A relative reliability approach for Direct Displacement-Based Design (DDBD) is first proposed in this paper, which is based on the average reliability level implicit in current Chinese design codes. By introducing a relative reliability coefficient
Real seismic response of structures shows that a traditional force-based seismic approach is difficult to accord with the expected structural displacement under the designed earthquake risk. Towards this, Displacement-Based Seismic Design (DBSD) was proposed and extensively studied in the last decades.
Nowadays, DBSD has been developed and employed in various structures. In 1995, a DDBD method [
According to statistical analysis, Gao et al. pointed out that, under the confidence 0.05 or 0.1, the ultimate drift ratio of a RC frame obeys a lognormal distribution or an extreme I distribution. They also studied the seismic reliability of ultimate deformation capacity of RC frames under rare earthquake risk [
The current studies on reliability approaches for DDBD are useful to instruct seismic design. However, the complete reliability theoretical system is still not established. Influences of structural reliability may differ from one analysis to another, and it is difficult to precisely determine them, such as variation coefficients,
The function
To assess the reliability in DDBD effectively, it is necessary to calculate the correlation standard deviation of the resistance and the effect of structure precisely, according to variation coefficients.
To establish the reliability approach for DDBD, evaluation of reliability at the given performance target is first transformed to the expected seismic risk level under frequent earthquake risk, while the section bearing capacity of structure is determined by a similar way as FBD. For instance, in Figure
Theory of the method.
As shown in Figure
To determine the nominal reliability
Similarly, the average reliability under the performance target can be evaluated by
For one performance target, with the same material parameters, structural model, and seismic measures, deviation coefficients from FBD based on current Chinese codes will be consistent with that from DDBD. Thus, the relative reliability coefficient can be introduced by
If there is the same performance level between FBD and DDBD (
Structural reliability depends on the expression of section bearing capacity and the seismic measures. Theoretically, once the concept of mode decomposition is introduced to DDBD [
The design steps of relative reliability approach for DDBD are given as follows: Determine the performance target of structure according to the proprietor. Preliminary design: determine the structural arrangement, section dimension, material strength and load action, and so forth. Evaluate the average resistance Based on the performance target, determine the relative reliability coefficient Determine the designed displacement Determine the equivalent parameters of the equivalent single degree of freedom system (ESDOF) for DDBD (for tall frames the method of mode decomposition should be used [ Calculate the base shear and seismic forces in DDBD. Use the same expression of section bearing capacity as in current Chinese codes for reinforcement design. Conduct the deformation capacity design and seismic measures design.
Given this paper is a part of the team’s research on the prestressed frame, the subject of this paper is the partial prestressed frame. However, there is no essential difference between the proposed DDBD procedures for ordinary frame and for PPRC frame.
Specifically, the main differences of the proposed DDBD procedure for PPRC frame are as follows: The prestressed reinforcements are determined in preliminary design (step 2) and not changed in the following process. In the process of determining the target displacement (from step 3 to step 5), the reference displacement demands are the displacement demands of PPRC frames under the same earthquake risk. After the base shear of DDBD is derived and the design force of the sections is determined, the design of sections, the seismic measures, and details of seismic design all follow the rules for PPRC frame in the codes of China.
To verify the suggested relative reliability approach for DDBD, 24 PPC frames were first designed by FBD based on current Chinese codes, as shown in Table
Basic parameters of PPC frames.
