A MULTI-SPRING HINGE ELEMENT FOR REINFORCED CONCRETE FRAME ANALYSIS
Brian H.H. Peng1; Rajesh P. Dhakal2*; Richard C. Fenwick2; Athol J. Carr2; and Des K. Bull2 1
Holmes Consulting Group, 50 Customhouse Quay, PO Box 942, Wellington 6140, New Zealand
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Department of Civil and Natural Resources Engineering, University of Canterbury, Private Bag 4800, Christchurch 8020, New Zealand *Corresponding author: Ph +64-3-364 2987 ext 7673; Email:
[email protected]
ABSTRACT This paper describes the development and validation of an analytical multi-spring plastic hinge element that can predict elongation of ductile reinforced concrete (RC) plastic hinges together with its flexural and shear responses. The element consists of layers of longitudinal and diagonal springs that represent the behavior of concrete, reinforcing bars and diagonal compression struts. Beam tests reported in literature for which elongation of plastic hinges was measured at different stages of the lateral cyclic loading were used to validate the effectiveness of the newly-developed plastic hinge element. Comparisons of the analytical predictions with experimental results show that the proposed element predicts elongation of plastic hinges satisfactorily. The ability of the model to predict elongation of a plastic hinge together with its flexural and shear deformations offers a significant advancement in seismic performance assessment of RC structures.
KEYWORDS RC frame; plastic hinge; multi-spring element; material models; diagonal struts; beam elongation; shear deformation; beam cyclic tests.
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INTRODUCTION Over the last three decades, experimental studies conducted in New Zealand on inelastic cyclic behavior of reinforced concrete (RC) beams [Fenwick et al 1981; Issa 1997; Matti 1998; Walker and Dhakal 2009] have shown that ductile beam plastic hinges typically elongate between 2 and 5 percent of the beam depth before strength degradation occurs. The elongation of beam plastic hinges forces the columns to move apart; which increases the interstory drift and consequently the shear and flexural actions in the columns. Cyclic loading test of 3D RC frame-floor sub-assembly [Matthews et al 2003] has shown that elongation of plastic hinges can cause precast floor units to unseat from their supporting beams. Furthermore, the beam elongation and its interaction with the floors increase the flexural strength of beam plastic hinges more than currently anticipated in the code due to a greater contribution from the slab reinforcement, which may subsequently lead to hinging in the columns in an event of a major earthquake. Hence, elongation of plastic hinges should be taken into consideration for reliable prediction of ductile RC frame response to seismic actions. To assess the seismic performance of RC frames, the common analytical approach is to use either nonlinear beam/column elements characterized by cyclic hysteresis rules (such as Takeda/ modified Takeda etc) to represent the beams and columns or elastic frame elements together with lumped plasticity elements to model the moment-rotation relationships [Rajashankar et al. 2009, Fabio and Mirko 2010]. Such models can be calibrated to closely capture the hysteresis response of frames, but they cannot capture elongation of the beams when subjected to inelastic cyclic actions. Hence, analysis using such models may not provide reliable seismic performance assessment as they fail to capture the additional inter-story drift demand and the beam strength enhancement from floor-frame interactions, which are important by-products of beam elongation. Although elongation of plastic hinges has been found to have a detrimental effect on seismic performance of RC buildings, it is generally overlooked in design and analysis due to a lack of satisfactory analytical models capable of predicting elongation of plastic hinges. Some methods
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have been proposed for predicting elongation of plastic hinges [Fenwick and Megget 1993; Lee and Watanabe 2003; Matthews et al 2004]. The elongation prediction equation proposed by Fenwick and Megget [1993] is only applicable for unidirectional plastic hinges which yield in only one direction of loading, and reversing plastic hinges, which are common in seismic frames, are not covered by the equation. Lee and Watanabe [2003] correlated elongation with reinforcement strain and beam rotation, and Matthews et al [2004] proposed the so-called rainflow method to estimate beam elongation; but these empirical methods were developed to estimate beam elongation outside finite element analysis domain and lacked necessary features to be readily incorporated into time-history analysis programs. An analytical elongation model for RC beams was proposed by Douglas et al [1996] and later refined by Lau et al [2003]. This model employs a filament type element to represent plastic hinge region. Although the model has shown to have promising elongation prediction, it requires the members to be calibrated prior to analysis. An elongation model for precast RC beams containing prestressed tendons has also been proposed by Kim et al [2004]. However, this model cannot be applied directly to predict elongation of plastic hinges in monolithic moment resisting frames where the behavior is more complex. When RC members are analysed using fiber-based finite element analysis approach [Spacone et al. 1996; Monti and Spacone 2000, Song et al. 2002], it is possible to capture elongation of the member if the used material models for reinforcing bars and concrete fibers are cyclic and fully path-dependent. However, a fiber-based approach is seldom used to analyze large RC frames because of its unrealistic computational demand. Therefore, it is desirable to have a fiber-model based plastic hinge element that can be used together with the traditional frame elements to assess the seismic performance of RC frames. This paper describes the development of a generic plastic hinge element that can account for elongation of plastic hinges in ductile RC members. Experimental data on elongation of plastic hinges in beams with different levels of axial forces are compiled from literature, and used to validate the model.
