Cyclic Elastoplastic Analysis of Metallic Plate Dampers

Dargush Gary F.(Jul 1999)

Background Constitutive Modeling Introduction Two-surface Model Defination Parametric Estimation

Finite Element Analysis Overview Baseline Analysis Parametric Studies References Appendix A :Dataset for Baseline Analysis

Background

One of the most effective mechanisms available for the dissipation of the energy input to a structure during an earthquake is through the inelastic deformation of metals. In traditional steel structures, this inelastic deformation is generally concentrated in the beam-column joints and is thus associated with damage to the primary structural elements. On the other hand, with the emergence of passive energy dissipation (PED) systems, special devices are incorporated within the structure to absorb or consume a portion of the input seismic energy. As a result, the energy dissipation demand on primary structural members is often considerably reduced, along with the potential for structural damage. These systems are reviewed in Soong and Dargush (1997) and Constantinou et al. (1998).

The idea of utilizing separate metallic PED dampers within a structure to absorb a large portion of the seismic energy began with the conceptual and experimental work of Kelly et al. (1972). During the ensuing years, considerable progress has been made in the development of metallic devices, most of which are made of mild steel or lead. Examples include flexural plate systems, torsional bar dampers, yield ring dampers, and extrusion devices (Skinner et al., 1980). An X-shaped plate damper or ADAS (Added Damping And Stiffness) device has been studied via experiments by Bergman and Goel (1987) and Whittaker et al. (1991), and subsequently employed in the aseismic retrofit projects discussed by Martinez-Romero (1993) and Perry et al. (1993). The hourglass shape produces nearly uniform curvature throughout the plate during infinitesimal deformation. Similar reasoning has also led to the development of triangular plate systems by Tsai et al. (1993).

It should be recognized that these plate dampers represent critical elements in the overall seismic protective system and therefore must be engineered to a high level of reliability. Previously, Dargush and Soong (1995) used a simple mechanics of materials approach to study the behavior of triangular metallic plate dampers under cyclic load. In that work, a one-dimensional two-surface plasticity model was employed, along with several simplifying kinematic assumptions. A finite element continuum mechanics approach is adopted here to study these same dampers under less restrictive assumptions. In order to accomplish that objective, a fully three-dimensional version of the two-surface plasticity model was developed and implemented as a user-defined routine in the general-purpose finite element code ABAQUS. This model is described in the next section, while the results of the finite element analyses are presented in Section 3.

Constitutive Modeling

Introduction

A vast number of theories of varying levels of sophistication have been developed to model the inelastic behavior of metals under cyclic conditions. Major classes include endochronic theories (Valanis, 1971, 1980), internal variable theories (Chaboche and Rousselier, 1983), and multisurface models (Mroz, 1969; Dafalias and Popov, 1975; Krieg, 1975). Since all of these theories attempt to simulate the same physical phenomena, it is not surprising that strong interrelationships exist (e.g., Watanabe and Atluri, 1986). The selection of a particular model often depends upon the particular aspects of the response that is judged to be important and, in some cases, is largely a matter of personal preference.

Two-surface Model Definition

The practical two-surface models, based primarily on the work of Krieg (1975), have found wide application in the computational mechanics literature, and are adopted here for analysis of the metallic damper. The specific formulation selected is a modified version of the model developed in Banerjee et al. (1987) and Chopra and Dargush (1994). The model is depicted in Fig. 1a. Two distinct, but nested, yield surfaces are defined in stress space. The inner or loading surface, separates the elastic and inelastic response regimes. It is characterized by its center and radius represented by the back stressand inner yield strength , respectively. On the other hand, the outer or bounding surface, which completely contains the smaller inner surface, is always centered at the origin of stress space with radius equal to a variable outer yield strength . Translation of the inner surface corresponds to kinematic hardening, while expansion of the outer surface produces isotropic hardening. This separation of kinmeatic and isotropic hardening mechanisms proves to be quite helpful in model formulation and implementation.

The yield criteria, flow rules, and hardening rules are established to ensure that the state of stressalways lies on or within both surfaces, that all transitions during loading are smooth, and that infinitesimal strain cycles do not cause anomalous behavior. The present model requires the determination of five inelastic material parameters, where is the yield strength of the loading surface, corresponds to the initial yield strength of the bounding surface, while, and n are hardening parameters. Details of the model will be provided in subsequent versions of this report.

Parameter Estimation

This two-surface constitutive model was used to represent ASTM A36 structural steel. The elastic modulus and Poisson ratio were equated with the usual handbook values, while the remaining five parameters were established from the stabilized cyclic data presented by Cofie and Krawinkler (1985). Parameter values, obtained from the Marquardt (1963) algorithm for nonlinear least-squares curve fitting are presented in Table 1. A comparison of the stress-strain response obtained from the two-surface model and the experimental data is displayed in Fig. 1b.