Number of frames | Site category | Number of spans | Fortification intensity | Number of stories | Total height (m) | Seismic level |
---|---|---|---|---|---|---|
YKJ-1 | II | 2 | 7-degree 0.1 g | 4 | 20.4 | 3 |
YKJ-2 | II | 2 | 8-degree 0.2 g | 4 | 20.4 | 2 |
YKJ-3 | II | 2 | 8-degree 0.2 g | 3 | 15.6 | 2 |
YKJ-4 | II | 2 | 7-degree 0.1 g | 3 | 15.6 | 3 |
YKJ-5 | II | 2 | 7-degree 0.1 g | 5 | 25.2 | 2 |
YKJ-6 | II | 2 | 8-degree 0.2 g | 5 | 25.2 | 1 |
YKJ-7 | II | 2 | 7-degree 0.1 g | 7 | 34.8 | 2 |
YKJ-8 | II | 2 | 8-degree 0.2 g | 7 | 34.8 | 1 |
YKJ-9 | II | 3 | 7-degree 0.1 g | 7 | 34.8 | 2 |
YKJ-10 | II | 3 | 8-degree 0.2 g | 7 | 34.8 | 1 |
YKJ-11 | II | 3 | 7-degree 0.1 g | 5 | 25.2 | 2 |
YKJ-12 | II | 3 | 8-degree 0.2 g | 5 | 25.2 | 1 |
YKJ-13 | II | 3 | 7-degree 0.1 g | 4 | 20.4 | 3 |
YKJ-14 | II | 3 | 8-degree 0.2 g | 4 | 20.4 | 2 |
YKJ-15 | II | 3 | 7-degree 0.1 g | 3 | 15.6 | 3 |
YKJ-16 | II | 3 | 8-degree 0.2 g | 3 | 15.6 | 2 |
YKJ-17 | III | 2 | 7-degree 0.1 g | 4 | 20.4 | 3 |
YKJ-18 | III | 2 | 8-degree 0.2 g | 4 | 20.4 | 2 |
YKJ-19 | III | 2 | 7-degree 0.1 g | 7 | 34.8 | 2 |
YKJ-20 | III | 2 | 8-degree 0.2 g | 7 | 34.8 | 1 |
YKJ-21 | III | 3 | 7-degree 0.1 g | 7 | 34.8 | 2 |
YKJ-22 | III | 3 | 8-degree 0.2 g | 7 | 34.8 | 1 |
YKJ-23 | III | 3 | 7-degree 0.1 g | 4 | 20.4 | 3 |
YKJ-24 | III | 3 | 8-degree 0.2 g | 4 | 20.4 | 2 |
With
Theoretically, the calculated base shear of structure under frequent earthquake risk from both DDBD and FBD is the same, due to the basic assumptions of ESDOF. However, inconsistence still exists due to the model errors of the DDBD method. To eliminate such errors in DDBD, the base shear derived from DDBD and from FBD has been compared by each example of 24 PPC frames, as shown in Table
Base shear comparison between force-based design and DDBD.
Number of frames | Base shear from FBD (kN) | Base shear from DDBD (kN) | Base shear (FBD)/base shear (DDBD) | Error |
---|---|---|---|---|
YKJ-1 | 383.632 | 310.68 | 1.23 | −19.02% |
YKJ-2 | 897.37 | 780.51 | 1.15 | −13.02% |
YKJ-3 | 666.58 | 684.59 | 0.97 | 2.70% |
YKJ-4 | 342.81 | 290.31 | 1.18 | −15.31% |
YKJ-5 | 464.48 | 363.24 | 1.28 | −21.80% |
YKJ-6 | 1031.15 | 891.58 | 1.16 | −13.54% |
YKJ-7 | 544.64 | 441.4 | 1.23 | −18.96% |
YKJ-8 | 1188.89 | 983.03 | 1.21 | −17.32% |
YKJ-9 | 834.33 | 676.5 | 1.23 | −18.92% |
YKJ-10 | 1782.51 | 1482.73 | 1.20 | −16.82% |
YKJ-11 | 693.87 | 545.57 | 1.27 | −21.37% |
YKJ-12 | 1549.93 | 1394.74 | 1.11 | −10.01% |
YKJ-13 | 561.615 | 419.65 | 1.34 | −25.28% |
YKJ-14 | 1315.35 | 1132.17 | 1.16 | −13.93% |
YKJ-15 | 493.85 | 404.59 | 1.22 | −18.07% |
YKJ-16 | 1013.83 | 1034.08 | 0.98 | 2.00% |
YKJ-17 | 486.19 | 402.47 | 1.21 | −17.22% |
YKJ-18 | 1136.12 | 985.24 | 1.15 | −13.28% |
YKJ-19 | 690.2 | 566.92 | 1.22 | −17.86% |
YKJ-20 | 1506.34 | 1239.11 | 1.22 | −17.74% |
YKJ-21 | 1057.1 | 863.32 | 1.22 | −18.33% |
YKJ-22 | 2258.44 | 1850.16 | 1.22 | −18.08% |
YKJ-23 | 711.57 | 542.73 | 1.31 | −23.73% |
YKJ-24 | 1666.55 | 1409.73 | 1.18 | −15.41% |
Mean | 1.20 | −15.85% |
Note: error = (base shear from DDBD − base shear from FBD)/base shear from DDBD.