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ANALYTICAL MODEL Model description In the proposed approach of RC frame analysis, the regions likely to undergo inelastic deformations are modeled using plastic hinge elements and the remaining portions of the frame are modeled using elastic beam/column elements. The plastic hinge element consists of a series of longitudinal and diagonal springs connected between the rigid links perpendicular to the member axis at the two ends. The longitudinal springs are used to represent flexural and axial response of the plastic hinge, and the diagonal springs are used to represent the shear response. The moment and shear are evaluated at the centre of the plastic hinge element and are extrapolated to obtain the nodal moment and forces at the two ends of the plastic hinge element. A schematic representation of the proposed plastic hinge element is shown in Fig. 1, where LP is the length of the element. In this element, two steel springs are located at the centroids of the top and bottom reinforcement to represent the reinforcing bars, two concrete springs are located at the centre of the top and bottom covers to represent unconfined concrete, eight concrete springs are distributed evenly between the centroids of the tensile and compressive reinforcing bars to represent the confined core concrete and two diagonal concrete springs are connected between the ends of the top and bottom steel springs to represent the diagonal compression struts. The cross section area of each steel spring is equal to the total area of reinforcing bars in that side of the cross section, and that of the longitudinal and diagonal concrete springs is calculated as the product of the beam width and the effective depth of the spring. While the effective depth of the longitudinal (cover and confined) concrete springs is obvious, the effective depth of the diagonal compression strut WD is taken as the perpendicular distance from the end-point of the reinforcement spring to the diagonal spring, which equals to LP sin, where is the angle of the diagonal struts to the horizontal plane. Due to the presence of diagonal cracks and tensile strains in the orthogonal direction, the effective concrete compressive strength of the diagonal spring is taken as 0.34f’c [To et al. 2001].
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Material models The average elongation at the centre of the cross-section of a plastic hinge consists of two parts: the extension of tensile reinforcement due to inelastic rotation and the irrecoverable extension of the compression reinforcement [Peng 2009]. The former is related to the cyclic stress strain behavior of reinforcing steel, and the later is related to the contact stress effect, which is a unique feature of cyclic stress strain relationship of concrete. Hence, the ability of the proposed plastic hinge element to capture the cyclic response and to predict the elongation depends mainly on how accurately the path-dependent cyclic behavior of the axial springs is modeled. The constitutive models adopted for the springs in the plastic hinge element are based on uni-axial average stress-strain relationship of concrete [Maekawa et al. 2003] and reinforcing steel [Dhakal and Maekawa 2002a]. The stress-strain envelope of concrete comprises a tension stiffening model in tension and the elasto-plastic and fracture model in compression. The concrete cyclic model takes into account the loss of stiffness due to fracture of concrete, and the unloading loop from tension into compression also allows for the contact stress effect (i.e. development of compressive stress before the cracks fully close due to concrete pieces dislodged in the cracks). As the stress strain relationships are generic, the different behaviors of cover concrete and the confined core concrete can be catered for by inputting different values of the input variables (compressive strength and peak strain) for the external (cover) and the internal (confined) concrete springs and activating the spalling criteria only for the cover concrete. The path-dependent cyclic model for steel consists of compression and tension envelopes, which can represent any combination of yield plateau and strain hardening, and the unloading and reloading loops represented by Giuffre-Menegotto-Pinto model which takes into account the Bauschinger effect. Although the original model accounted for buckling of reinforcing bars inside RC members, buckling related features were switched off for the steel springs in the plastic hinge
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element because of incompatibility with the finite element analysis program. Buckling reduces the average stress carried by reinforcing bars in compression [Dhakal and Maekawa 2002b], and the extent of the compressive stress reduction depends on the stiffness and spacing of the stirrups [Dhakal and Maekawa 2002c]. Because transverse reinforcement in plastic hinge regions have to satisfy strict confinement and anti-buckling requirements, the effect of buckling will be less pronounced in ductile plastic hinges. Furthermore, as elongation occurs because the concrete diagonal strut contributes to the compressive force and the steel bars do not fully unload in compression as they are required to carry only a small compression force, buckling (i.e. reduction of compressive stress in bars) is unlikely to significantly affect elongation.
Length of plastic hinge element The length of the plastic hinge element, LP, (see Fig. 1) is chosen to represent the inclination of the diagonal compression struts, , in the plastic hinge region. This is an important parameter as it controls the proportion of the longitudinal component of the force in the diagonal springs, thereby influencing the magnitude of the flexural compression force in the reinforcement and hence the overall elongation that develops. This length does not represent the length commonly used for calculating the curvature from plastic hinge rotation or ductile detailing length as used in concrete structures codes in different countries [NZS3101 2006; ACI318 2005]. Using the truss analogy, it is hypothesized that the diagonal cracks will form in an angle such that it crosses just enough stirrups to resist the shear force in the web. Consequently, LP is equal to the product of the number of stirrups required to resist the shear force and the stirrup spacing, s. This can be expressed as shown in Equation 1 where Vyc is the shear force corresponding to the flexural strength of the beam, Myc, given by Equation 2 (assuming the resultant compression force acts at the compression reinforcement and the axial force acts mid way between the tensile and compression reinforcement), Vc is the shear resistance of concrete, and (d - d’) is the distance between the centroids of
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reinforcing bars, Av, As, fvy, fy are the area and yield stress of the shear and longitudinal reinforcement, respectively, and P is the applied axial force. LP
V
yc
Vc s
(1)
Av f vy
M yc As f y d d' P
d d'
(2)
2
It should be noted that the theoretical flexural strength, Myc, is used to calculate LP instead of the nominal flexural strength, Mn. This is because under cyclic loading where the compression reinforcement has been yielded in tension in the previous cycles, the majority of the compression force is resisted by the compression reinforcement unless compression bars yield back and the cracks close fully. It should also be noted that Vyc is generally smaller than the maximum shear force sustained in the beam due to strain hardening of the longitudinal reinforcement. It is commonly assumed that the shear resistance of concrete in beam plastic hinges is negligible [NZS3101 2006; ACI318 2005]. However, as the axial compression force increases the shear resistance of concrete should also increase. Unfortunately, there are no appropriate guidelines in these codes that specify the shear resistance of concrete in plastic hinges with different levels of axial force. Therefore, despite acknowledging that the concrete contribution to shear resistance may not be insignificant in the presence of axial compression, the concrete shear resistance is taken as zero in this study.
