4th International Conference on Earthquake Engineering Taipei, Taiwan October 12-13, 2006 Paper No. 094
MIXED LOAD / DISPLACEMENT CONTROL STRATEGY FOR HYBRID SIMULATION Narutoshi Nakata1, B. F. Spencer, Jr. 2, and Amr S. Elnashai3
ABSTRACT Owing to multiple actuator interaction and the nonlinear nature of coordinate transformations, spatial mixed load-displacement control algorithms are still major theoretical and practical challenges. This paper presents a mixed load and displacement control strategy for coupled multi-axial control systems that can be employed for static and hybrid simulation. The method is based on an incremental iterative approach employing the Broyden update of the stiffness Jacobian of the test specimen. Experimental verification is performed on reinforced concrete (RC) columns employing a state-of-the-art six-degrees-of-freedom self-reacting box at the University of Illinois at UrbanaChampaign. Experimental results indicate that the proposed mixed-mode testing method exhibits excellent control performance and robustness even when the specimens are highly inelastic. The framework and example application are transferable to other simulation platforms. Keywords: Mixed Load and Displacement Control, Multi-Axial Control System, Hybrid Test, Broyden update, Jacobian Matrix
INTRODUCTION Seismic performance of structures under strong earthquake ground motion is a highly coupled causeeffect problem. Understanding the cause (load) and its effect (deformation) is key for seismic assessment. Gravity loads can have a significant effect on the behavior of vertical structural elements (i.e. walls and columns) because of the second-order moment resulting from the deflection of the member, referred to as P-∆ effect. Moreover, axial loads affect noticeably the shear capacity of reinforced concrete members due mainly to the contribution of concrete under compression to the shear resistance mechanisms. Thus, accounting for effect of gravity loads is important to the assessment of the seismic performance of structures. Gravity load effects have been carefully considered in structural tests by many researchers. Lynn et al. (1996) evaluated pre-70’s RC building columns under constant axial and cyclic lateral loadings. In the test setup, an axial load was supplied by two vertical actuators where one actuator was in force control to maintain constant axial load and the other was in displacement control to maintain zero rotation. Lateral loading was supplied by a horizontal actuator in displacement control. Kawashima et al. reported a series of RC column tests subjected to bi-lateral loading (2004) and flexural-torsional loading (2005). The axial loading was applied by one vertical actuator in force control. Bidirectional and flexural-torsional actions were supplied by two in-plane actuators. In the loading system, the 1
Ph. D. Candidate, University of Illinois at Urbana-Champaign, IL, USA,
[email protected] Newmark Professor, University of Illinois at Urbana-Champaign, IL, USA,
[email protected] 3 Hall Professor, University of Illinois at Urbana-Champaign, IL, USA,
[email protected] 2
effect of the load in the vertical actuator on the lateral load was taken into account using a geometric relationship linking the instantaneous angles of inclination of both actuators. However, the effect of the load in the lateral actuators on the vertical load was ignored using simplified geometric assumptions. Pan et al. (2005) tested a base-isolated structure using two hydraulic actuators. A vertical actuator was primarily in force control and a lateral actuator was in displacement control. Later researchers introduced a displacement-force switching control for the vertical actuator for on-line test applications. Thus, gravity load effects have been traditionally applied or considered only through hydraulic actuators in the vertical direction. Because actuators in other axes are attached perpendicular to the vertical actuators at the initial loading point, the assumption of decoupling of the actuator forces is made for the vertical load. Therefore, the vertical actuators can be in force control. It should be noted that under large deformation, lateral actuators will have a force component in the vertical direction that should be considered. If a control system has coupling between actuator and Cartesian coordinates (i.e., x, y, z, θx, θy, θz), mixed load and displacement control, including gravity loading and lateral displacement control, cannot be achieved with independent control of each actuator. The challenge is due to the contribution of unknown displacements in load-controlled actuator and unknown forces in displacement-controlled actuator to the target mixed load and displacement. In other words, mixed load and displacement commands cannot be explicitly decomposed into each actuator command in the coupled multi-axial control system. Therefore, versatile and generally-applicable mixed-mode control algorithms need to be developed to take into account instantaneous and spatial coupling in the control system. This paper presents a generic mixed-mode control method which incorporates load-to-displacement command conversions in a mixed-mode feedback loop. The conversion is based on the incremental iteration process employing the Broyden (1965) update of the stiffness Jacobian of the tested structure. Because all actuator servo loops are closed with displacement output, the control system is extremely robust for mixed-mode control. Following a description of mixed-mode control algorithm, experimental verification is performed for reinforced concrete (RC) columns using the state-of-the-art 6DOF loading unit, the load and boundary condition box (LBCB) at the University of Illinois at UrbanaChampaign. Experimental results confirm that the proposed method successfully achieves highprecision control of the axial load, as well as the lateral and torsional displacements, in a mixed fashion, even when the tested specimen undergoes inelastic deformations.
