP99, BS8, N-., S6) are given in Figure 6. for the commonly used high ductility reinforcing bar steels. For these high ductility steels, it can be assumed that the strain capacity of the steels is sufficiently large to ensure that they never fracture prior to concrete crushing. Therefore, the ultimate failure of the R beam is always governed by concrete crushing at a strain εc that is often assumed to range between. to.4. % moment redistribution % B German mean value European British % ustralian anadian neutral ais parameter k u Figure 6. oment redistribution dependence on neutral ais parameter k u s shown in Figure 6., it can be seen that there is a general agreement for an upper bound of % to the amount of moment redistribution that can occur. However, there is a fairly wide divergence between predictions. For eample, no moment redistribution is allowed when ku.4 for the ustralian Standard requirements but this is substantially increased to ku.6 for both the anadian and British Standards; the mean value for no moment redistribution from the five approaches is ku.48 and is shown as point. t the other etremity, the British Standard approach allow % redistribution when ku. whereas the anadian approach uses % as an upper bound as ku ;

10 Theoretical Studies on oment Redistribution the mean value for % redistribution is ku.5 and is shown as point B. line joining these mean values at the etremities is shown as the mean value in Figure 6.. The effect of the variation -B in Figure 6. of the neutral ais parameter ku on the strain profile of a reinforced concrete (R) section at failure is illustrated in Figure 6.4. For ease of eplanation, let us consider the case of a deep R beam in which the effective depth d approaches the depth of the beam h as shown in Figure 6.4a (which is the cross-section of a beam in the hogging or negative region). It has been assumed that the concrete crushing strain εc is.5, as shown in Figure 6.4b. s the strain capacity of the reinforcing bars is assumed to be very large, as previously eplained, concrete crushing always controls failure so that the strain at the compression face of εc.5 is common to all the strain profiles shown and can be considered to be a pivotal point. It can be seen in Figure 6.4b that the neutral ais parameter ku controls the maimum tensile strain at the tension face εtf for any depth of beam d, as εtf εc(-ku)/ku and, hence, εtf is independent of d. It needs to be pointed out that the neutral ais parameter ku does not control the curvature χ for any beam depth as this depends on the actual depth of the beam, that is χ εc/kud and, hence, it depends on d. However, the ku factor controls the rotation of the plastic hinge, of length hinge, as this is given by χhinge where the curvature at failure χ εc/kud; for eample, if hinge d then the rotation is equal to εc/ku. It can, therefore, be seen that the ability to redistribute moment depends on the maimum tensile strain εtf. tension face (ε tf ).5d. (ε tf ).48d.8 % % to % % ε ductile reinforcing bars d h EB steel plates NS FRP strips mean % (point B) k u d.48d k u d.5d EB FRP plates mean % (point ) pivotal point compression face (a) (b) ε c.5 Figure 6.4 oment redistribution dependence on tension face strains The strain profile associated with the mean % redistribution, that is point in Figure 6., is shown in Figure 6.4b as the line mean % ; as the depth of the neutral ais from the compression face is.48d, the strain at the tensile face is (εtf).48d.8. The strain profile for the mean % redistribution at

