Dynamic Stability and Design of Cantilever Bridge Columns

Transcription

1 Proceedings of the Ninth Pacific Conference on Earthquake Engineering Building an Earthquake-Resilient Society April, 211, Auckland, New Zealand Dynamic Stability and Design of Cantilever Bridge Columns T.Z. Yeow, G.A. MacRae & V.K. Sadashiva Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand K. Kawashima Department of Civil Engineering, Tokyo Institute of Technology, Tokyo, Japan. ABSTRACT: In congested metropolitan areas it is often difficult to build bridge columns directly and concentrically below the bridge due to space limitations. Columns with a horizontal cantilever in one direction, forming an inverted "L", are sometimes used. Due to the presence of eccentric loads with this setup, some codes, such as the Japanese Road Association code, require the flexural strength in the direction of eccentricity to be larger than that in the opposite direction by the size of the eccentric moment. However, application of the Hysteresis Centre Curve concept indicates that the strength difference should be doubled for structures with bilinear hysteresis loops. To evaluate the strength difference required, inelastic dynamic time history analyses were conducted using a suite of ground motion records. The columns had different strength differences, periods and capacity reduction factors. The columns were modelled using Takeda and elastoplastic hysteresis loops to observe hysteretic shape effects. From analyses, the optimum strength difference that causes the smallest average residual displacements and the smallest average maximum displacement was found to be 2.5 and 2.3 times the eccentric moment respectively. It is recommended to use 2.5 times the eccentric moment as the strength difference in design. 1 INTRODUCTION Cantilever bridge columns are bridge columns with a cantilever in one direction, such as that shown in Figure 1. These are used in locations where little space is available to install new road infrastructure. The eccentric load, P, causes a static eccentric moment, M e, about the base of the column, possibly resulting in a tendency for the column to deform more in one direction compared to the other. In design, columns are generally provided with more reinforcing bars on one side than the other in recognition of this effect, as shown in Figure 1. MacRae and Kawashima (1993) developed and applied a concept called the Hysteresis Centre Curve (HCC) to explain the possibility of more deformation in one direction than the other for a cantilever bridge column. The HCC is the curve joining the averages of the positive and negative yield points (including signs) along the same load/unload line. By using this concept, they demonstrated that the dynamic stability range of the column was reduced in the direction of the eccentricity but increased in the other. This causes the column to have a greater tendency to collapse in the direction of the eccentricity compared to the other. The HCC concept was also used to devise a countermeasure against the decrease in the stability range. The countermeasure involves increasing the flexural strength in the direction of the eccentricity by providing significantly more reinforcing bars than that currently used in design. While the HCC concept seems to be rational and powerful, there is a need to (i) run simple and robust analyses to either verify or disprove the concept for the bilinear structures for which it was developed, (ii) to quantify the behaviour for columns with more realistic hysteresis curves and to (iii) develop a procedure for designing such cantilever bridge columns. Paper 97

2 Steel Reinforcing Bars F More bars required on this side due to higher tension demand Figure 1 Typical Cross Section of Cantilever Bridge Column (left most), Simplified Model of Cantilever Bridge Column (middle left), Static Bending Moment Diagram (middle right) and Photograph of Cantilever Bridge Column (right most) In order to address the needs described above, answers will be sought to the following questions regarding cantilever bridge columns: 1) What is the optimum increase in flexural strength for bilinear columns without P-delta effects? 2) What is the optimum increase in flexural strength which would result in the smallest average residual displacements of columns subject to earthquake action? 3) What is the optimum increase in flexural strength which would result in the smallest average maximum displacement of columns subject to earthquake action? 4) What is the sensitivity of the optimal flexural strengths in the different directions with respect to the column s capacity reduction factor (instead of applying ductility), natural period and hysteretic loop shape? 5) How can the prediction of optimal response of columns be improved? 6) How should these columns be designed for optimal performance? 2 RELATED STUDIES 2.1 Application of the Hysteresis Centre Curve Concept The HCC can be used to assess the dynamic stability of a segment of a hysteresis loop. This is demonstrated in Figure 2. As a result of the eccentric moment Pd shown in Figure 1, the baseline of the hysteresis loop shifts up by Pd /L (the equivalent lateral force) where L is the height of the column. The intercept of this new baseline with the HCC is the new point of stability as shown in the leftmost diagram in Figure 2. It can be seen that the stability range in the positive direction decreases while the stability range in the negative direction increases. This implies that there is a higher probability for the column to deform in the positive direction. Note that the shakedown displacement labelled in the diagram is the distance between the zero displacement position and the point of stability. In theory, to have the new point of stability at the zero displacement position, the HCC line should be shifted up by Pd /L as shown on the rightmost diagram in Figure 2. Therefore, the intersect between the HCC line and the new baseline is at the zero displacement position. The simplest way is to increase the strength in the positive direction by 2Pd. This would shift the top yield line up by 2Pd /L, resulting in an average increase by Pd /L. This is twice the required strength increase in the Japanese Road Association code (JRA, 199). By doing this, the stability range in both directions is equal. The diagram on the right in Figure 2 shows graphically how the countermeasure works. Increasing the strength by more than 2Pd would however shift the point of stability in the negative direction and decrease the stability range in the negative direction. Therefore, the column would have a tendency to collapse in the negative direction. Hence, 2Pd is the optimal strength increase for an idealized hysteresis loop. 2