MIXED LOAD / DISPLACEMENT CONTROL METHOD Mixed load and displacement (mixed-mode) control in this study is defined as a combined control with either a load or displacement control in each Cartesian axis at a connection point. Mixed-mode control here also accounts for actuator configuration and orientation regardless of the number of load control axes and patterns. However, as mentioned in the previous section, mixed-mode commands cannot be directly decomposed into each actuator load or displacement command in coupled multiaxial control systems. Therefore, the described herein incorporates load-to-displacement command conversion using an approximation of the stiffness Jacobian in an outside control loop. Due to the load-to-displacement command conversion, all the actuator servo loops are in the displacement control mode regardless of the number of load control axes and patterns. The proposed method requires an incremental iterative process for force convergence. Thus, it is specifically designed for static and hybrid simulation with discrete ramp-hold testing procedures. Control Architecture Figure 1 (a) shows the data flow and process in the proposed mixed-mode control strategy. Vectors u and f are the displacement and force vectors in Cartesian coordinates, respectively, whereas vectors d and r are the displacement and force vectors in the actuator space. Superscript t , c , and m denote target, command, and measured data, respectively. Accents " " and " " denote data in displacementand force-controlled axes, respectively. Finally, vector p is a geometric parameter vector for actuator pin location.
Figure 1 (b) shows detailed data flow for the force-to-displacement conversion. Here, ef and ∆u c are the force error and displacement increment vectors in the load control axes, respectively, while K and G are the stiffness Jacobian and mixed-mode gain matrices, respectively. In the command execution process, Cartesian mixed load and displacement command is converted into a Cartesian displacement command. Thereafter, the Cartesian displacement command is decomposed into the actuator displacement command, which is in turn executed in a ramp-hold fashion. In the measurement process, actuator displacements and loads are converted to the Cartesian displacements and loads. The rectangular blocks in the figure represent data processes. A brief description of each process is as follows: 1) Force-to-Displacement Conversion: Cartesian target force is converted into Cartesian command displacement using an approximation of the stiffness Jacobian. The details of the iterative procedure are described in the next section. 2) Cartesian-to-Actuator Conversion: Cartesian command displacement is converted into each actuator command displacement. This process is a geometric transformation using the coordinates of the actuator pin locations with respect to the attachment point. 3) Actuator-to-Cartesian Conversion: Measured actuator displacement and load from the LVDT and load cell are converted into the Cartesian measured displacement and load at the attachment point. In general, this conversion is an iterative process to solve a set of nonlinear kinematics relationships. A modified Newton-Raphson method is implemented in the controller. 4) Ramp-Hold: Ramp is for loading and hold is for averaging data. These are discrete events. 5) Digital PI Loops: A discrete proportional-integral (PI) control loop is implemented digitally for each actuator to improve the control performance. This discrete loop is activated only during the hold phase. 6) Analog PID Loops: An analog proportional-integral-derivative (PID) (inner) control loop is employed for continuous feedback on each actuator.