Stiffness of steel springs As concrete cracks at an early stage and the post-cracking tangent modulus of concrete (i.e., concrete springs) is insignificant, the overall flexural behavior of the plastic hinge element is largely governed by the axial behavior of the steel springs. The axial stiffness of the steel springs is calculated as the product of the tangent modulus and cross-section area divided by the length of the springs, which is equal to LP. However, as the length of the plastic hinge element is decided based on the equilibrium between shear force and the capacity of stirrups; it has nothing to do with the 7
beam length over which the reinforcing bars yield, and the stiffness of the steel spring calculated using LP would lead to inaccurate predictions. Therefore, the stiffness of the steel springs is calculated using the actual length over which the reinforcement yields, Lyield, rather than the length of the plastic hinge element LP. The components of the reinforcement yielding length Lyield are illustrated in Fig. 2 and it can be calculated using Equation 3 [Dhakal and Fenwick 2008].
L yield
M M max M yc Lts Le V M max
(3)
In this equation, M/V is the moment to shear ratio, Mmax is the maximum moment sustained in the beam, Lts is the length of tension shift effect and Le is the length of yield/strain penetration into the support. The maximum moment can be assessed from the experimental results or estimated giving due consideration to the strain hardening of reinforcing bars. Intuitively, the yielding length of the reinforcing bars changes with the level of loading. However, for simplicity a constant yield length corresponding to the maximum flexural capacity of the member, Mmax, is used in the element. In general, the length of yield penetration, Le, is assumed to be a portion of the development length. For a beam with no axial force, the length of tension shift, Lts, can be approximated using Equation 4 [Paulay and Priestley 1992]. Lts
d d' 2
(4)
Equation (4) is based on the assumption that the diagonal crack extends over a distance (d - d’) along the plastic hinges away from the column face as illustrated in Fig. 3, where C1 is the flexural compression force at section 1, T2 is the flexural tension force at section 2 and Vs is the shear force resisted by the shear reinforcement. As the shear force in the plastic hinge is assumed to be carried solely by the stirrups crossing the crack (Vs = V), the moment at section 1 can be expressed as M 1 T2 d d' 0.5d d' Vs
(5)
M 2 V d d'
(6)
Rearranging the above equations; a relationship between the tension force and moment at section 2 can be derived as 8
T2
M2 0.5V d d'
(7)
In Equation 7, the term 0.5V implies that the flexural tension force at section 2 is proportional to the moment at a distance 0.5(d – d’) to the left of the section. For beams with axial compression force, the diagonal crack angle would decrease and the length of tension shift Lts would increase. Unfortunately, the relationship between the crack angle and the applied axial force in the low moment end of plastic hinges under reversing cyclic actions is not readily available in literature.
EXPERIMENTAL CANTILEVER BEAM TESTS Experimental data was obtained from cantilever beam tests carried out in New Zealand [Fenwick et al 1981; Issa 1997; Matti 1998; Walker and Dhakal 2009]. These tests were conducted to examine the effect of axial load, shear span, area of top and bottom reinforcement, concrete and steel strength and stirrup spacing on the performance of RC members subjected to inelastic cyclic loading. The typical set up of the cantilever beam tests is illustrated in Fig. 4a where L is the span length and and P are the displacement at the load point and axial force applied to the beams. The central block was bolted down to the strong floor with two cantilever beams extended to both sides, where each beam was tested separately. Additional steel bars (two 10 mm diameter round bars) were welded to the longitudinal reinforcement (starting at 50 mm inside the block from the interface) passing through the central block to prevent yield penetrating into the support. Welded bar samples were tested to verify that the effect of welding on the strength and ductility was negligible. The typical beam sections employed in these experiments are illustrated in Fig. 4b and Fig. 4c where R and D stand respectively for round and deformed bars and H stands for higher strength reinforcing bar with a design yield stress of 500 MPa. The number following the letters represents the diameter of reinforcing bar in millimeter. A grid of linear potentiometers was mounted on each beam as illustrated in Fig. 5. These potentiometers were fixed to studs that were welded on the beam bars. This was crucial in accurately measuring the deformation of the reinforcing bars, which is central to the concept of 9
beam elongation. Measurements from these linear potentiometers were used to find the deformation patterns along the beam. The deformation can be divided into three categories; deflection due to flexural rotation, f, deflection due to shear deformation, s, and elongation of the beam. Equations for calculating these deformation components are given below where bi and ti are the deformation of the top and bottom linear potentiometers at the ith grid, li is the distance between the center of the ith grid to the load point on the beam, h is the distance between the top and bottom reinforcement,
Dbi and Dti are the deformations of the diagonal potentiometers at the ith grid, and i is the angle of the diagonal potentiometer to the horizontal plane at the ith grid. These symbols are also illustrated in Fig. 5 where a positive value indicates that the linear potentiometer is extending. n
bi ti
i 1
h
f li
(8)
Dbi Dti 2 sin i i 1 n
s
(9) n
bi ti
i 1
2
Beam elongaiton
(10)
A typical displacement loading sequence applied to the specimens is illustrated in Fig. 6. The tests were terminated when force sustained at a peak displacement was less than 80% of the maximum force. The measured material properties of the selected beams are summarized in Table 1, where fvy is the yield stress of shear reinforcement, fy and fu are the yield and ultimate stress of the longitudinal bars respectively, and fc’ is the concrete compressive strength. Beams AA1 and AA2 were tested by Walker and Dhakal [2009], beams 2A, 1A, 1B were tested by Fenwick et al [1981], beams M1 and M2 were tested by Matti [1998] and beams S1A, S1B, S2A and I1B were reported by Issa [1997]. Table 2 summarizes the calculated material and section properties of the beams where Myc is the theoretical flexural strength of the beam as prescribed by Equation 2, L is the length of shear span, Vyc is the shear force corresponding to the theoretical flexural strength, Myc, and y, sh and u are the yield, strain hardening and ultimate strains of the longitudinal reinforcement, respectively. The coordinates of the onset of strain hardening point and the ultimate 10
point are listed for all specimens in Tables 1 and 2 because the model requires the post yield properties of the reinforcing bars as input to account for the effect of strain hardening. The concrete tensile strength, ft, and the Young’s modulus of concrete, Ec, are calculated based on NZS3101 [2006] and are given below. Note that the units in these equations are in MPa.