11 Theoretical Studies on oment Redistribution point B in Figure 6. is also shown in Figure 6.4 and this requires a tensile face strain of (εtf).5d.. Hence, when the concrete crushes and the tensile face strain εtf <.8 then no moment redistribution is allowed, when.8 < εtf <. then between % and % redistribution is allowed, and when εtf >. then % redistribution is allowed. Generally speaking, ductile reinforcing bars can easily accommodate these strains. However, if a plate is adhesively bonded to the tension face as shown in Figure 6.4a, then these strains have also to be accommodated by the plate. The strain capacity of FRP tension face plates depends on either its fracture strength or its I debonding resistance (hapter ). For most cases, I debonding controls the strain capacity of eternally bonded FRP plates which commonly debond at less than half of the fracture capacity, ecept for very thin plates used in the wet lay up process. Tests reported in hapter 5 and also those published elsewhere (Oehlers et al. ) conducted on. mm pultruded carbon FRP (FRP) plated beams, found that the I debonding strains ranged from.5 to.5. This range of strains is shown as the shaded region labelled EB FRP plates in Figure 6.4b. The bounds of this range just fall either side of the mean % profile which suggests that in general pultruded carbon FRP plated structures have little capacity for moment redistribution. This, however, is not the case for NS FRP plates, as can be seen in Figure 6.4b. Due to the strong bond that eist at the plate/concrete interface, NS strips are found to debond at much large strains (hapter 5). Depending on the plating configuration, NS FRP plates can accommodate around % redistribution (Oehlers et al. 5), as indicated by the unshaded region labelled NS FRP plates in Figure 6.4b. etal plates can be designed to I debond prior to yielding in which case the behaviour is similar to that of FRP plates. However and in contrast to FRP plates, metal plates can be designed to yield prior to I debonding; although it should be remembered that s have shown that in the majority of cases the metal plated beam will still eventually debond but at a much larger strain than if it remained elastic. EB beam s in hapter 5 have shown that the I debonding strains for mm steel plates range from.45 to. which is shown as the hatched region in Figure 6.4b, and which suggests that metal plates that have been designed to yield prior to I debonding may have adequate capacity to redistribute moment. It has been shown that moment redistribution based on the ku approach is controlled by the strain at the tension face εtf when the concrete crushes at εc. Hence, it is not the moment/curvature, /χ, relationship that is important in moment redistribution but the moment/tension-face-strain, /εtf,

12 Theoretical Studies on oment Redistribution relationship such as those shown in Figure 6.5; Figure 6.5 was derived from a standard non-linear fullinteraction sectional analysis of the eternally plated beams ed in this research (hapter 5). oment (knm) Plate yield O FRP plated J concrete crushes at ε c steel plated I plate fracture εp.fr F D E G H I debonding ε p.db I debonding ε p.db yield of reinforcing bars ε y unplated concrete crushes at ε c B concrete crushes at ε c tension face strain (ε tf ) Figure 6.5 Typical moment/ tension face strain behaviours The /εtf relationship for the unplated R beam is shown as O--B in Figure 6.5. s the ductile reinforcing bars are assumed to have almost unlimited strain capacity in comparison with the finite concrete strain capacity εc, the beam can only fail by concrete crushing at point B. Hence, there is typically a very long tensile strain plateau commencing at yield of the reinforcing bars εy at point and terminating when the concrete crushes at a strain εc at point B. Over this plateau, -B, the moment capacity remains almost constant. It may be worth noting that the national standards use of ku to control the amount of moment redistribution implicitly applies to sections with the behaviour represented by the curve O--B, that is a long tensile strain plateau that is terminated by concrete crushing. The /εtf relationship O--D-E-F in Figure 6.5 applies to a steel plated beam in which I debonding has not occurred prior to concrete crushing at F. The plate yields at and the reinforcing bars at D, after which the moment remains fairly constant until the concrete crushes at F. This steel plated beam behaviour, O--D-E-F has almost identical characteristics to that of the unplated beam O--B and, hence, the ku factor used in standards can be used to control the moment redistribution. et us now assume that I debonding occurs at point E in Figure 6.5 that is prior to concrete crushing at F but after yielding of both the tension face plate and tension reinforcing bars at D. In this case, ku cannot be used to control the moment redistribution as the ku approach implicitly requires the concrete