Mux
ut f t
2)
uc 1)
Force to Disp. Conversion
Cartesian to Actuator Conversion
u c p
u
dc
4)
5)
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6) Analog PID Loops
Digital PI Loops
3)
m
dm
Actuator to Cartesian Conversion
fm
LVDTs Load Cells
rm
Cartesian Space
Digital Domain
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(a) Data flow and process in the mixed-mode control 1)
f t
+ ef - K f m
⋅ ef K x Demux
G
x
∆u c + +
Demux
fm um
Servo Valves
Stiffness Jacobian Update
K
(b) Force to displacement conversion Figure 1. Control architecture.
u c
Analog Domain
Iterative Procedure for Mixed Load and Displacement Convergence In the proposed method, target mixed load and displacement are achieved through a process that comprises directional and iterative ramps. First, the directional ramp is executed with an updated command in the displacement controlled axes whereas the command in the load control axes remains that at the end of the previous step. Then, iterative ramps are repeated with an updated command in the load controlled axes until convergence to the target load is achieved. After each ramp, an approximation of the stiffness Jacobian is updated using the Broyden’s method. A single step in the mixed mode control procedure is described below: (i) Directional ramp at step N:
u cN (0) = u tN
(1)
u cN (0) = u cN −1
(2)
where u Nc (0) and u cN (0) are the command displacement at the directional ramp in the N-th step in displacement- and load-controlled axes, respectively. u Nt is a target displacement in displacement controlled axes at step N. u cN −1 is a command displacement in load controlled axes at step N–1. (ii) Update the stiffness Jacobian after the directional ramp:
K N (0) = K N −1
( ∆f +
m N (0)
)(
− K N −1 ⋅ ∆u mN (0) ⋅ ∆u mN (0) ∆u mN (0)
)
T
(3)
2
where K N (0) is a updated stiffness Jacobian after the directional ramp at the N-th step. ∆u mN (0) and ∆f Nm(0) are the measured incremental displacement and load vectors, respectively, and are written as follows: ∆u mN (0) = u mN (0) − u mN −1 (4)
∆f Nm(0) = f Nm(0) − f Nm−1
(5)
Eq. 3 is known as Broyden update of the Jacobian. It satisfies the following relationship. ∆f Nm(0) = K N (0) ∆u mN (0)
(6)
(iii) Iterative Ramp at i-th iteration at step N:
u cN ( i ) = u tN
(7)
(
⋅K t m u cN (i ) = u cN ( i −1) + G N ( i −1) ⋅ f N − f N ( i −1)
)
(8)
where G is a mixed-mode gain matrix in the load-controlled axes and K N ( i −1) is the updated stiffness Jacobian after the i-th iteration at the N-th step in the load controlled axes. (iv) Update the stiffness Jacobian after the i-th iterative ramp
K N ( i ) = K N ( i −1)
( ∆f +
m N (i )
)(
− K N ( i −1) ⋅ ∆u mN ( i ) ⋅ ∆u mN ( i ) ∆u mN ( i )
2
)
T
(9)
where K N ( i ) is an updated stiffness Jacobian after the i-th iterative ramp at the N-th step. The displacement and force increment vector ∆u mN ( i ) and ∆f Nm( i ) are written as follows: ∆u mN ( i ) = u mN ( i ) − u mN ( i−1) (10)
∆f Nm(i ) = f Nm( i ) − f Nm( i−1)
(11)
fNt − fNm( i ) ≤ ef
(12)
(v) Convergence evaluation:
where ef is a load tolerance vector in the load controlled axes. If Eq. 12 is not satisfied, the process goes back to (iii) and is repeated until the convergence criterion is satisfied. Following convergence, the process goes to the next step N+1 with following relationships:
u cN = u cN ( i )
(12)
fNm = fNm(i )
(13)
Because of the updating feature, the proposed method takes into account material inelasticity and geometric nonlinearity and other abrupt effects, such as cracking, in the control process. Therefore, the proposed method is robust and efficient in terms of the load control in multi-axial control systems. Most importantly, with small increments in the directional ramp, this method is capable of producing the desired mixed load and displacement load history even for path-dependent structures.
