Study of Rotational Column with Plastic Hinge
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1 Study of Rotational Column with Plastic Hinge Presented by: Michael Long, Rice University Corey Bergad, Bates College REU Interns, SUNY at Buffalo Advisor: Andrei M. Reinhorn, Ph.D., P.E. Professor and Co-director of Structural Engineering and Earthquake Simulation Laboratory Department of Civil, Structural and Environmental Engineering SUNY at Buffalo REU Symposium: August 7, 2004
2 Overview: Discuss FAST MOST experiment and function of plastic hinge Describe test setup Discuss analytical model and design considerations Present experimental results Conclude on behavior and design of plastic hinge
3 Section I: Background and Test Setup
4 Background Information: Column setup was built for the FAST-MOST hybrid testing experiment One of five scaled down columns modeling supports for a bridge Goal is to replicate a complete bridge in real time through networking with other test sites N E1 E2 E3 E4 E5 E6 N2 N3 N4 N5 N6 COL01 Berkeley COL02 Boulder COL03 UIUC COL04 Buffalo COL05 Lehigh
5 Test Setup: Column is subjected to lateral displacement at the top in quasi-static loading and is fixed at the base Top forces and deflections sent as input to other test sites Simulates earthquakeinduced ground motion, including pseudo-dynamic response of bridge The column-hinge system, with scale
6 Our Focus: Plastic Hinge Pinned connection between column and foundation Rotation prevented by small vertical members These small bars act as fuses, allowing rotation before yielding of column Called Fuse Bars Fuse Bars
7 Function of Plastic Hinge: Type of energy dampening device Yielding and subsequent deflection of fuse bars absorb energy Reduces fatigue in column during repeated ground motion
8 Function of Plastic Hinge: Post-earthquake, fuse bars can be replaced relatively easily Column-hinge system is then restored to original condition
9 Section II: Analytical Model and Design Considerations
10 Research Proposal: Develop an accurate model of column-hinge system in the elastic range of the fuse bars Consider two design conditions for plastic hinge 1. Hinge must become plastic before column yields 2. Limit deflection at top of column for stability of the overlying structure Plastic behavior of fuse bars not tested
11 Model of Hinge: Effective Length of Fuse Bars Length able to bend is portion between inside nuts (l = 2 1 / 4 ) Axially deformable length depends on direction of elongation Length in tension (blue) Length in compression (red) 2 1 / 4 Take effective area of threads into account Average is the effective length for axial elongation (l e = 3.46 )
12 Model for Hinge Motion Moment about pin will cause rotation of hinge Resisted by internal forces of fuse bars Rigid Plate Rigid Connection Internal Forces in Fuse bar F M axial = bending EA l e = θ r EI l θ
13 Buckling Analysis Buckling Analysis Eqs P L cr E = = 2 π EI L kl e 2 E P cr is the buckling load L E is the Euler length Consider the two extreme cases of Buckling k P cr (kips) Cantilever Fixed-Fixed At a load of 17.4 kips (lower extreme), axial stress in the fuse will equal ksi. Fuse-bars yield at approx. 100 ksi Buckling will not occur.