13 Theoretical Studies on oment Redistribution to crush as illustrated in Figure 6.4b. If the steel plate debonds prior to yielding at point, then the behaviour is similar to that of an FRP plated section described in the following paragraph. The /εtf relationship for an FRP plated slab in which debonding, that is I, PE and D, does not occur prior to concrete crushing is given by O-G-H-I-J in Figure 6.5. The moment continues to increase after the reinforcing bars yield at point G because FRP is a linear elastic material that does not yield prior to fracturing, so that the plate keeps attracting more force, thereby, increasing the moment. Hence, an FRP plated section does not have a near horizontal plateau, such as D-E-F or - B, that is ideal for accommodating moment redistribution away from the plated section. For this reason, the FRP plated section keeps attracting more moment even though the moment is being redistributed. Because of this rising plateau (G-I-J in Figure 6.5), FRP plated sections are less capable of redistributing moment as compared to metal plated sections with a horizontal plateau. Furthermore, I debonding of FRP plates such as at point H often occurs soon after the reinforcing bars yield and generally well before the plate fractures at point I in Figure 6.5 or the concrete crushes at point J so that the length of the rising plateau is relatively short. In conclusion, it is suggested that the use of the ku factor in standards to control moment redistribution should not be applied to FRP plated structures because invariably the concrete does not crush and there is no ductile horizontal plateau; both of which are implicitly required in national standards. Furthermore, there is usually only a short rising plateau. Hence, it is suggested that the ku factors in national standards for controlling moment redistribution should only be used for metal plated sections in which the concrete crushes prior to the plate debonding. 6. FUNDENT ONEPT OF FEXUR RIGIDITY () PPROH Through the literature review performed in Section 6., it can be seen that the eisting plastic hinge approach has its limitations, especially for members retrofitted with eternal plates. This led to the development of the fleural rigidity () approach in this research. The approach assumes that the slope at the supports is zero, as shown in line in Figure 6.6d, and which is accommodated by variations in the fleural rigidity along the length of the beam, such as that shown in Figure 6.6c where hog represents the fleural rigidity of the hogging region and sag that of the sagging region. It is worth noting that it is not the magnitudes of these fleural rigidities that control the moment redistribution but their relative values or proportions, that is hog/sag. The minimum fleural rigidity

14 Theoretical Studies on oment Redistribution ()min of hog depends on the ultimate sectional curvature capacity χu as shown in Figure 6.a. In this approach, the region where the fleural rigidity is reducing is referred to as the plastic hinge for ease of comparison with the eisting moment redistribution approaches although the hinge approach is not adopted here. In the fleural rigidity approach it is assumed that the hinge is bounded by the points of contrafleure as shown in Figure 6.6b. (a) w hog (b) ( static ) el sag hog / sag hog B (c) approach: hog Δ static sag hog (d) overall deformation: dy/d; approach D dy/d > : plastic hinge approach Figure 6.6 Fundamental concept of fleural rigidity approach In summary, the plastic hinge approach assumes that a small plastic hinge region of unknown length hinge occurs in a statically indeterminate beam, where within the hinge the curvature is assumed to be constant and the rotation of the hinge is greater than zero to allow for the constant stiffness along the beam. In contrast, the approach assumes a larger plastic hinge region, where variation in curvature is allowed for within the hinge, such that the stiffness along the beam is not constant and that there is zero rotation at the hinge. 6.4 SIPIFIED FEXUR RIGIDITY () PPROH Based on the fleural rigidity concept developed, a simplified fleural rigidity approach is proposed where it is assumed that the in the hogging and the sagging regions varies, while the within each

15 Theoretical Studies on oment Redistribution of these regions remains constant as discussed in the journal paper in Section lso presented in the paper is the verification of the theoretical approach using the results obtained in hapter 5 on specimens with eternally bonded plates. pplication of the model to the NS specimens reported in hapter 5 is described in Section 6.4.., along with further discussions of the comparison between the eperimental and theoretical results of the EB specimens. In addition, the derivation of the mathematical equations developed for the simplified fleural rigidity approach is presented in Section for various loading systems JOURN PPER: OENT REDISTRIBUTION IN ONTINUOUS PTED R FEXUR EBERS. PRT FEXUR RIGIDITY PPROH In the following paper, the simplified fleural rigidity approach is presented and verified using the results for eternally bonded plated beams presented in hapter 5. To demonstrate the application of the model, an encastre beam was analysed which was subjected to uniformly distributed loading and where the beam was plated over the hogging regions with different plating materials

16 Theoretical Studies on oment Redistribution

17 Theoretical Studies on oment Redistribution oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach *Oehlers, D.J., **iu, I., ***Ju, G., and ****Seracino, R. orresponding author *Dr. D.J. Oehlers ssociate Professor School of ivil and Environmental Engineering entre for Infrastructure Diagnosis, ssessment and Rehabilitation The University of delaide delaide S55 USTRI Tel Fa doehlers@civeng.adelaide.edu.au **s. I. iu Postgraduate student School of ivil and Environmental Engineering The University of delaide ***Dr. G. Ju ecturer Department of rchitectural Engineering University of Yeungnam South Korea ****Dr. R. Seracino Senior ecturer School of ivil and Environmental Engineering The University of delaide Published in Engineering Structures 4, vol. 6, pg

