Keywords

1 Introduction

Although an elastic method is suitable for the seismic analysis of many regular structures, it cannot predict the failure mechanisms and the redistribution of forces during a progressive yielding. Therefore, for complex structures, an advanced seismic analysis that employs a nonlinear method should be used. The continuum finite element model, distributed plasticity model, and lumped plasticity model are among techniques used for the nonlinear analysis of structures. However, due to its simplicity, computational efficiency, and supports from seismic design codes, the lumped plasticity model has been broadly used by practice engineers and researchers [1,2,3,4]. Despite its wide application for seismic analysis of different structures, the accuracy of the lumped plasticity model requires further investigations [5]. It has been shown that the plastic hinge length, stiffness reduction factors, and the location of plastic hinges can significantly affect the results of a nonlinear analysis that use the lumped plasticity model [5,6,7,8]. Besides, the plastic hinges’ moment-rotation relationships and their damage levels’ acceptance criteria can later the results of a nonlinear analysis that make use of the lumped plasticity model. In this study, the results of an experiment conducted on two reinforced concrete frames (RC) are used to examine the efficiency of the lumped plasticity model in estimating their force-displacement relationships. The first frame represents a special moment resisting RC frame and the second one is a low-ductile RC frame with inadequate lap splice length.

2 Details of the Selected Frames and Conducted Experiment

The selected frames for this study are show in Fig. 1. As the Fig. 1 shows, the RC frames have a similar geometry, reinforcement ratio, and material properties. However, their seismic detailing is quite different. While the first frame (referred to as Frame 1) satisfies the requirements of ACI 318 [9] for a special moment frame, the second frame (referred to as Frame 2) suffers from many deficiencies. Inadequate lap splice length in the reinforcing bars of columns, the use of a 90-degree hook in stirrups, large distance between stirrups at critical locations, and joints without any shear link are among the main deficiencies of the second frame. The 28-days compressive strength of the employed concrete measured on the standard cylinder was 30 MPa. Table 1 displays the mechanical properties of the employed reinforcing bars in frames. The frames were subjected to a similar quasi-static loading and their force-displacement responses were recorded at their beam level. Figure 2 depicts the obtained force-displacement relationships for the frames. As can be seen, Frame 1 has a slightly larger ultimate load than Frame 2 and its displacement at the ultimate load is also significantly larger than that of the Frame 2. Both frames have shown a linear response up to the lateral displacement of 3.2 mm. Frame 1 has reached the ultimate load of 61.48 kN at the lateral displacement of 90.3 mm. On the other hand, Frame 2 has reached an ultimate load of 54.5 kN at the lateral displacement of 66.8 mm. More detail about these frames can be found in [10].

Fig. 1.
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Details of the tested frames (a) Frame 1 (b) Frame 2.

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Table 1. Mechanical properties of the frames’ reinforcing bars
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Fig. 2.
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The backbone curves of tested frames (a) Frame 1 (b) Frame 2.

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3 Finite Element Simulation

As mentioned earlier, the lumped plasticity model was employed to simulate the inelastic response of the frames under a gradually increasing lateral load (i.e., Pushover analysis). Figure 3 shows the considered locations for the plastic hinges in both frames. As can be seen from Fig. 3a, two plastic hinges were assigned to the both ends of beams and columns. The moment-rotation relationships of plastic hinges followed the shown multiline representation in Fig. 3b. In this figure, line AB shows the elastic stiffness of the element, with point B as the yield point. Besides, line BC displays the post-yield stiffness, with point C as the ultimate capacity of the plastic hinge. Line CDE represents the strength degradation of the plastic hinge after passing its ultimate capacity. In order to determine the required parameters of Fig. 3b (i.e., a, b and c) two different methods were employed. In the first approach, these parameters were determined based on the recommendation of ASCE/SEI 41 [11] as shown in Tables 2 and 3. The yield and ultimate capacities of the plastic hinges were calculated based on the elements’ cross-sectional and material properties using the given equations in ACI 318 [9]. As shown in the tables, the beams were classified based on their stirrup spacing into conforming (i.e., stirrup spacing was less than 1/3 of beams’ effective depth) and non-conforming (i.e., stirrup spacing was larger than 1/2 of beams’ effective depth). In the second approach, the required parameters in Fig. 3b were determined based on the moment-curvature analysis of elements’ cross-section. For this purpose, at first, the cross-section of elements was divided into small fibres of steel and concrete materials. Then, nonlinear material properties were assigned to these fibres. For steel fibres, the values shown in Table 1 were used to determine the stress-strain relationships. However, the concrete’s stress-strain relationships were developed based on Mender’s equations [12] for confined and unconfined concrete elements. The plastic hinges’ length was estimated using the equations suggested by Paulay and Priestley [13]. Table 3 displays the obtained results from the moment-curvature analysis. It should be motioned that the effect of inadequate lap splice length in the columns of Frame 2 was taken into account using the proposed method in ASCE/SEI 41 [11]. In this method, the yield stress of the reinforcing bars with inadequate lap splice length is reduced proportional to the ratio of the provided lap length to the required lap length. Based on this method, the yield stress of longitudinal reinforcing bar in the columns of Frame 2 (only at the location of lap length) was reduced from 537 MPa to 475 MPa. The effective stiffness of frames were taken into account using two different methods. First, the recommended values by ASCE/SEI 41 [11] were used (i.e., 0.3EI and 0.7EI for the flexural action of columns and beams, respectively), and then the proposed value by Eurocode 8 [14] (i.e., 0.5EI for beams and columns) was examined.