Table 1 - Model Parameters for A36 Structural Steel

Numerical Algorithm: Marquardt Method (Press et al., 1992)
Experimental Data: Cofie and Krawinkler (1985)
Number of data points:36
Assumed standard deviation:15 MPa
Optimal Parameter Values:
E=200,000MPa
V=0.3MPa
=198MPa
=427MPa
=6450MPa
=-8.47MPa
n =-10.4MPa

Figure 1(a & b) Two-surface Constitutive Model Definition

Fig 1(a)

Fig 1(a)

Finite Element Analysis

Overview

The two-surface elastoplasticity model discussed in the previous section is now applied to study the cyclic response of metallic triangular plate dampers. In order to perform the required analysis, the constitutive model has been implemented as a user-defined material model within the general purpose finite element program ABAQUS. The most recent version of that code (ABAQUS Version 5.8) also contains a two-surface cyclic plasticity model based upon the work of Lemaitre and Chaboche (1990). While the Lemaitre-Chaboche model has significant applicability in certain situations, it is difficult to achieve stabilized hysteresis loops for continued cycling due to the inherent coupling between kinematic and isotropic hardening. Stabilization of the hysteresis loops is an important aspect in the response analysis of plate dampers. Consequently, the user-defined material model outlined in Section 2 is more appropriate, as will be shown below. Considerable care in needed to ensure that the incremental constitutive equations are properly integrated. In the literature, single step backward Euler methods are most often used. Here an adaptive step-size Runge-Kutta method is employed to ensure that a high level of accuracy is maintained.

For the ABAQUS analysis, a solid element model for one-half of one plate is used as illustrated in Fig. 2. The particular triangular-shaped damper plate studied here has an overall length of 0.304 m, a base width of0.1333 m and a 0.0361m thickness. This damper is designated 2B2 by Tsai et al. (1993). The wide end of the plate is clamped, symmetry is enforced about the centerline and a constant amplitude displacement-controlled loading history is applied normal to the plate at the tip. The default automatic time stepping algorithms included in ABAQUS were utilized to analyze five complete loading cycles assumed to take place at 1Hz. Except where noted, large deformation theory was invoked.

Figure 2 ABAQUS Finite Element Model

Baseline Analysis

In the baseline analysis, the amplitude of the enforced displacement at the tip of the plate is 0.0912m. This corresponds to a nominal angle =0.30 as defined in Tsai et al. (1993) and Dargush and Soong (1995). The complete ABAQUS dataset is provided in Appendix A. The damper force-displacement response, using the Lemaitre-Chaboche model with approximate material parameters is shown in Fig. 3. Clearly, as discussed above, the response does not stabilize. On the other hand, results from the present two-surface model are displayed in Fig. 4. Not only does the response stabilize, but the shape is quite similar to that obtained in experiments (Tsai et al., 1993). This certainly suggests that the present constitutive model is at least reasonable. The magnitude of the forces obtained from the analysis are approximately 25% higher than those obtained in the experiments. While further investigation is still in order, one cause for this difference may be the hardening estimates for mild steel at large strains. The Cofie-Krawinkler data, upon which the model parameters have been established, only consider strains below 2%.

Figure 3 Baseline Force-Displacement Response with Lemaitre-Chaboche Model

Figure 4 Baseline Force-Displacement Response with Present Model

Additional results for the baseline analysis are provided in Figs. 5-9. The deformed shape plot at time t=4.25s is shown in Fig. 5. This corresponds to an instant of maximum displacement. The deformation is shown at actual scale. Figures 6 and 7 contain contour plots of vonMises equivalent stress  and the longitudinal stress, respectively, at that same time. As expected, maximum stress generally occurs on the outer fibers. Under small deformation theory for a perfect triangular plate, stresses would be uniform across any laminate parallel to the mid-surface. Clearly, this is not the case for the actual large deformation response. The longitudinal plastic strains are displayed in Fig. 8, while the yield strength of the bounding surface is shown in Fig. 9. The latter contour plot indicates that significant hardening has taken place over a relatively large portion of the plate. As mentioned above, the present model tends to somewhat overpredict this effect.

Figure 5 Baseline Deformed Shape at Maximum Displacement

Figure 6 Baseline vonMises Stress Contours at Maximum Displacement

Figure 7 Baseline Longitudinal Stress Contours at Maximum Displacement

Figure 8 Baseline Longitudinal Plastic Strain Contours at Maximum Displacement

Figure 9 Baseline Bounding Yield Strength Contours at Maximum Displacement

Parametric Studies

Several additional analyses were conducted to further study the cyclic response of the TPEA damper. Figure 10 presents the force-displacement hystersis loops for constant amplitude cycling at=0.36. Once again, the response begins to stabilize after the first cycle. The response for =0.30 and =0.36 during the fifth cycle is compared in Fig. 11. The additional isotropic hardening component is clearly visible, as the expanded loop for=0.36.