18 Theoretical Studies on oment Redistribution oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach Statement of uthorship OENT REDISTRIBUTION IN ONTINUOUS PTED R FEXUR EBERS. PRT : FEXUR RIGIDITY PPROH Published in Engineering Structures 4, vol. 6, pg.9-8 IU, I.S.T. (andidate) Performed all analyses, interpreted data and co-wrote manuscript. Signed Date OEHERS, D.J. Supervised development of work, co-wrote manuscript and acted as corresponding author. Signed Date SERINO, R. Supervised development of work, and manuscript review. Signed Date

19 Theoretical Studies on oment Redistribution oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach OENT REDISTRIBUTION IN ONTINUOUS PTED R FEXUR EBERS. PRT : FEXUR RIGIDITY PPROH Oehlers, D.J., iu, I., Ju, G., and Seracino, R. BSTRT dhesive bonding plates to the surfaces of reinforced concrete members is now frequently used to increase both the strength and stiffness. However, because of the brittle nature of the plate debonding mechanisms, plating is often assumed to reduce the ductility to such an etent that guidelines often preclude moment redistribution. Tests on seven full-scale fleural members have shown that significant amounts of moment can be redistributed from steel and carbon fibre reinforced polymer (FRP) plated regions. In this paper, a procedure is developed for quantifying the amount of moment redistribution that can occur in eternally bonded steel or FRP plated members which can be used to design plated members for ductility. Keywords: Retrofitting; reinforced concrete beams; eternally bonded plates; ductility; moment redistribution. INTRODUTION It was suggested in the companion paper that the neutral ais parameter (ku) approach used in international standards for controlling the moment redistribution in reinforced concrete structures depends on both the concrete crushing and the eistence of a horizontal plateau in the moment/curvature relationship. Both requirements seldom occur in plated structures due to intermediate crack, I, debonding of the plate so the ku approach is felt to be unsuitable for this new form of plated structure. Instead, an alternative approach based on fleural rigidities has been developed to quantify moment redistribution in plated members in which I debonding controls the ultimate strength.. OENT REDISTRIBUTION ONEPT In order to illustrate the phenomenon of moment redistribution, that is the ability of statically indeterminate beams to redistribute moment, let us consider the encastre or built in beam of length in Fig.(c), which can also be considered to represent an internal span of a continuous beam. For convenience, let us assume that the same longitudinal reinforcing bars are in the top and bottom of the beam. Hence, the hogging (hog) and sagging (sag) regions have the same moment/curvature (/χ) relationships as shown in Fig.(a), where: the idealised perfectly elastic portion has a fleural rigidity of ()elas up to a moment capacity of u at a curvature χy; after which there is a perfectly plastic ductile plateau in which the secant stiffness ()sec reduces up to a curvature of χu at which failure occurs when the secant stiffness is at its minimum ()min. et us also assume that the beam is subjected to a uniformly distributed load w, as shown in Fig.(c), so that whilst the fleural rigidity of the whole beam remains at, the moment at the supports hog is twice that at mid-span sag. Hence for this specific beam, there is no moment redistribution whilst the maimum hogging moment hog is equal to twice the maimum mid-span moment sag. onversely, when hog sag, then there is