f t 0.36 f c'
(11)
Ec 3320 f c' 6900
(12)
COMPARISONS BETWEEN ANALYTICAL AND EXPERIMENTAL RESULTS The newly-developed plastic hinge element is encoded into a time history nonlinear finite element analysis program RUAUMOKO2D [Carr 2004]. To simulate the aforementioned experiments, an analytical model of the tested cantilever beam is set up as illustrated in Fig. 7. The analytical cantilever beam is divided into two regions, the elastic region and the plastic hinge region. The elastic region is modeled using an elastic beam element. The effective moment of inertia of the elastic beam is taken as 0.4Ig as recommended in NZS3101 [2006] where Ig is the second moment of area based on the gross section. The input parameters for the plastic hinge element in the different tested beams are summarized in Table 3, where LP is the element length calculated using Equation 1; is the angle of the diagonal struts to the horizontal plane which is dependent on the element length LP, WD is the effective depth of the diagonal springs calculated as LPSin as prescribed before, Mmax is the maximum moment capacity of the beam cross-sections calculated considering the strain hardening effect, Lts is the tension shift length calculated using Equation 4, and Lyield is the reinforcement yielding length calculated using Equation 3 (with the yield penetration into the support taken as the un-welded length within the central block, which is equal to 50 mm). These variables are the key input parameters for the plastic hinge element, and will govern the predicted results. For representative specimens with and without axial force, comparisons between the experimental and analytical results are presented below. Detailed
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comparison between experimental and analytical results for all specimens can be found elsewhere [Peng 2009].
Beams with no axial force Comparisons between the analytical and experimental elongation history for beam 2A are shown in Fig. 8a. It can be seen in this figure that the analytical elongation matches well with the experimental results. In general, the following trends can be observed: (i) in the elastic cycles, the beam elongates by a small amount but the elongation is fully recovered when the load is released; (ii) elongation increases when more than one displacement cycle of the same amplitude is applied, but with a reduced rate; and (iii) during unloading from a peak displacement, elongation remains more or less constant until the displacement is reversed back into the opposite direction, after which the elongation starts to increase. This observation, however, is not true for beams with substantial axial compression force, as discussed later (see Fig. 13b). Fig. 8b shows the comparisons between the experimental and analytical shear force versus displacement relationships at the loading point. It can be seen that the analysis predicts the elastic stiffness, yield displacement, yield force and ultimate force accurately. However, strength degradation observed in the experiment in the large displacement cycles is not captured satisfactorily in the analysis. Pinching in the force-displacement relationship is also under-estimated in the analysis after 4 cycles. This is due to the inability of the proposed plastic hinge element in capturing slip of rebar and the full shear deformation mechanisms in the plastic hinge. Shear deformation in the plastic hinges arises due to two main mechanisms: (i) elongation of plastic hinges; and (ii) inelastic extension of stirrups. These deformation components can be described using truss analogy as shown in Fig. 9, where part (a) shows the shear deformation component arising from elongation of plastic hinges. The solid line shows the shear deformation corresponding to elongation at the peak applied displacement. When the applied displacement reverses (i.e. moving downwards), the diagonal, D1, gets stretched. Since concrete does not carry
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any tension force, opening of the diagonal cracks (i.e., extension of diagonal D1) does not resist any shear force. On the other hand diagonal D2, which had been stretched substantially in tension, has to shorten until the diagonal cracks close so that it can resist compression and contribute to shear resistance. The closure of the diagonal cracks results in a shear displacement without much change in force; thereby inducing a pinched hysteresis loop in the force-displacement diagram as observed in Fig. 8b. Shear deformation from stirrup extension is illustrated in Fig. 9b. In this case when the applied displacement reverses, the two diagonals (D1 and D3) elongate and cannot resist any shear force, whereas the other two diagonals (D2 and D4) shorten; thereby slowly closing the diagonal cracks. Before these cracks close completely for the shear force to pick up, there is a phase where the vertical movement of the beam does not sustain appreciable force. This also contributes to the shear pinching behavior as observed in Fig. 8b. Tests have shown that the inelastic extension of stirrups can be appreciable [Issa 1997; Matti 1998; Walker and Dhakal 2009]. Fig. 10a compares the experimental and analytical shear force versus shear deformation response for beam 2A. It can be seen from the experimental results that shear deformation can be substantial in a ductile RC plastic hinge under large inelastic cyclic loading. The measured shear deformation is roughly 50% of the total deformation at 3.5% drift. The proposed elongating plastic hinge element automatically takes into account shear deformation from beam elongation, but it does not model shear deformation from extension of stirrups. Subsequently, shear deformation is under-predicted beyond the 2.2% drift cycles. Fig. 10b shows the experimental and analytical moment rotation relationship of the plastic hinge. As the model under-predicts the total shear deformation at large displacement cycles, the rotation is over-estimated. The deformation of the top and bottom longitudinal reinforcement within the plastic hinge region at each peak displacement ductility cycle is plotted in Fig. 11. It can be seen that the analysis predicts the reinforcement deformation satisfactorily up to the first cycle of displacement ductility 6. The predicted rotation after this displacement cycle is larger than the experimentally measured
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rotation. It should be noted that a larger rotation in the analysis does not necessarily lead to a larger elongation. In this case, a larger predicted rotation reduces the predicted elongation because it forces the compression steel to yield back rather than the tension steel to extend. This is because the force required to extend the tension reinforcement, in the strain hardening region, is much greater than that to shorten the compression reinforcement and the concrete. Comparisons of the elongation history, force-displacement and moment-rotation relationships for the other five beams tested with no axial force led to similar observations. The analytical and experimental elongations for all six beams are summarized in Fig. 12. In the figure, each data point represents the measured and predicted elongations at the peak of one drift cycle in a test. In this figure with a perfect match, the points would all lie on top of the solid line. In general, the analytical elongation for ductile beams matches satisfactorily with the experimental results till near the end of the test where elongation is slightly under-predicted in the analysis except for a nominally ductile beam AA1. This is due to the rotation being over-predicted in the analysis as explained before.