Table
To verify the effectiveness of the suggested relative reliability approach for DDBD, 10 PPC frames were tested. Aimed at the performance target of “being intact after frequent earthquakes,” the structure was designed by the relative reliability approach for DDBD (with the same material properties, dimensions of members used in FBD based on current Chinese codes). Subsequently, the reliability level at both “being intact after frequent earthquakes” and “not collapsed after rare earthquakes” based on DDBD was compared with that from FBD.
The basic information of 10 examples is shown in Table
Basic parameters of 10 PPC frames.
Number of frames | Site category | Number of spans | Fortification intensity | Number of stories | Total height (m) | Seismic level |
---|---|---|---|---|---|---|
YKJ-1 | II | 2 | 7-degree 0.1 g | 4 | 20.4 | 3 |
YKJ-2 | II | 2 | 8-degree 0.2 g | 4 | 20.4 | 2 |
YKJ-7 | II | 2 | 7-degree 0.1 g | 7 | 34.8 | 2 |
YKJ-8 | II | 2 | 8-degree 0.2 g | 7 | 34.8 | 1 |
YKJ-9 | II | 3 | 7-degree 0.1 g | 7 | 34.8 | 2 |
YKJ-10 | II | 3 | 8-degree 0.2 g | 7 | 34.8 | 1 |
YKJ-13 | II | 3 | 7-degree 0.1 g | 4 | 20.4 | 3 |
YKJ-14 | II | 3 | 8-degree 0.2 g | 4 | 20.4 | 2 |
YKJ-19 | III | 2 | 7-degree 0.1 g | 7 | 34.8 | 2 |
YKJ-20 | III | 2 | 8-degree 0.2 g | 7 | 34.8 | 1 |
As all these examples are regular PPC frames with total height less than 40 m, it is reasonably assumed that the first mode of vibration will dominate the structural response during the analysis. Numerical results for performance points and reliability indices are shown in the following sections.
Tables
Performance point coordinates and related parameters of frame designed according to DDBD under frequent earthquake risk.
Number of frames | Performance point ( |
Performance point ( |
Story drift ratio of performance point | Target drift ratio | Relative error |
---|---|---|---|---|---|
YKJ-1 | (0.00946, 0.3081) | (277.10, 0.01139) | 1/1330 | 1/1399 | 4.93% |
YKJ-2 | (0.01538, 0.7411) | (679.72, 0.01853) | 1/826 | 1/850 | 2.82% |
YKJ-7 | (0.01271, 0.2370) | (368.04, 0.01585) | 1/1733 | 1/1399 | −19.3% |
YKJ-8 | (0.02148, 0.5505) | (873.99, 0.02680) | 1/969 | 1/850 | −14.0% |
YKJ-9 | (0.01193, 0.2506) | (579.10, 0.01488) | 1/1827 | 1/1399 | −23.4% |
YKJ-10 | (0.02089, 0.5642) | (1328.4, 0.02610) | 1/995 | 1/850 | −17.0% |
YKJ-13 | (0.01000, 0.2933) | (390.00, 0.01204) | 1/1226 | 1/1399 | 12.4% |
YKJ-14 | (0.01523, 0.7473) | (1009.6, 0.01835) | 1/821 | 1/850 | 3.41% |
YKJ-19 | (0.01626, 0.2975) | (461.99, 0.02028) | 1/1383 | 1/1399 | 1.14% |
YKJ-20 | (0.02714, 0.6991) | (1109.9, 0.03387) | 1/769 | 1/850 | 9.53% |
Note: relative error = (story drift ratio of performance point − target drift ratio)/story drift ratio of performance point.
Performance point coordinates and related parameters of frame designed according to DDBD under rare earthquake risk.