20 Theoretical Studies on oment Redistribution oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach moment redistribution. We will, therefore, define moment redistribution as occurring when the distribution of moment within a beam is not given by elastic analyses that assume is constant within the beam. We will use this simple definition for convenience, as designers generally assume in their preliminary analyses that is constant within a beam in determining the initial distribution of moment which can then be redistributed. (a) u elastic sag hog ductile plateau hog (b) hog u ( static ) u w /8 ( static ).5 u w /8 elastic u / sag () elas () sec () min χ y χ χ u (c) continuous beam sag u / sag u B non-elastic w (kn/m) hogging joint Figure oment redistribution concept s the uniformly distributed load w is gradually applied to the beam in Fig.(c), the beam is initially elastic so that hog sag and there is no moment redistribution. When the support moment first reaches its moment capacity u as shown as the point hog in Fig.(a), then the mid-span moment reaches a value of u/ which is shown as sag. t this stage, the static moment is (static).5u w /8 as shown in Fig.(b) and the distribution of moment is given by line which is labelled elastic. Up to this point, the beam behaviour remains linear elastic. s the load is increased, the beam deflects further resulting in an increase in sag above u/ in Fig.(b). However, the moment at the support remains at u. The only way that the increase in deflection or deformation, due to the increased load, can be accommodated is for the curvature at the supports to be increased from hog to hog as shown in Fig.(a) and the hogging curvature will keep increasing until the sagging curvature sag reaches sag in Fig.(a), that is the mid-span moment has reached its capacity u whilst the behaviour of the hogging region is no longer elastic. The static moment has now reached (static) u w /8 in Fig.(b), which is the maimum static moment and, hence, the maimum load w that can be applied as all the joints have reached their moment capacities and a collapse mechanism has formed. The distribution of moment within the beam is now given by line B which has been labelled non-elastic as shown in Fig. (b). It can be seen in the eample shown in Fig., that it is the hogging joints that are required to maintain the moment whilst their curvature is increasing. Hence in this eample, it is the hogging joints that have to redistribute moment and it is their ductility that governs the amount of moment redistribution. If for eample, it was necessary for hog in Fig.(a) to eceed the curvature capacity of the section χu, to achieve the static moment (static) in Fig.(b) then sag in Fig.(a) cannot achieve u and the continuous beam would fail before reaching its theoretical plastic capacity. It can be seen in this eample that the sagging moment joint has only to reach its moment capacity, u in Fig. (a) at point sag, that is its curvature has only to reach χy. Hence its ductility, that is its capacity to etend along the plateau in Fig.(a), is of no consequence. Unless of course the beam is required to absorb energy

21 Theoretical Studies on oment Redistribution oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach such as under seismic loads, in which case it may be a requirement that point sag is also etended into the plastic zone to allow the beam to deflect further and absorb energy without an increase in load.. OENT REDISTRIBUTION PPROH FOR PTED BES To determine whether a beam is ductile enough to redistribute moment is an etremely comple problem and there is much good ongoing research -6 to develop a comprehensive and simple solution. The problem is to understand how the beam can deform to accommodate the non-elastic distribution of moment (line B in Fig.(b) and also shown in Fig. (a)) and then to determine whether the deformation capacity of the beam can accommodate this required deformation. Two approaches can be followed: (i) assume that there is a discontinuity of the slope at the supports as shown in line D in Fig.(f) and this will be referred to as the hinge approach; or (ii) assume that there is no discontinuity, such as at line, and this will be referred to as the fleural rigidity () approach. In many ways, these approaches can be combined. hog static elastic (a) sag hog / sag > hog / B non-elastic approach: () hog ()sag () hog (b) hinge approach: hinge w (kn/m) (c) static deformations: (dy/d) static plastic hinge (d) static support moment deformations: (e) (dy/d) support hog hog deformation: dy/d, approach (f) D dy/d > : plastic hinge approach Figure ompatibility in moment redistribution