Beams with different levels of axial force To cover the effect of a wide range of axial force on the general behavior of RC plastic hinges, the analytical and experimental results from two tests are discussed in detail. The first one, i.e. beam S1B, was subjected to an axial compression force of 500 kN (equal to 0.14Agf’c where Ag is the gross area of the beam cross-section, and the second, i.e. beam I1B, was subjected to an axial tension force of 125 kN. Comparisons between experimental and analytical results for these two beams are presented in Fig. 13. It can be seen from the shear force versus shear displacement relationship in Fig. 13c that the pinching behavior reduces dramatically in beam S1B where there is an axial compression force. In this case, the measured shear deformation is only 15% of the total deflection at the end of 6 cycles. Most of the shear deformation arises due to elongation of plastic hinges, which is captured by the analysis. Therefore the predicted force-displacement relationship matches well with the experiment.
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On the other hand, in beam I1B where an axial tension force was applied, the shear pinching becomes much more prominent. The shear deformation at the end of 6 cycles, as illustrated in Fig. 13c, is more than 60% of the total deflection. In this case, the contribution of the extension of the stirrups to the total shear deformation is more than in the previous case. Hence, pinching is under-estimated in the analysis as shown in Fig. 13a. The analysis is still able to predict the initial loading/unloading stiffness, yield/ultimate force and yield displacement accurately as they are not affected by the shear pinching mechanism. Fig. 13b compares the predicted and measured elongation for these two beams. The elongation of beam with axial compression (i.e. S1B) is significantly less than that of the beams with no axial force or beams with axial tension force. The elongation trend for beam I1B (with axial tension) is similar to beam 2A (with no axial force), whereas beam S1B (with axial compression) shows a distinctly different elongation behavior. In the presence of axial compression force, the beam elongates during loading in either direction, but a major portion of the elongation is recovered during unloading; thereby resulting in a small residual elongation at zero displacement. These general trends are captured by the analysis; however elongation is over-predicted by 2.5 mm in S1B and 5 mm in I1B. The over-prediction of elongation for beam I1B is due to the rotation being over-predicted at large displacement cycles in the analysis. This differs from that observed in beam A1 where a larger predicted rotation resulted in a smaller elongation. This is because the axial tension force in the beam forces the tension reinforcement to extend rather than the compression reinforcement to shorten. The experimental elongation histories for beams with different levels of axial force are summarized in Fig. 14a. It can be seen that elongation results vary appreciably between the experiments. However, there is a trend between the amount of elongation and the applied axial force. In general, elongation increases with increasing axial tension force and decreases with increasing axial compression force. This is because the applied axial compression force increases the magnitude of the flexural compression force relative to the flexural tension force in the
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reinforcement. Hence the compression reinforcement has to yield back further and the extension of tensile reinforcement is restrained during load reversals. This consequently reduces the overall beam elongation. The analytical and experimental elongation comparison for beams with different levels of axial force is illustrated in Fig. 14b. The figure shows that the predicted elongation matched reasonably well with the experimental results with the exception of beam M2. The elongation measured in beam M2, which sustained an axial tension force, is smaller than the average elongation measured for beams with no axial force, which defies the general trend. Hence, it is suspected that there may be some errors with the experimental measurements in this test.
DISCUSSION AND CONCLUSIONS A plastic hinge element has been developed and validated using cantilever beam cyclic test results available in literature. Comparisons of the analytical and experimental results have indicated that the proposed plastic hinge element reasonably predicts the inelastic cyclic elongation response of plastic hinges in ductile RC members. This element is also found to be capable of predicting important aspects of hysteresis response such as yield and peak forces, yield displacement, and initial unloading/reloading stiffness for beams sustaining different levels of axial force. The model is also able to predict the relative contribution of shear deformations to the overall drift capacity of RC members and the effect of axial load on it. It may have limited importance in beams (which typically have small or no compressive force); but this could be important for other components, including columns and walls.
Although validation for beams with different levels of axial force
has shown promising results, the proposed plastic hinge element has rooms for improvement. As shear deformation from stirrup extension is not considered in the model, the shear deformation and pinching behavior in the force-displacement response are under-estimated and the rotation is over-predicted by the analysis at large displacement cycles. This leads to larger energy dissipation, which increases hysteretic damping and reduces the peak displacement in time-history analysis.
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Incorporating shear deformation due to stirrup yielding will substantially reduce the predicted rotation, but it will only have minor effect on the elongation predictions. The length of plastic hinge element LP in the current model is calculated assuming that the shear resistance of concrete in plastic hinges is negligible. This may not be the case when an appreciable axial compression force is applied to the beam. More research is required to examine the shear contribution of concrete in plastic hinges with different levels of axial force. The element is developed to predict the behavior of RC plastic hinges up to the peak response (before strength degradation occurs). Therefore, buckling of longitudinal reinforcement, which mainly affects the post-peak response of plastic hinges, is not considered in this model. In addition, it was not possible to quantify the variation of bond slip of beam reinforcement in the beam column joint within the scope of this project. Consequently, the variation in bond slip (strain penetration) with applied displacement cycles was not considered in developing the plastic hinge element. Unlike the conventional analysis approach using nonlinear beam/frame elements, which require calibration to yield a desired hysteresis response, the use of the proposed plastic hinge element enables cyclic behavior of RC plastic hinges to be predicted based on generic concrete and reinforcing bar models that do not require any calibration. In addition, as conventional analytical approaches are unable to predict elongation and inelastic shear deformation, the proposed element offers a significant advancement in conducting analytical seismic performance assessments of RC structures.