Number of frames | Performance point ( |
Performance point ( |
Story drift angle of performance point |
---|---|---|---|
YKJ-1 | (0.06463, 1.361) | (1223.9, 0.07784) | 1/160 |
YKJ-2 | (0.10150, 2.501) | (2293.9, 0.12230) | 1/111 |
YKJ-7 | (0.08813, 1.158) | (1798.3, 0.10991) | 1/235 |
YKJ-8 | (0.14140, 2.109) | (3348.3, 0.17644) | 1/138 |
YKJ-9 | (0.08282, 1.241) | (2868.0, 0.10329) | 1/238 |
YKJ-10 | (0.13840, 2.175) | (5120.8, 0.17272) | 1/145 |
YKJ-13 | (0.06858, 1.256) | (1670.1, 0.08259) | 1/153 |
YKJ-14 | (0.10080, 2.580) | (3485.5, 0.12143) | 1/119 |
YKJ-19 | (0.10710, 1.273) | (1976.9, 0.13357) | 1/186 |
YKJ-20 | (0.17800, 2.505) | (3977.0, 0.22205) | 1/104 |
As shown in the last column of Table
If the effect was defined by the maximum story drift ratio at the performance point of the expected earthquake, while the resistance was represented by the maximum story drift ratio corresponding to the expected performance level, the reliability indices for each performance level can be calculated (as shown in Tables
The reliability index of 10 PPC frames under frequent earthquake.
Number of frames | Methods | The design base shear | Resistance |
Effect |
Reliability index | ||
---|---|---|---|---|---|---|---|
Story drift ratio | Coefficient of variation | Story drift ratio | Coefficient of variation | ||||
YKJ-1 | DDBD | 373.092 | 1/486 | 0.22 | 1/1330 | 0.31 | 2.760 |
FBD | 383.632 | 1/480 | 0.22 | 1/1398 | 0.31 | 2.927 | |
|
|||||||
YKJ-2 | DDBD | 905.484 | 1/482 | 0.22 | 1/826 | 0.31 | 1.504 |
FBD | 897.370 | 1/517 | 0.22 | 1/883 | 0.31 | 1.495 | |
|
|||||||
YKJ-7 | DDBD | 420.360 | 1/609 | 0.22 | 1/1733 | 0.31 | 2.864 |
FBD | 544.640 | 1/578 | 0.22 | 1/1814 | 0.31 | 3.127 | |
|
|||||||
YKJ-8 | DDBD | 1022.93 | 1/320 | 0.22 | 1/969 | 0.31 | 2.811 |
FBD | 1188.89 | 1/311 | 0.22 | 1/998 | 0.31 | 3.187 | |
|
|||||||
YKJ-9 | DDBD | 625.572 | 1/655 | 0.22 | 1/1827 | 0.31 | 2.811 |
FBD | 834.330 | 1/587 | 0.22 | 1/1875 | 0.31 | 3.174 | |
|
|||||||
YKJ-10 | DDBD | 1516.80 | 1/360 | 0.22 | 1/995 | 0.31 | 2.786 |
FBD | 1782.51 | 1/367 | 0.22 | 1/1017 | 0.31 | 2.793 | |
|
|||||||
YKJ-13 | DDBD | 551.832 | 1/438 | 0.22 | 1/1226 | 0.31 | 2.820 |
FBD | 561.615 | 1/402 | 0.22 | 1/1263 | 0.31 | 3.130 | |
|
|||||||
YKJ-14 | DDBD | 1334.77 | 1/448 | 0.22 | 1/821 | 0.31 | 1.684 |
FBD | 1315.35 | 1/478 | 0.22 | 1/868 | 0.31 | 1.66 | |
|
|||||||
YKJ-19 | DDBD | 657.432 | 1/562 | 0.22 | 1/1383 | 0.31 | 2.475 |
FBD | 690.200 | 1/568 | 0.22 | 1/1454 | 0.31 | 2.581 | |
|
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YKJ-20 | DDBD | 1599.89 | 1/311 | 0.22 | 1/770 | 0.31 | 2.491 |
FBD | 1506.34 | 1/309 | 0.22 | 1/783 | 0.31 | 2.553 |
The reliability index of 10 PPC frames under rare earthquake.