14 Angular Stiffness of Hinge Moment induced about pin by external load Resisting moment generated by axial force and bending moment Σ M pin = 0 M hinge M 4( M + M ) = 0 hinge axial bending 2r M M bending axial = = EI θ l EAr l e 2 θ Angular Stiffness K θ 2 M hinge EAr EI = = 4 + θ le l
15 Flexural Stiffness of Column K flex = P/δ flex Properties of S3x5.7 column taken from ASCI manual Additional flanges and connecting plates assumed to be infinitely stiff Flexural Flexibility Equation δ flex 3 3 PL c L R 3h R = 1 EI Lc L c
16 1 st Design Consideration: Elastic Limits of Hinge and Column Compare maximum stresses developed in column and fuse bars Moment Diagram Maximum moment in column Moment about hinge
17 Comparing Strength of Hinge to Column Stress in both fuse and column can be expressed as f(p) Fuse bar stress σ + σ = σ axial bending max NTS σ σ axial = bending Faxial Er PL = A l K M bending = = S σmax = σaxial + σbending = f ( P) e θ EI PL ls K θ Column Stress σ max M σ max = = S PL S xx c
18 2nd Design Consideration: Deflection of column and overlying structure Determine deflection at top of column Stiffnesses of hinge and column act in series δ θ defl. due to hinge rotation 2 PL δθ = θ L = Kθ δ flex defl. due to flexure of column δ flex = P K flex NTS
19 Justification of Simplified Model Assumed small angle changes when relating hinge rotation to axial strain of fuse bars Compare axial elongation equations: Small Angle Approx. =θr General Solution r θ θ = 2 sin cos + β cos β 2 2 Comparison of Results at Theoretical Elastic Limit of Hinge Small Angle Approx General Case % Error (in) % δtot (in) % P (lb) %
20 Section III: Experimental Results
21 Overall Setup: Displacementdriven actuator attached to top of column Actuator pinned at either end No axial force generated in column
22 Initial Test Setup: Gauge Location and Interpretation L1 = Load P D1= δ tot x 2 3 arctan D + θ D = x
23 Experimental Test Pattern Experiment ran using a quasi-static cyclic deflection pattern 0.2 Actuator Displacement (in) Time (seconds) Maximum top displacement: cycles of increasing amplitude Total test run time 125 seconds Fuse bars were not yielded in test due to limited moment capacity of bottom load cell
24 Initial Experimental Results Stiffness of column-hinge system lower than expected Theoretical Experimental
25 Analysis of Initial Results Rotational stiffness much less than expected from model
26 Analysis of Initial Results Flexural stiffness matched analytical predictions Comparison of analytical and experimental flexural stiffness Experimental Analytical % Error K flex (lb/in) % Model of column flexure accurate Rotation is the source of extra deflection Rotational behavior requires more data
27 2nd Test Setup: Potentiometers added between the top and bottom of hinge Measured relative motion of top and bottom plates of hinge Separates hinge rotation from remainder of base fixture Potentiometer 3/4 View
28 Analysis of Results: 2 nd Test Rotation of hinge now comparable to model Comparison of Hinge Rotation Experimental Model % Error Kθ (kip/rad) % Locating source of additional rotation requires more data
29 3rd Test Setup: Krypton Coordinate Measurement Machine 12 LED s placed throughout structure to monitor displacement Camera system tracks motion of LEDs in 3-D
30 Analysis of Results: 3 rd Test Separately tracked motion of each component below hinge Located source of extra rotation! Load cell at base
31 Energy Dissipation During Testing Hysteresis present in loading/unloading cycles despite absence of fuse-bar yielding Possibly caused by some small component yielding or slipping Location of component is unknown Large number of interfaces bolted together
32 Energy Dissipation Work done on setup by actuator is equal to area within hysteresis Yielding/slipping of component absorbs this energy Energy dissipated is the net work done on the column-hinge system
33 Section IV: Conclusions
34 Conclusion: Energy Dissipation Energy dissipation protects column from excess strain and fatigue Fuse-bars designed to be replaced easily after they wear out Additional energy can be absorbed by small component yielding or slipping Not desired, may wear out component over time This should be subtracted out when analyzing function of hinge
35 Conclusion: Design of Plastic Hinge Position, axial and bending stiffness, and yielding strength of fuse bars determines moment capacity of hinge Compare to moment capacity of column given a lateral load at the top Effective lengths for different types of stress vary; sensitive to manner of attaching fuse bars
36 Conclusion: Deflection at Top of Column Calculation of deflection requires: Rotational stiffness of hinge Flexural stiffness of column Sum deflections in series Complicated by: Effective length of fuse-bars Determined by method of attaching bars Non-rigid components Load cell
37 Special Thanks to: Dr. Andrei Reinhorn Dr. Avigdor Rutenburg Dr. Mettupalayam Sivaselvan Hwasung Roh, Ph.D. candidate Scot Weinreber MCEER Education Program
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