3 Figure 2 - Effect of Eccentric Moment (left) and Possible Countermeasure (right) 2.2 Previous Work Considerable and extensive work has been done by on conducting an experimental and analytical study on the seismic performance of cantilever bridge columns (Kawashima et al. 21). The conclusions drawn from the study showed that extensive failure of the column on the compression column face during an earthquake acting in the transverse direction resulted from the static eccentric moment, M e. Several retrofit methods were tried, including steel bearings, steel jackets, RC compression jacket and prestressing tendons. However, only the RC compression jacket with prestressing tendons showed acceptable performance with limited residual displacement. Hence, there is still a need to mitigate residual displacements. 3 STRUCTURAL MODELLING AND ANALYSIS A large number of inelastic time history analyses (THA) was carried out to assess what is the optimum strength increase required which results in the smallest residual and maximum displacements. The + following cases was considered, where M is the moment capacity in the direction of the eccentricity and M - is the moment capacity in the opposite direction. 1) No flexural increase: M + = M -. 2) Japanese Road Association recommendations: M + = M - + M e (JRA 199). 3) HCC approximation: M + = M M e (MacRae & Kawashima 1993). 4) Over-design: M + = M M e. The following scenarios were selected to examine the effects of the hysteresis loop shape and P-delta on the response of the column: 1) An elastoplastic hysteresis loop ignoring P-delta effects. 2) An elastoplastic hysteresis loop including P-delta effects. 3) A Takeda hysteresis loop including P-delta effects. The procedure used when running the analyses can be found in Figure 3. T, the natural period of the column, was modified by varying the stiffness of the column. The earthquake records were scaled by a scale factor SR, which is the ratio of the elastic moment capacity to the maximum moment caused by an earthquake when carrying out an elastic time history analysis. The capacity reduction factor, R, is the ratio of the moment capacity to the elastic moment capacity in the direction opposite the eccentricity. R is applied instead of ductility as it is easier to specify. is the ratio of the difference between the column capacity in either direction to the eccentric moment. 3

4 . Base Model (height of column = 1m) Deck weight = 12,kN Base moment capacity. M(R=1) = 72,kNm Damping ratio = 5% Bi-linear factor = 4% Ductility = 1 (or ) Specify Model Elastoplastic without P-delta, elastoplastic with P- delta or Takeda with P-delta Specify T Natural Period of Bridge Column T = to 1.5s with step size of.1s Specify Earthquake Record Select earthquake record and scale by factor (SR) Base Moment Capacity, M ( R 1) SR Max Moment caused by Unscaled Earthquake in ElasticTHA Specify α Eccentric Moment Raito α = to.4 with step size of.1 Specify R Capacity Reduction Factor, 2, 4 or 6 Specify Ratio of Strength Difference to Eccentric Moment = to 3 with step size of 1 Specify EQ direction Initial or reversed Outputs Maximum and Residual Displacement Figure 3 - Flowchart for Analysis Since M(R=1) = 72,kNm, M + and M - would be defined as in Equations 2 & 3: M ( R 1) M R (2) M M M add (3) where M add = M e = αm - and M e =αm -. The earthquake records used for the THA was from SAC suite of 2 records with a 1% probability of exceedance in 5 years (SAC ). Further information on the earthquake records can be found in Sadashiva et al. (9). To test the dynamic stability of the column, the column was subjected to each earthquake record in one direction, and then subjected to the same earthquake in the opposite direction. This cancels out the effect of directionality. The computer program used to run the inelastic THA using Newmark constant average acceleration was RUAUMOKO 2N (Carr 4). MATLAB (The MathWorks 8) was used to run the framework shown in Figure 3 by modifying values in input files and extracting values from the outputs. 4 RESPONSE OF COLUMNS A preliminary analysis of the columns was carried out for T = 1 second and α =.4. Figures 4-6 shows the response of the cantilever columns for the three models. Each figure contains two plots; the first is 4