REFERENCES American Concrete Institute, (2005). “Building code requirements for structural concrete and commentary (ACI 318M-05),” ACI, Farmington Hills, MI. Carr, A. J. (2004). “RUAUMOKO2D - Inelastic Dynamic Analysis,” Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand.
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Dhakal R. P., and Fenwick R. C. (2008). “Detailing of plastic hinges in seismic design of concrete structures,” ACI Structural Journal, V. 105, No.6, pp. 740-749. Dhakal R. P., and Maekawa, K. (2002a). “Path-dependent cyclic stress-strain relationship of reinforcing bar including buckling,” Engineering Structures, V. 24, No. 11, pp. 1383-1396. Dhakal, R.P., and Maekawa, K. (2002b). “Modeling for postyield buckling of reinforcement.” Journal of Structural Engineering, V. 128, No. 9, pp. 1139-1147. Dhakal, R.P., and Maekawa, K. (2002c). “Reinforcement stability and fracture of cover concrete in reinforced concrete members.” Journal of Structural Engineering, V. 128, No. 10, pp. 1253-1262. Douglas, K. T., Davidson, B. J., and Fenwick, R. C. (1996). “Modelling reinforced concrete plastic hinges,” Eleventh World Conference on Earthquake Engineering, Acapulco, Mexico. Fabio, M., and Mirko, M. (2010). “Nonlinear analysis of spatial framed structures by a lumped plasticity model based on the Haar-Karman principle,” Computational Mechanics, V. 45, No. 6, pp. 647-664. Fenwick, R. C., and Megget, L. M. (1993). “Elongation and load deflection characteristics of reinforced concrete members containing plastic hinges,” Bulletin of the New Zealand Society for Earthquake Engineering, V. 26, No. 1, pp. 28-41. Fenwick, R. C., Tankut A. T., and Thom, C. W. (1981). “The deformation of reinforced concrete beams subjected to inelastic cyclic loading: experimental results,” School of Engineering Report No. 268, University of Auckland, Auckland, New Zealand. Issa, M. S. (1997). “The deformations of reinforced concrete beams containing plastic hinges in the presence of axial compression or tension load,” Master Thesis, University of Auckland, Auckland, New Zealand. Kim, J., Stanton, J., and MacRae, G. (2003). “Effect of beam growth on reinforced concrete frames,” Journal of Structural Engineering, V. 130, No. 9, pp. 1333-1342.
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Lau, D. B. N., Davidson, B. J., and Fenwick, R. C. (2003). “Seismic performance of RC perimeter frames with slabs containing prestressed units,” Pacific Conference on Earthquake Engineering, Christchurch, New Zealand. Lee, J. Y., and Watanabe, F. (2003). “Predicting the longitudinal axial strain in the plastic hinge regions of reinforced concrete beams subjected to reversed cyclic loading,” Engineering Structures, V. 25, No. 7, pp. 927-939. Maekawa, K., Pimanmas, A., and Okamura, H. (2003). “Nonlinear Mechanics of Reinforced Concrete,” 1st Edition, Spon Press, London, 721 pp. Matthews, J. G., Mander, J. B., and Bull, D. K. (2004). “Prediction of beam elongation in structural concrete members using a rainflow method,” Annual Conference of the New Zealand Society of Earthquake Engineering, Rotorua, New Zealand. Matthews, J., Bull, D., and Mander, J.B. (2003). "Hollow-core floor slab performance following a severe earthquake," FIB Conference, Athens, Greece, May 2003. Monti, G., and Spacone, E. (2000). “Reinforced concrete fiber beam element with bond slip,” ASCE Journal of Structural Engineering, V. 126, No. 6, pp. 654-661. Matti, N. A. (1998). “Effect of axial loads on the behaviour of reversing plastic hinges in reinforced concrete beams,” Master Thesis, University of Auckland, Auckland, New Zealand. Paulay, T., and Priestley, M. J. N. (1992). “Seismic design of reinforced concrete and masonry buildings,” Wiley, New York, 744 pp. Peng, B.H.H. (2009). “Seismic performance assessment of reinforced concrete buildings with precast concrete floor systems” http://hdl.handle.net/10092/3103, PhD Thesis, Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand. Rajashankar, J., Iyer Nagesh, R., and Prasad, A.M. (2009). “Modeling inelastic hinges using CDM for nonlinear analysis of reinforced concrete frame structures,” Computers and Concrete, V. 6, No. 4, pp. 319-341.
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Song, H.W., You, D.W., and Byun, K.J. (2002). “Finite element failure analysis of reinforced concrete T-girder bridges,” Engineering Structures, V. 24, No. 2, pp. 151-162. Spacone, E., Filippou, F.C., and Taucer, F.F. (1996). “Fibre beam-column model for nonlinear analysis of RC frames: 1. Formulation,” Earthquake Engineering and Structural Dynamics, V. 25, No. 7, pp. 711-725. Standards New Zealand, (2006). “Concrete Structures Standard: NZS3101:2006,” Standards New Zealand, Wellington, New Zealand. To, N.H.T., Ingham, J.M., and Sritharan, S. (2001). “Monotonic non-linear analysis of reinforced concrete knee joints using strut-and-tie computer models.” Bulletin of the New Zealand Society for Earthquake Engineering, V. 34, No.3, pp. 169-190. Walker, A., and Dhakal, R. P. (2009). “Assessment of material strain limits for defining plastic regions in concrete structures,” Bulletin of the New Zealand National Society for Earthquake Engineering, V. 42, No. 2, pp. 86-95.