22 Theoretical Studies on oment Redistribution oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach. Hinge approach In the hinge approach, it is assumed that most of the beam of length remains linear elastic at a fleural rigidity as shown in Fig.(c), and that there are small hinge regions at the joints of length hinge where moment redistribution requires ductility. The hinge length hinge <<, being of the order of magnitude of the depth of the beam. It is assumed that the hinge, often referred to as the plastic hinge, accommodates the discontinuity of slope at the supports in line D shown in Fig.(f). This discontinuity is caused by the non-elastic moment distribution (line B in Fig.(a)) where sag > hog/. The discontinuity of slope can be determined from the static moment in Fig.(d) and the redundant moment shown in Fig.(e). The slope at the supports (dy/d)static and (dy/d)support in Figs.(d) & (e) can be derived by integration of the curvature along the length of the beam. Hence the discontinuity of the slope in line D in Fig.(f) is equal to the difference between (dy/d)static and (dy/d)support which is accommodated by the plastic hinge in Fig.(c). s the length of the hinge is very small, it is often assumed that the curvature within the hinge is constant so that the rotation capacity of the hinge is simply χuhinge where χu is the curvature capacity of the section as illustrated in Fig.(a). The main problem with this approach is deciding what is the length of the plastic hinge region.. Fleural rigidity approach In contrast to the plastic hinge approach, the fleural rigidity approach assumes that the slope at the supports is zero, as shown in line in Fig.(f). This can only be accommodated by allowing variations in the fleural rigidity along the length of the beam such as shown in Fig.(b), where ()hog represents the fleural rigidity of the hogging region and ()sag that of the sagging region. It is not the magnitudes of these fleural rigidities that control the moment redistribution, but their relative values or proportions, that is ()hog/()sag. For eample when ()hog ()sag, that is ()hog/()sag, then, in this eample, the elastic distribution of moment is achieved so that hog sag and consequently there is no moment redistribution. Even if one were to double both fleural rigidities, ()hog/()sag would still remain at unity and, therefore, hog would remain at sag so there would still be no moment redistribution. However, if the secant fleural rigidity ()sec is taken in the hogging region, it reduces as χ increases along the plateau in Fig.(a). onsequently ()hog/()sag also reduces. s hog is constant whilst sag is increasing, hog < sag that is moment redistribution is occurring. The minimum fleural rigidity ()min of hog depends on the ultimate sectional curvature capacity χu as shown in Fig.(a).. hoice of approach for plated sections In order to determine which of the two moment redistribution approaches, that is the hinge approach or the fleural rigidity approach in Fig., is suitable for plated sections, let us first consider their moment/curvature responses. The theoretical non-linear full-interaction moment/curvature response for the steel plated sections in the companion paper is shown in Fig. and those for the FRP specimens in Fig.4. These relationships were derived from standard sectional analyses that allowed for the non-linear properties of the materials and which assumed full interaction. The points marked to G in Figs. and 4 occurred when the strains in the plate were equal to their maimum recorded strains εp.ma; this occurred either at debonding or just prior to debonding when there was virtually full interaction between the plate and the concrete, as discussed in the companion paper. These strains are shown

23 Theoretical Studies on oment Redistribution oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach for all the s in column 4 in Table. lso shown are the maimum concrete strains in the compression face at plate debonding in column 5, and in column 6 the curvatures at debonding in terms of the curvature at yield of the reinforcing bars which were derived from the /χ analyses. oment (knm) oment (knm) 5 5 tensile reinforcing bars yield plate yields SF debonds fleural cracking SF debonds Figure. /χ for steel specimen F SS debonds tensile reinforcing bars yield D SS debonds fleural cracking curvature (mm - ) E SF4 debonds SF SF&SF SF4 5 SF debonds curvature (mm ) G SS debonds B ( sec ) SS α( sec ) SS α( sec ) SS ( sec ) SS. Figure 4. //χ for FRP specimens Table nalysis of results Spec. b pt p plate χ ma/ %R tot α () sec/ ε (mm) material p.ma ε c.ma χ yield (ε p.ma) () yield () () () (4) (5) (6) (7) (8) (9) SS 75 steel SS steel SS 4 steel SF 5.4 FRP SF 5. FRP SF 8. FRP SF4.4 FRP From Table, three of the specimens (SF to SF) debonded before the reinforcing bars yielded and one specimen (SF4) debonded at about yield; for these specimens, the concrete compressive strains, measured on the compression face adjacent to the central support, were very low ranging from