Fig. 3.
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(a) Location of plastic hinges in the FE models (b) Employed forced-displacement relationship for the plastic hinges.

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Table 2. Moment-rotation parameters used for nonlinear analysis of beams
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Table 3. Moment-rotation parameters used for nonlinear analysis of columns
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4 Results and Discussions

Figures 4 and 5 compare the experimentally obtained force-displacements relationships of Frame 1 and Frame 2 with those predicted by the finite element models. As shown in the figures, the employed methods for the calculations of the plastic hinge parameters (i.e., ASCE 41–17 and moment-curvature approaches) have had an insignificant effect on the obtained force-displacement relationships from the finite element models. However, the selected effective stiffness for structural elements has had a significant effect on the obtained force-displacement relationships. In both frames, regardless of the employed method for the calculation of plastic hinges parameters and the selected values for the effective stiffness of structural members, the ultimate loads have been estimated with good accuracy. The maximum difference between the obtained ultimate loads for Frame 1 and Frame 2 and that of the experiment are, respectively, 13.38% and 6.37%. The obtained results for the ultimate loads also show that the proposed method by ASCE/SEI 41–17 for the consideration of inadequate lap splice length has been efficient as the ultimate load of Frame 2 has been estimated accurately. On the other hand, the displacement corresponding to the ultimate loads have been underestimated in all models. Although the use of ASCE/SEI 41–17’s effective stiffness has reduced the difference between the displacements at ultimate loads, the differences in the results are around 33.9% for Frame 1 and 22.9% for Frame 2. It is also noteworthy that, although ASCE/SEI 41–17’s effective stiffness values estimate the displacement at the ultimate load better than that of Eurocode 8’s effective stiffness values, it has predicted the initial stiffness of frames with a lower accuracy. Figure 6 displays the predicted damage level of the plastic hinges at the ultimate load of frames. As can be seen, all models predict severe damage to the base of columns (i.e., CP hinges) while they expect a minor damage to the beam (i.e., IO hinges). This prediction correlates well with the observed damage to the beams and columns of both frames. In order words, the finite element models have been able to predict the damaged zone and the intensity of damage with a good precision.

Fig. 4.
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Comparison between force-displacement relationships of finite element models and experiment for Frame 1 (a) using ASCE/SEI 41 effective stiffness values (b) using Eurocode 8 effective stiffness values.

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Fig. 5.
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Comparison between force-displacement relationships of finite element models and experiment for Frame 2 (a) using ASCE/SEI 41 effective stiffness values (b) using Eurocode 8 effective stiffness values.

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Fig. 6.
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Damage level of the plastic hinges (a) Frame 1 using ASCE/SEI 41 effective stiffness values (b) Frame 1 using Eurocode 8 effective stiffness values (c) Frame 2 using ASCE/SEI 41 effective stiffness values (2) Frame 2 using Eurocode 8 effective stiffness values.

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5 Conclusion

This study investigated the accuracy of two different lumped plasticity models for estimating the force-displacement relationships of two similar RC frames but with different seismic detailing. Comparisons with the experimental results indicated that both ASCE/SEI 41–17 and moment-curvature lumped plasticity models estimated the ultimate loads of both frames with less than 15% error. However, both models underestimated the displacements corresponding to the ultimate loads by 33.9%. Therefore, the lumped plasticity model should be used in the performance-based seismic design methods like the capacity spectrum approach with precaution as the target displacement may be estimated inaccurately. The finite element models were able to predict the damage level and failure mode similar to the obtained results from the experiment.