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TABLES AND FIGURES List of Tables: Table 1 – Stirrup arrangement and measured material properties for the selected beams Table 2 – Calculated material and section properties for the selected beams Table 3 – Calculated analytical plastic hinge parameters for the selected beams
List of Figures: Fig. 1 – Schematic illustration of the plastic hinge model Fig. 2 – Schematic diagram showing the reinforcement yield length in a RC beam Fig. 3 – Internal forces at low moment end of a RC plastic hinge Fig. 4 – Typical test arrangement and beam configuration (unit in mm): (a) Test set-up; (b) All specimens except AA1 & AA2; (c) Specimens AA1 & AA2; (d) Details of round bars welded to longitudinal reinforcing bars within support block Fig. 5 – Instrumentation attached on the beam Fig. 6 – Typical loading history applied in the beam tests Fig. 7 – Analytical model for cantilever beam Fig. 8 – Analytical and experimental comparisons for beam 2A: (a) Elongation history; (b) Force-displacement response Fig. 9 – Schematic diagram showing shear deformation components in the plastic hinge: (a) Shear deformation arising from beam elongation; (b) Shear deformation arising from stirrup extension Fig. 10 – Shear and flexural behavior of beam 2A: (a) Shear force-shear displacement response; (b) Moment-rotation response Fig. 11 – Reinforcement extension within the plastic hinge of beam 2A: (a) Analysis; (b) Experiment Fig. 12 – Elongation comparison for beams with no axial force Fig. 13 – Analytical and experimental comparisons for beams S1B and I1B: (a) Force-displacement
21
response; (b) Elongation history; (c) Force-shear displacement response Fig. 14 – Elongation for beams with different levels of axial force: (a) Experimental elongation summary; (b) Elongation comparison
Table 1 – Stirrup arrangement and measured material properties for the selected beams Test 2A S1A 1A 1B AA1 AA2 S1B M1 S2A M2 I1B
Cross fc’ Axial force section (MPa) P (kN) 200×500 37.6 0 200×500 37 0 200×500 33.2 0 200×500 42.1 0 250×400 41.5 0 250×400 42.2 0 200×500 37 -500* 200×500 29.4 -200* 200×500 37.8 -100* 200×500 29.4 75 200×500 40 125
Main reinforcement 5D20 top & bottom 5D20 top & bottom 5D20 top & bottom 5D20 top & bottom 3D25 top & bottom 3D25 top & bottom 5D20 top & bottom 5D20 top & bottom 5D20 top & bottom 5D20 top & bottom 5D20 top & bottom
fy fu Stirrups arrangement fvy(MPa) (MPa) (MPa) 306 459 2R10 + R6 @ 100c/c 298(1) 357(2) 331.6 476 2R10 + R6 @ 100c/c 344(1) 391(2) 311 460 2R10 + R6 @ 100c/c 298(1) 357(2) 311 460 2R10 + R6 @ 100c/c 298(1) 357(2) 350 525 HR10 @ 175c/c 445 350 525 HR10 @ 100c/c 445 331.6 478 2R10 + R6 @ 100c/c 344(1) 391(2) 318 577 3R6 @ 55c/c 377 331.6 478 2R10 + R6 @ 100c/c 344(1) 391(2) 318 577 3R6 @ 55c/c 377 320.7 474 3R6 @ 55c/c 331
(1)
Yield stress for R10 stirrup.
(2)
Yield stress for R6 stirrup.
(3)
Shear capacity of the stirrups ∑Asfvyz/s (assuming z = section depth-100).
*
Negative value implies axial compression force.
Table 2 – Calculated material and section properties for the selected beams
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Vs(3) (kN) 227.7 260.5 227.7 227.7 119.9 209.8 260.5 232.7 260.5 232.7 204.3
Test 2A S1A 1A 1B AA1 AA2 S1B M1 S2A M2 I1B
Myc (kNm) 185 200 188 188 155 155 296 230 219 177 170
L (mm) 1500 1500 1500 1500 1420 1420 1500 1500 1500 1500 1500
Vyc (kN) 123 133 125 125 109 109 197 153 146 118 113
sh / y
u / y
13 14 14 14 10 10 14 14 14 14 20
130 62 130 130 80 80 62 116 62 116 154
Table 3 – Calculated plastic hinge element parameters for the selected beams Test 2A S1A 1A 1B AA1 AA2 S1B M1 S2A M2 I1B
Lp (mm) 220 210 220 220 280 160 300 264 224 204 221
(degree) 60 61.3 60 60 47 61.9 52 55.5 59.7 62 60.1
WD (mm) 190.5 192.9 190.5 190.5 205 141 236 218 193 180 192
23
Mmax (kNm) 217 233 233 227 170 179 326 286 265 228 243
Lts (mm) 192 192 192 192 150 150 192 192 192 192 192
Lyield (mm) 463 452 532 498 328 390 380 535 502 578 696
Diagonal concrete spring
Steel spring
Confined concrete spring
WD Rigid link
Cover concrete spring
LP
Fig. 1 – Schematic illustration of the plastic hinge model
Support interface Actual tension force Mmax Theoretical tension force
Myc Le
Lts Lyield
M/V
Fig. 2 – Schematic diagram showing the reinforcement yield length in a RC beam
24
1
2
C1
d - d’
M V
Vs T2 d - d’
Fig. 3 – Internal forces at low moment end of a RC plastic hinge
L High tensile bolts Cantilever beam
Central block
P
Strong floor
(a) Test set-up 200
250 25 mm clear spacing HR10 400
500
R6 (for M1, M2, I1B) R10 (for others) R6