24 Theoretical Studies on oment Redistribution oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach to.8, so that these sections were still pseudo-elastic and would not have formed a plastic hinge. The remaining three specimens debonded after yield; specimens SS and SS had concrete compressive strains of about. which was still well below the crushing strain of the concrete of about.5 so that it is felt that a plastic hinge would not have formed here either. It was only the remaining specimen SS that debonded at a concrete strain of.6 that approached the concrete crushing strain. Because most of the s debonded at relatively low concrete compressive strains and often whilst the sections were pseudo-elastic, it was felt that the plastic hinge approach that requires rotation to be concentrated over very small plastic regions as in Fig.(c) would not be suitable. Hence the fleural rigidity approach in Fig.(b) where the change in slope is accommodated over the whole hogging region, has been adopted for plated structures in the following analyses. 4. NYSIS OF TEST RESUTS 4. Fleural rigidity model In order to apply the fleural rigidity approach, the specimens have been idealised as propped cantilevers about the line of symmetry, as in Fig.5, where the fleural rigidity in the hogging region and in the sagging region vary but are constant within a region. This distribution of is not meant to represent the general behaviour, such as would be required for determining the deflection, but it is only meant to represent moment redistribution where the differences in between regions affect the amount of moment redistributed. P / / hog sag Figure 5. Idealised structure for moment redistribution stiffness analysis software package, with two elements of stiffnesses and, could be used to find a solution to the beam in Fig.5; an iterative procedure is required to adjust the length of each element, by varying the length of the hogging region, until the point of contrafleure also occurs at distance. lternatively, an elastic solution (for the beam in Fig.5 with two fleural rigidities and a single concentrated load at mid-span) can be derived using the force method and conjugate beam theory as given by Eqs. and ; this can be used in an iterative analysis to determine the position. hog () P hog 6 hog [ ( )( )] P () [ ( )(8 )] s an eample of the iterative approach, for a fied applied load P, the hogging moment hog could be estimated or guessed and, from Eq., the position of the point of contrafleure determined. Inserting this value of into Eq. would give a value for P and hog which would be adjusted until P from Eq. equalled the fied value

25 Theoretical Studies on oment Redistribution oment redistribution in continuous plated R fleural members. Part : Fleural Rigidity pproach 4. alibration of fleural rigidity model In the following analyses, the fleural rigidities at the positions of the maimum hogging and sagging moments were used to represent the fleural rigidities in the idealised beam shown in Fig.5. s in the beams, moment was redistributed from the hogging plated region to the sagging unplated region, the fleural rigidity of the sagging region was fied at the fleural rigidity of the cracked unplated section. The fleural rigidity of the hogging region was taken as the secant fleural rigidity of the specimen at maimum plate strain prior to debonding εp.ma at the points to in Fig. and points D to G in Fig. 4. s an eample for specimen SS in Fig., the stiffness of the hogging region () equals (()sec)ss which is the secant of the hogging region of the beam at εp.ma in column 4 of Table. The secant fleural rigidities at the maimum strain ()sec are compared with those at yield of the reinforcing bars ()yield in column 9 of Table. It can be seen that the secant fleural rigidities are as low as 9% of that at yield of the reinforcing bars. The secant stiffness of the hogging region was adjusted to a value of α()sec (as shown in Fig. for specimens SS and SS where the arrows indicate the magnitude and direction of the adjustment) so that the theoretical moment redistribution obtained from Eqs. and was equal to the eperiment moment redistribution listed in column 7 of Table ; the derived α factors are given in column 8. The variation of the α factor with the maimum plate strain prior to debonding is shown in Fig.6 where the average α value of all s is.96. The α factor remains fairly constant over debonding strains up to about.6 and within this range the mean value is.9. It can be seen that a value of α would tend to overestimate the fleural rigidity of the hogging region and, consequently, underestimate the moment redistributed ecept at high curvatures, such as for specimen SS, where it is slightly unconservative. The variation of the α factor with curvature is shown in Fig.7; it is fairly constant over a wide range of curvatures from about 5% to 5% of the curvature at yield of the reinforcing bars. The use of α in predicting the moment redistribution in the specimens is shown in Fig.8 and gives a safe design with a mean of α.8 and a standard deviation of.. Using α.9 improves the redistribution prediction by reducing the mean value of α to.7 with a standard deviation of.8. However for a safe design, an α of is suggested as it is only slightly unconservative at high curvatures. It can be seen from Figs.6-8 that the moment redistribution estimated using the fleural rigidity approach compares well with the eperimental results for a wide range of debonding strains and curvatures. α.4..8 SF SS SF SF4 SF SS SS maimum plate strain ε p.ma Figure 6. Variation of fleural rigidity adjustment factor α with maimum plate strain