Also there is an additional stiffening of the plate at the higher displacement magnitudes. This is caused by geometric stiffening. Figure 12 compares the response obtained under the assumptions of large and small deformation theories. The stiffening that occurs beyond m is due to a form of geometric locking. This must be considered in design of these metallic dampers, because significantly higher forces can develop in some cases.

Figure 10 Force-Displacement Response with =0.36

Figure 11 Comparison of Force-Displacement Response with Displacement Amplitude

Figure 12 Force-Displacement Response for Large and Small Deformation Theories

References

Banerjee, P.K., Wilson, R.B. and Raveendra, S.T. (1987), ``Advanced Applications of BEM to Three-dimensional Problems of Monotonic and Cyclic Plasticity,'' Int. J. Mech. Sci., 29(9), 637-653.

Bergman, D.M. and Goel, S.C. (1987), Evaluation of Cyclic Testing of Steel-Plate Devices for Added Damping and Stiffness, Report No. UMCE 87-10, The University of Michigan, Ann Arbor, MI.

Chaboche, J.L. and Rousselier, G. (1983), ``On the Plastic and Viscoplastic Constitutive Equations - Parts I and II,'' J. Pressure Vessel Tech., ASME,105, 153-164.

Chopra, M.B. and Dargush, G.F. (1994), ``Development of BEM for Thermoplasticity,'' Int. J. Solids Struct., 31, 1635-1656.

Cofie, N.G. and Krawinkler, H. (1985), ``Uniaxial Cyclic Stress-Strain Behavior of Structural Steel,'' J. Engrg. Mech., ASCE, 111(9), 1105-1120.

Constantinou, M.C., Soong, T.T. and Dargush, G.F. (1998), Passive Energy Dissipation Systems for Structural Design and Retrofit, MCEER Monograph Series 1, Multidisciplinary Center for Earthquake Engineering Research, Buffalo, NY.

Dafalias, Y.F. and Popov, E.P. (1975), ``A Model of Nonlinearly Hardening Materials for Complex Loading,'' Acta Mechanica, 21, 173-192.

Dargush, G.F. and Soong, T.T. (1995), ``Behavior of Metallic Plate Dampers in Seismic Passive Energy Dissipation Systmes,'' Earthquake Spectra, 11, 545-568.

Kelly, J.M., Skinner, R.I. and Heine, A.J. (1972), ``Mechanisms of Energy Absorption in Special Devices for Use in Earthquake Resistant Structures,'' Bull. N.Z. Soc. Earthquake Engrg., 5(3), 63-88.

Krieg, R.D. (1975), ``A Practical Two Surface Plasticity Theory,'' J. Appl. Mech., ASME, E42, 641-646.

Lemaitre, J. and Chaboche, J.-L. (1990), Mechanics of Solid Materials, Cambridge University Press, Cambridge, UK.

Marquardt, D.W. (1963), ``An Algorithm for Least-Squares Estimation of Nonlinear Parameters,'' J. Soc. Ind. Appl. Math., 11(2), 431-441.

Martinez-Romero, E. (1993), ``Experiences on the Use of Supplemental Energy Dissipators on Building Structures,'' Earthquake Spectra, 9(3), 581-625.

Mroz, Z. (1967), ``On the Description of Anisotropic Work Hardening, J. Mech. Phys. Solids, 15, 163-175.

Perry, C.L., Fierro, E.A., Sedarat, H. and Scholl, R.E. (1993), ``Seismic Upgrade in San Francisco Using Energy Dissipation Devices,'' Earthquake Spectra, 9(3), 559-579.

Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992), Numerical Recipes in FORTRAN, Cambridge University Press, Cambridge, UK.

Skinner, R.I., Tyler, R.G., Heine, A.J. and Robinson, W.H. (1980), ``Hysteretic Dampers for the Protection of Structures from Earthquakes,'' Bull. N.Z. Soc. Earthquake Engrg., 13(1), 22-36.

Soong, T.T. and Dargush, G.F. (1997), Passive Energy Dissipation Systems in Structural Engineering, Wiley, Chichester, UK and New York, NY.

Tsai, K.C., Chen, H.W., Hong, C.P. and Su, Y.F. (1993), ``Design of Steel Triangular Plate Energy Absorbers for Seismic-Resistant Construction,'' Earthquake Spectra, 9(3), 505-528.