Number of frames | Methods | Resistance |
Effect |
Reliability index | ||
---|---|---|---|---|---|---|
Story drift ratio | Coefficient of variation | Story drift ratio | Coefficient of variation | |||
YKJ-1 | DDBD | 1/116.5 | 0.359 | 1/160 | 0.407 | 0.636 |
FBD | 1/116 | 0.359 | 1/169 | 0.407 | 0.749 | |
|
||||||
YKJ-2 | DDBD | 1/98 | 0.359 | 1/111 | 0.407 | 0.268 |
FBD | 1/93 | 0.359 | 1/120 | 0.407 | 0.517 | |
|
||||||
YKJ-7 | DDBD | 1/117 | 0.359 | 1/235 | 0.407 | 1.362 |
FBD | 1/115 | 0.359 | 1/248 | 0.407 | 1.497 | |
|
||||||
YKJ-8 | DDBD | 1/101 | 0.359 | 1/138 | 0.407 | 0.632 |
FBD | 1/101 | 0.359 | 1/136 | 0.407 | 0.598 | |
|
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YKJ-9 | DDBD | 1/120 | 0.359 | 1/238 | 0.407 | 1.338 |
FBD | 1/117 | 0.359 | 1/252 | 0.407 | 1.495 | |
|
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YKJ-10 | DDBD | 1/106 | 0.359 | 1/146 | 0.407 | 0.642 |
FBD | 1/103 | 0.359 | 1/141 | 0.407 | 0.630 | |
|
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YKJ-13 | DDBD | 1/112 | 0.359 | 1/153 | 0.407 | 0.626 |
FBD | 1/113 | 0.359 | 1/164 | 0.407 | 0.742 | |
|
||||||
YKJ-14 | DDBD | 1/93 | 0.359 | 1/119 | 0.407 | 0.501 |
FBD | 1/94 | 0.359 | 1/120 | 0.407 | 0.497 | |
|
||||||
YKJ-19 | DDBD | 1/115 | 0.359 | 1/186 | 0.407 | 0.948 |
FBD | 1/116 | 0.359 | 1/193 | 0.407 | 1.002 | |
|
||||||
YKJ-20 | DDBD | 1/99 | 0.359 | 1/105 | 0.407 | 0.143 |
FBD | 1/102 | 0.359 | 1/108 | 0.407 | 0.139 |
The average reliability indices from FBD between the performance level of “being intact after frequent earthquakes” and the performance level of “not collapsed after rare earthquakes” are 2.517 and 0.669 [
The contrast of the reliability index under frequent earthquake risk.
The contrast of the reliability index under rare earthquake risk.
The triangular points shown in Figures
Comparing results in Figure The reinforcement design of a structure may not be dominated by the seismic effect. Seismic design of PPC frames is strongly dependent on the anticrack measures and seismic measures. The lateral displacement mode of DDBD in this paper is based on the cantilever mode [ Current approach of DDBD is directly based on Priestley’s work [
Based on the average reliability level in current Chinese codes for the design of PPC frame structures, a relative reliability approach for DDBD is proposed. The calculation of reliability at any performance level can be transformed to the nominal reliability of frequent earthquake for the expected earthquake risk level. In this approach, the displacement demand (maximum story drift ratio) under frequent earthquake risk is used as effect, while the maximum displacement (maximum story drift ratio) corresponding to the performance level is used as resistance. By introducing the relative reliability coefficient Based on the assumption of elasticity, if the displacement corresponding to the performance point of frequent earthquake from FBD was adopted as the target displacement in DDBD, the calculated base shear from FBD (mode decomposition method) will be equal to that from DDBD. Meanwhile, the error analysis was conducted between the base shear from DDBD and that from FBD, and a correction factor was suggested for calculation of the base shear in DDBD. Under frequent earthquake risk, the section bearing capacity for the suggested relative reliability approach for DDBD can also be expressed by the formula of section bearing capacity of FBD in current Chinese codes. The displacements of the PPC frames designed by the suggested relative reliability approach for DDBD are in effective control, and the reliability of the performance target is much closer to the mean of reliability in current Chinese codes. The suggested approach is verified to be effective and superior.
A relative reliability approach for DDBD is first proposed in this paper; therefore, significant improvements of its theoretical framework are still required. The numerical examples for selection of parameters (such as target displacement) in the suggested approach are limited. The suggested approach is based on the bilinear model of ESDOF, which may seriously affect the calculation accuracy of DDBD. During the theoretical derivation, the structure is assumed in elasticity under frequent earthquake risk, which may make errors in reality. Towards this, a correction factor has been introduced in this paper to revise the base shear in DDBD. In this paper, the relative reliability approach for DDBD is introduced based on Chinese codes, but it is obvious that this approach can also be operable on the other codes.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by “the Fundamental Research Funds for the Central Universities” of China (Grant no. 106112015CDJXY200004). The authors are grateful to Professor Hong Yang for enlightening discussions.