37.5 mm clear cover to reinforcement
30 mm clear cover to reinforcement D25
D20
(b) All specimens except AA1 & AA2
(c) Specimens AA1 & AA2
25
50mm
(d) Details of round bars welded to longitudinal reinforcing bars within support block
Fig. 4 – Typical test arrangement and beam configuration (unit in mm): (a) Test set-up; (b) All specimens except AA1 & AA2; (c) Specimens AA1 & AA2; (d) Details of round bars welded to longitudinal reinforcing bars within support block
l1 l2 l3 l4
t1 h
t2
t3
Dt1 Db1
t4 4
b1
b2
b3
b4
load point linear potentiometers
Fig. 5 – Instrumentation attached on the beam
26
6i 6ii 4i 4ii
4 4
Elastic
2 2
2i 2ii
0
Ductility
Displacement Ductility
6 6
0
-2 -2
0
-4 -4
-2i -2ii -4i -4ii
-6 -6
-6i -6ii
Fig. 6 – Typical loading history applied in the beam tests
L LP Rigid link P
Diagonal spring
Concrete spring
Steel spring
Fig. 7 – Analytical model for cantilever beam
27
16
Analysis
Experiment
Elongation (mm)
14 12 10 8 6 4 2 0 -60
-40
-20
0
20
40
60
Applied Displacement (mm)
(a) Elongation history 200
Analysis
Experiment
Shear Force (kN)
150 100 50 0 -50 -100 -150 -200 -60
-40
-20
0
20
40
60
Applied Displacement (mm)
(b) Force-displacement response Fig. 8 – Analytical and experimental comparisons for beam 2A: (a) Elongation history; (b) Force-displacement response
28
D1 D2
Gap due to diagonal cracks
Beam elongation
(a) Shear deformation arising from beam elongation Stirrup extension
Stirrup extension D3
D1 Gap due to diagonal cracks
D4 D2 Segment 1
Segment 2
(b) Shear deformation arising from stirrup extension Fig. 9 – Schematic diagram showing shear deformation components in the plastic hinge: (a) Shear deformation arising from beam elongation; (b) Shear deformation arising from stirrup extension
29
200
Analysis
Experiment
Shear Force (kN)
150 100 50 0 -50 -100 -150 -200 -30
-20
-10
0
10
20
30
Shear Displacement (mm)
(a) Shear force-shear displacement response 300
Analysis
Experiment
Moment (kNm)
200 100 0 -100 -200 -300 -0.04 -0.03 -0.02 -0.01
0
0.01 0.02 0.03 0.04
Rotation (rads)
(b) Moment-rotation response Fig. 10 – Shear and flexural behavior of beam 2A: (a) Shear force-shear displacement response; (b) Moment-rotation response
30
2i
2ii
4i
4ii
6i
6ii
Vertical height up the beam (mm)
442
6i
250
6ii
58 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
2i 2ii
Elongation (mm)
4i
4ii
(a) Analysis
2i D2i
-D2i2i
2ii D2ii
-D2ii2ii
D4i4i
-D4i4i
4ii D4ii
-D4ii4ii
D6i6i
6i -D6i
6ii D6ii
-D6ii6ii
Vertical height up the beam (mm)
442
250
58 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Elongation (mm)
(b) Experiment Fig. 11 – Reinforcement extension within the plastic hinge of beam 2A: (a) Analysis; (b) Experiment
31
18
Analytical Elongation (mm)
Analytical Elongation (mm)
16 14 12 10
2A (0kN) 8
S1A (0kN) 1A (0kN)
6
1B (0kN) 4
AA1 (0kN) AA2 (0kN)
2
Theoretical line 0 0
2
4
6
8
10
12
14
16
18
Experimental Elongation Elongation (mm) Experimental (mm)
Fig. 12 – Elongation comparison for beams with no axial force Beam S1B with Axial Compression Force Analysis
250
Beam I1B with Axial Tension Force
Experiment
200
Experiment
150
Shear Force (kN)
150
Shear Force (kN)
Analysis
200
100 50 0 -50 -100
100 50 0 -50 -100
-150
-150
-200 -250
-200 -60
-40
-20
0
20
40
60
-80 -60 -40 -20
Applied Displacement (mm)
0
20
40
60
80
Applied Displacement (mm)
(a) Force-displacement response Analysis
7 6
Experiment
25
Elongation (mm)
Elongation (mm)
Analysis
30
Experiment
5 4 3 2
20 15 10 5
1 0
0 -60
-40
-20
0
20
40
60
-80
Applied Displacement (mm)
-40
0
40
Applied Displacement (mm)
32
80
(b) Elongation history Analysis
250
Experiment
200
Experiment
150
Shear Force (kN)
150
Shear Force (kN)
Analysis
200
100 50 0 -50 -100
100 50 0 -50 -100
-150
-150
-200 -250
-200 -10
-5
0
5
10
-60
-40
Shear Displacement (mm)
-20
0
20
40
60
Shear Displacement (mm)
(c) Force-shear displacement response Fig. 13 – Analytical and experimental comparisons for beams S1B and I1B: (a)
25
25
S1B (-500kN) M1 (-200kN) Analytical Elongation (mm)
Experimental Measured Elongation (mm)
Force-displacement response; (b) Elongation history; (c) Force-shear displacement response
20 S2A (-100kN) 15
M2 (+75kN) I1B (+125kN)
10
5
20
15 S1B (-500kN) M1 (-200kN)
10
S2A (-100kN) M2 (+75kN)
5
I1B (+125kN) Theoretical line
0
2i
4i
6i
0
6iii
0
Applied Displacement Cycle
5
10
15
20
25
Experimental Elongation (mm)
(a) Experimental elongation summary
(b) Elongation comparison
Fig. 14 – Elongation for beams with different levels of axial force: (a) Experimental elongation summary; (b) Elongation comparison
33