TESTING SETUP The Load and Boundary Condition Boxes (LBCBs) The state-of-the-art 6 DOF loading unit, referred to as Load and Boundary Condition Box (LBCB) at the University of Illinois at Urbana-Champaign (UIUC), is used as a platform for implementing and verifying the developed control strategies. Figure 2 (a) and (b) show the full-scale and 1/5th-scale LBCBs, respectively. Both units are servo hydraulic systems each consisting of a reaction box, a loading platform and six actuators with a servo valve, load cell and LVDT. The specifications of each LBCB are summarized in Table 1. Two different sizes of L-shape reaction walls are available at UIUC to support full- and 1/5thscale LBCBs. Those walls enable multiple LBCBs to be mounted at any location and orientation. Thus, facilities at UIUC have a capability to perform multi-axial loading test at multiple connection points at both full- and 1/5th-scales.
Z X
(a) Full-scale LBCB
Z X
Y
(b) 1/5th-scale LBCB
Figure 2. The Load and Boundary Condition Boxes (LBCBs).
Y
Table 1. Specifications of full- and 1/5th-scale LBCBs 1/5th-scale LBCB
Full-scale LBCB Displace ment (mm)
x
+-254
y
+-127
z
+-127
x Force (kN)
x +-16.0 Rotation (degree)
y +-11.8 z
Moment (kN*m)
+-16.0
+-2402
y
+-1201
z
+-3603
x
+-862
y
+-1152
z
+-862
Displace ment (mm)
x +-50.8 y +-25.4 z
Force (kN)
+-25.4
x +-16.0 Rotation (degree)
y +-12.0 z
+-16.0
Moment (kN*m)
x
+-8.9
y
+-4.5
z
+-12.3
x
+-1.13
y
+-2.03
z
+-1.13
Control System Architecture Figure 3 shows the main hardware components for slow-rate hybrid tests using the LBCB. In the test system, a hydraulic pump and accumulator, analog and digital controllers, a coordinator PC, and a digital camera/server are employed. Analog Controller & Signal Conditioner
An analog controller manufactured by Shore Western is used to close the inner displacement-control servo loop with a PID scheme and sends drive current to all the servo valves. It also acts as a signal conditioner for all the LVDTs and load cells in the actuator. Analog signals from the LVDT and load cell, as well as the servo error, are fed through the analog controller into the digital controller; the analog displacement command from the digital controller goes into the analog controller as an external input source. All signals in the analog controller are at the actuator level. Digital IOs to the lock-up valves on the LBCB and switches for low and high pressures on the accumulator are also controlled by the analog controller. Digital Controller & Data Acquisition
The digital controller consists of a personal computer, analog-to-digital and digital-to-analog converters, and associated hardware. A National Instruments (NI) PCI-6281 board is used with SCXI1001 signal conditioner for data acquisition as analog to digital converter. A NI PCI-6330 board is used with BNC-2110 terminal block for displacement command generation as digital-to-analog converter. All of the digital signal processing, including mixed-mode algorithm and coordinate conversions, are implemented in the digital controller. Additional sensor data can also be collected at the same rate with the LBCB data and stored on the hard drive in the digital controller. In addition, the digital controller has a gateway application for the network communication with a coordinator PC. The gateway application is compatible with the NEESgrid tele-control protocol (NTCP) and can exchange target and measured mixed load and displacement data in the NEES cyber-infrastructure environment. Coordinator PC
The coordinator PC is a client machine in slow-rate hybrid tests that sends a command to the gateway application in the digital controller and other experimental and computational modules. UI-SIMCOR (Kwon et al, 2005) is one of the coordinator applications for hybrid testing using NTCP and can be used with the digital controller developed in this study.