5 the average residual displacement and the second is the average maximum displacement, both obtained from results of all 2 earthquake records. Both of these include signs. Therefore, if the average is, this shows the column behaves as if it is perfectly straight (same stability range in both directions). It can be seen that the analyses with an elastoplastic hysteresis loop ignoring P-delta effects had an optimum of 2. The optimum for the elastoplastic hysteresis loop including P-delta effects was 2.1. The optimum for the Takeda hysteresis loop including P-delta effects was 2.3 for the maximum displacement and varies from 2.4 to 2.8 for the residual displacement. This shows that there is affected by both P-delta effects and the hysteresis shape as will be discussed later. Note that for, there is a residual displacement at =. This is because the earthquake scale factor, SR, was selected such that the cantilever column would just yield. Adding an eccentric moment in addition to SR would cause the column to yield in the positive direction, leading to a positive average residual displacement. Also, there were a number of columns which collapsed for R > 4 and < 1 which were not included in the results below, so the average residual and maximum displacements will be larger for those cases. For the elastoplastic hysteresis model including P-delta effects, all columns with and = collapsed. Average Residual Displacement (mm) Average Maximum Displacement (mm) (a) Maximum Displacement (b) Residual Displacement Figure 4. Average Displacements for Elastoplastic Hysteresis Excluding P-delta Model (T = 1 s & α =.4) Average Residual Displacement (mm) (a) Maximum Displacement Average Maximum Displacement (mm) (b) Residual Displacement Figure 5. Average Displacements for Elastoplastic Hysteresis Including P-delta Model (T = 1 s & α =.4) Average Residual Displacement (mm) Average Maximum Displacemnet (mm) (a) Maximum Displacement (b) Residual Displacement Figure 6. Average Displacements for Takeda Hysteresis Including P-delta Model (T = 1 s & α =.4) 5

6 5 ESTIMATION OF DISPLACEMENTS The plot on the left in Figure 7 compares the Maximum Displacement Ratio (MDR) to. Here, MDR is the ratio of the average absolute maximum displacement at a given value to the average absolute maximum displacement at = 2.3. The plot on the right in Figure 7 compares the Residual to Maximum Displacement Ratio (RMDR) to. Here, RMDR is the ratio of the average absolute residual displacement to the average absolute maximum displacement. Both plots can be used to estimate the maximum and residual displacement of a bridge column. The work undertaken could also be used for perfectly straight columns where eccentric loading is expected (lane closures or heavy vehicle lanes). An analysis can be carried out on a perfectly straight column, and multiplied by the MDR at = to get the maximum displacement. Maximum Displacement Ratio (a) MDR (b) RMDR Figure 7 Displacement Ratio Plots Residual to Maximum Displacement Ratio As mentioned before, for < 1 and R > 4, a large number of collapses occurred. Therefore the likely MDR is higher than predicted from the above plots, especially for higher R where collapse is more likely. Work is being done to estimate the displacements using a statistical approach to incorporate the data for collapsed columns into the displacement estimation. Currently, the displacements estimated using Figure 7 could be treated as a lower bound approximation. The following is an example of how to use the above charts. Consider a theoretical column designed for = 1.5 with and that column has been modelled as a perfectly straight column with a maximum displacement of mm. From Figure 7, the MDR is 2.75 and the RMDR is.78. The expected maximum displacement is therefore 275mm (mm 2.75) and the expected residual displacement would be 215mm (275mm.78). 6 DISCUSSION 6.1 Effects of P-delta Figure 8 shows the comparison between the elastoplastic hysteresis loop with P-delta to the elastoplastic hysteresis loop without P-delta. When = 2, the positive yield strength relative to the new baseline, x 1, is equal to the negative yield strength relative to the new baseline, x 2. However, to be dynamically stable, y 1 must be equal to y 2. Thus, = 2 is not optimum when P-delta effects is considered. Taking this into account, Equation 3 was derived (Yeow 21) to calculate the optimum required, accounting for P-delta effect. P/KL is always positive and less than or equal to 1. Therefore the optimal is always greater than 2 for elastoplastic hysteresis loops including P-delta effects. This is consistent with the value of 2.1 obtained from Figure 5. 2 (3) P 1 KL 6