Valanis, K.C. (1971), ``A Theory of Viscoplasticity without a Yield Surface,'' Arch. Mech., 23, 517-534.

Valanis, K.C. (1980), ``Fundamental Consequence of a New Intrinsic Time Measure-Plasticity as a Limit of the Endochronic Theory,'' Arch. Mech., 32, 171.

Watanabe, O. and Atluri, S.N. (1986), ``Internal Time, General Internal Variable, and Multi-Yield-Surface Theories of Plasticity and Creep: A Unification of Concepts,'' Int. J. Plasticity, 2, 37-57.

Whittaker, A.S., Bertero, V.V., Thompson, C.L. and Alonso, L.J. (1991), ``Seismic Testing of Steel Plate Energy Dissipation Devices,'' Earthquake Spectra, 7(4), 563-604.

Appendix A: ABAQUS Dataset for Baseline Analysis

*HEADING

triangular plate metallic damper under constant strain range cycling

*NODE

1, 0.0000, 0.0000, -0.01805

9, 0.0750, 0.0000, -0.01805

321, 0.0000, 0.2690, -0.01805

329, 0.0175, 0.2690, -0.01805

361, 0.0000, 0.3040, -0.01805

369, 0.0175, 0.3040, -0.01805

1001, 0.0000, 0.0000, -0.00903

1009, 0.0750, 0.0000, -0.00903

1321, 0.0000, 0.2690, -0.00903

1329, 0.0175, 0.2690, -0.00903

1361, 0.0000, 0.3040, -0.00903

1369, 0.0175, 0.3040, -0.00903

2001, 0.0000, 0.0000, 0.00000

2009, 0.0750, 0.0000, 0.00000

2321, 0.0000, 0.2690, 0.00000

2329, 0.0175, 0.2690, 0.00000

2361, 0.0000, 0.3040, 0.00000

2369, 0.0175, 0.3040, 0.00000

3001, 0.0000, 0.0000, 0.00903

3009, 0.0750, 0.0000, 0.00903

3321, 0.0000, 0.2690, 0.00903

3329, 0.0175, 0.2690, 0.00903

3361, 0.0000, 0.3040, 0.00903

3369, 0.0175, 0.3040, 0.00903

4001, 0.0000, 0.0000, 0.01805

4009, 0.0750, 0.0000, 0.01805

4321, 0.0000, 0.2690, 0.01805

4329, 0.0175, 0.2690, 0.01805

4361, 0.0000, 0.3040, 0.01805

4369, 0.0175, 0.3040, 0.01805

*NGEN,NSET=BASE

1, 9

1001, 1009

2001, 2009

3001, 3009

4001, 4009

*NGEN,NSET=PIN0

321, 329

1321, 1329

2321, 2329

3321, 3329

4321, 4329

*NGEN,NSET=PIN

361, 369

1361, 1369

2361, 2369

3361, 3369

4361, 4369

*NFILL

BASE, PIN0, 32, 10

*NFILL

PIN0, PIN , 4, 10

*NSET,NSET=XSYM

1, 361, 10

1001, 1361, 10

2001, 2361, 10

3001, 3361, 10

4001, 4361, 10

*NGEN,NSET=PINL

2361, 2369

**

*ELEMENT,TYPE=C3D8 ,ELSET=PLATE

1, 1, 3, 23, 21,1001,1003,1023,1021

*ELGEN,ELSET=PLATE

1, 4, 2, 1, 18, 20, 20, 4, 1000, 1000

**

*SOLID SECTION, ELSET=PLATE, MATERIAL=STEEL

**

*MATERIAL,NAME=STEEL

*USER MATERIAL, CONSTANTS=8

200.E+09, 0.3, 198.E+06, 427.E+06, 6.45E+09, -8.47, -10.4, 1.

*DEPVAR

24,

*USER SUBROUTINE, INPUT=twotol21.f

**

*AMPLITUDE, NAME=CYCLIC, DEFINITION=PERIODIC

1, 6.283, 0.0, 0.0

0.0, 1.00

**

*RESTART,WRITE,FREQUENCY=2

**

*STEP, INC=1000, NLGEOM=YES

*STATIC

0.01, 5.00, 0.001, 0.02

*BOUNDARY

XSYM , 1, 1

BASE , 1, 3

*BOUNDARY, AMPLITUDE=CYCLIC

PINL , 3, 3, 0.0912

**

*EL PRINT, FREQUENCY=10, ELSET=PLATE

*NODE PRINT, FREQUENCY= 1, NSET =PINL , TOTALS=YES

*ENERGY PRINT, FREQUENCY=10, ELSET=PLATE

*NODE FILE , FREQUENCY= 1, NSET =PINL

U,RF

**

*MONITOR, NODE=2361, DOF= 3

**

*END STEP

**