Target mixed load and displacement Trigger
Coordinator PC Measured load and displacement
Digital Camera Server
Digital Controller & Data Acquisition
Still Camera Displacement Command
LVDT Signals Load Cell Signals
Test System
Reaction Wall
Digital IO
LBCB
Valve Inputs
Servo Valves LVDTs
LVDT Signals Load Cell Signals
Load Cells
Analog Controller & Signal Conditioner
Pressure
Specimen
Return
Hydraulic Pump
Digital IO Return Pressure
Accumulator
Figure 3. Schematic layout of the test setup.
Digital Camera Server
Digital cameras are used to capture high-resolution images in slow-rate hybrid tests. A server program was developed by the NEESgrid System Integration team and is compatible with the NTCP. Thus digital cameras can be triggered from the coordinator PC at every step in the test.
EXPERIMENTAL VERIFICATION Experimental verification of the proposed mixed-mode control method was conducted using the 1/5thscale LBCB with reinforced concrete (RC) specimens. All the tests presented in this section were performed using a slow loading rate.
Test Specimen The test specimens are 1/20th-scale RC columns of the skew bridge in the FHWA Seismic Design of Bridges, Design Example No.4 (Report No. FHWA-SA-97-009. 1996). The specimens have a circular section with 51 mm diameter and a height of 305 mm. Micro-concrete with 31 MPa compressive strength and heat treated threaded rods with 345 MPa yield strength are used to model the concrete and longitudinal reinforcement, respectively. Transverse spiral reinforcements are made of annealed steel wire with 414 MPa tensile strength.
Lateral Displacement and Axial Force Controlled Test
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4000
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Displacement X (mm)
The first mixed-mode experiment is a mixed lateral displacement and axial load controlled test. The lateral displacement input is a sinusoidal wave with increasing amplitude and the axial load input is a 4448 N constant compression throughout the test. Note that positive Z displacement is downwards from the loading platform. The other four DOFs, i.e. rotations and out–of-plane displacement are fixed. In other words, the latter DOFs are under displacement control with constant zero inputs. Figure 4 shows a sample results from the lateral and axial loading test case. Figure 4 (a) and (b) are the controlled lateral displacement and axial force, respectively. Regardless of the lateral displacement, the axial force is controlled within 1.0% of the reference value of 4448 N during the test. The axial displacement in Figure 4 (d) changes according to the control axial force based on the proposed mixed-mode control method. Despite concrete cracking (See Figure 6 (a)) and associated sudden reduction in stiffness in both the lateral and axial directions, the control objective is successfully achieved with a maximum error 0.02 mm and 4.4 N in lateral displacement and axial force, respectively. The proposed mixed-mode control method clearly exhibits very good performance and robustness in lateral and axial loading test even in the inelastic range of the response.
400
600
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1000 1200
0.5
0
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(c) Lateral Force X
(d) Axial Disp. Z
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-10
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(e) Lateral Disp. X – Lateral Force X
15
0
-0.5 -15
-10
-5
0
5
10
Displacement X (mm)
(f) Lateral Disp. X – Axial Disp. Z
Figure 4. Test results under lateral and axial loadings.
15
Torsional Rotation and Axial Force Controlled Test
10 4000 5
Force Z (N)
Rotation Z (degree)
The second mixed-mode experiment is a mixed torsional rotation and axial load controlled test. The torsional rotation input is a sinusoidal wave with increasing amplitude; the axial load input is a 2224 N constant compression throughout the test. The other DOFs are under displacement control with constant zero inputs. Figure 5 shows sample results from the torsional rotation-axial loading test. Figure 5 (a) and (b) show the controlled torsional rotation and axial force, respectively. The axial load is well controlled throughout the test. Figure 5 (c) and (d) depict the torsional moment and axial displacement, respectively. As observed in Figure. 6 (b), cracks and strength reduction are also observed in the torsional loading test; the axial displacement is adjusted to control the axial force based on the proposed method. Figure 5 (e) and (f) show the torsional rotation and moment, and torsional rotation and axial displacement relationships, respectively. Ductile behavior and energy dissipation can be seen in the torsional rotation and moment relationship. Note that axial strength reduction and inelastic behavior can be seen from the torsional rotation and axial displacement relationship. Similar to the lateral loading test, the proposed mixed-mode control method shows very good control performance and robustness in torsional and axial loading test even in the inelastic range of the response.