7 Figure 8 - Effect of P-delta on Yield Strength Relative to New Baseline 6.2 Effect of Hysteretic Shape Figure 9 shows the comparison of the two different hysteresis models, ignoring P-delta effects. Figure 9 - Comparison of Hysteretic Shape [α =.2, = 2, T =.5 s,, LA1 record] (Yeow 21) The Takeda hysteresis model shape, as can be seen above, is more complex than that of the elastoplastic hysteresis model. The HCC can be applied to the elastoplastic hysteresis model as it is just an average of the top and bottom yield lines. It is more complex to apply it to Takeda, which is a trilinear hysteresis model. A possible reason for the value of increasing with the Takeda loop is due to the amount of energy required for the column to yield (shown by shaded areas in the hysteresis loop plots). In the elastoplastic case, the energy required to yield in either direction are the same. In the case of the Takeda model however, the energy required to fully yield is less in the positive direction than in the negative direction. Thus, the column is more likely to deform in the positive direction, resulting a larger. 7 RECOMMENDATION Based on the initial and preliminary analyses undertaken to date with the Takeda hysteresis loop, it seems as though the strength capacities of a cantilever column should be provided according to Equations 4 & 5 in order to minimize the likely residual displacements where F = design base shear caused by earthquake. It is anticipated that a larger number of analyses with a more realistic hysteresis loop may slightly modify these recommendations. M FL 2. 5 M e (4) M FL (5) 7

8 8 CONCLUSION The following findings address the questions raised in the introduction regarding cantilever bridge columns. 1) The optimum increase in flexural strength is twice the eccentric moment for a bilinear column excluding P-delta effects. This confirms the HCC theory. However the optimum is higher when P-delta is considered. 2) The optimum increase in flexural strength which resulted in the smallest average maximum displacement of columns subjected to earthquake action is 2.3 times the eccentric moment, based on the results from the Takeda hysteresis models including P-delta effects. 3) The optimum increase in flexural strength which resulted in the smallest average residual displacement of columns subjected to earthquake action is 2.5 times the eccentric moment, based on the results from the Takeda hysteresis models including P-delta effects. 4) It was found that the capacity reduction factor, R, had no significant effect on the optimum strength increase. However, P-delta and hysteresis effects increase the required optimum strength increase. 5) A new equation for the optimum was developed to improve the prediction of the strength increase required to optimise the performance of the columns. This equation, Equation 4, builds on the previous prediction by allowing for the effects of P-delta for elastoplastic hysteresis models. No improvements were made for Takeda hysteresis models. 6) From the limited analyses undertaken, columns able to be modelled with Takeda type hysteresis loops should be designed with an optimum strength increase of 2.5 times that of the eccentric moment on one side to minimise the likely residual displacements. 9 ACKNOWLEDGEMENTS This paper is same as that published in the SEE6 Tehran Conference, REFERENCES Carr AJ. (4) Ruaumoko 2N Inelastic dynamic analysis program. Department of Civil Engineering, University of Canterbury, Christchurch, 4. Japan Road Association. (199). Design Specifications for Road Bridges Part V Seismic Design. Kawashima K., Seiji N., & Watanabe G. (21). Seismic Performance of a Bridge Supported by C-bent Columns. Journal of Earthquake Engineering, vol 14, issue 8, 21, MacRae G. A, Kawashima K. (1993). The Seismic Response of Bilinear Oscillators Using Japanese Earthquake Records, Public Works Research Institute Ministry of Construction, Asahi, Tsukuba-shi, Ibaraki-ken, 35 Japan. MATLAB 7.6 (R8a). The Maths Works, Inc.: Natick, MA, 9. SAC. ().SAC (SEAOC-ATC-CUREE) Steel Project. Sadashiva VK, MacRae GA, Deam BL. (9). Determination of Structural irregularity limits mass irregularity example. Bulletin of the New Zealand Society for Earthquake Engineering 9; 42(4): Yeow T. (21). "Seismic Behaviour of Cantilever Bridge Columns", Undergraduate Report, University of Canterbury, Christchurch, New Zealand. 8