0 -5 -10
0
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3000 2000 1000 0
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(a) Torsional Rot. Z
(b) Axial Force Z
Moment Z (N*mm)
x 10 2 1 0 -1 -2 0
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5
0.2 0.1 0 -0.1 -0.2 0
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(c) Torsional Moment X
(d) Axial Disp. Z
5
x 10
0.2
Displacement Z (mm)
Moment Z (N*mm)
2 1 0 -1 -2 -10
-5
0
5
10
Rotation Z (degree)
(e) Torsional Rot. Z – Torsional Moment
0.1
0
-0.1 -10
-5
0
5
Rotation Z (degree)
(f) Torsional Rot. Z – Axial Disp. Z
Figure 5. Test results under torsional and axial loadings.
10
(a) Specimen under lateral and axial loading
(b) Specimen under torsional and axial loading
Figure. 6 Test specimens under mixed load-displacement control.
CONCLUSIONS This paper presented a mixed load-displacement control method for coupled multi-axial control systems. The method is suitable for static and hybrid simulation. The control algorithm utilizes an iterative approach using Broyden update of the stiffness Jacobian. Experimental verification is performed on RC column specimens using the NEES Load and Boundary Condition Box at the University of Illinois at Urbana-Champaign. Experimental results demonstrate that the proposed mixed-mode control method exhibits very good performance and robustness even when the specimens are highly inelastic.
ACKNOWLEDGMENTS This research described in the paper is supported by the George E. Brown, Jr. Network for Earthquake Engineering Simulation (NEES) Program (NSF Award No. CMS-0217325) funded by the National Science Foundation (NSF) and the Mid-America Earthquake Center, an Engineering Research Center funded by the National Science Foundation under cooperative agreement reference EEC 97-01785.
REFERENCES Lynn, A.C., Moehle, J.P., Mahin, S.A., and Holmes, W.T., (1996). “Seismic Evaluation of Existing Reinforced Concrete Building Columns,” Earthquake Spectra, 12(4), 715-739. Kawashima, K., Gakuho, W., Nagata, S., and Ogimoto, H., (2004). “Effect of Bilateral Excitation of the Seismic Performance of Reinforced Concrete Bridge Columns,” Japan-Europe Seismic Risk Workshop, University of Bristol, UK Tirasit, P., Kawashima, K., and Watanabe, G., (2005). “An Experimental Study on the Performance of RC Columns Subjected to Cyclic Flexural-Torsional Loadings,” 2nd International Conference on Urban Earthquake Engineering, 357-364, Tokyo, Japan Pan, P., Nakashima, M., and Tomofuji, H., (2005). “Online test using displacement-force mixed control,” Earthquake Engineering and Structural Dynamics, 34, 869-888. Broyden, C.G., (1965). “A class of methods for solving nonlinear simultaneous equations.” Math. Comp. 19, 577-593. Federal Highway Administration, (1996). “Seismic Design of Bridges -Design Example No.4 Three-Span Continuous CIP Concrete Bridge,” FHWA-SA-97-009, Federal Way, Washington Kwon, O. S., Nakata, N., Elnashai, A. S., and Spencer, B. F., (2005). "A Framework for Multi-Site Distributed Simulation and Application to Complex Structural Systems." Journal of Earthquake Engineering, 9(5), 741753.
