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Values of g and sgn(z) per segment. Table 1 A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968

Table 1 Values of g and sgn(z) per segment. Table 1 A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968

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Fig. 1. Bouc-Wen model. where A, f, y, m are dimensionless quantities controlling the behavior of the model, sgn(-) is the signum function and the overdot denotes the derivative with respect to time.
Fig. 1. Bouc-Wen model. where A, f, y, m are dimensionless quantities controlling the behavior of the model, sgn(-) is the signum function and the overdot denotes the derivative with respect to time.
Fig. 2. Response of Bouc-Wen model under cyclic excitation.
Fig. 2. Response of Bouc-Wen model under cyclic excitation.
Fig. 3. (a) Displacement drift and (b) force relaxation of hysteretic spring. A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 3. (a) Displacement drift and (b) force relaxation of hysteretic spring. A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
The hysteretic spring exhibits force relaxation when cycled between two unequal displacements uw, uw with u,>u>. Referring to Fig. 3b, the force relaxation f can be quantified by utilizing Eqs. (6) and (7) over the transition A + B—- C in which the hysteretic force does not change sign. When n = 1, it is given by:
The hysteretic spring exhibits force relaxation when cycled between two unequal displacements uw, uw with u,>u>. Referring to Fig. 3b, the force relaxation f can be quantified by utilizing Eqs. (6) and (7) over the transition A + B—- C in which the hysteretic force does not change sign. When n = 1, it is given by:
In case f = y the unloading branch is covered by Eq. (8) and hence Eqs. (13) and (14) are modified, respectively, as follows: Regarding the Bouc-Wen model, the work of the elastic spring over a closed strain loop is zero. Thus, in violation of Ilyushin’s postulate, the total work W over the transition A > B— C of Fig. 3b is attributed to the hysteretic spring only and it is expressed by the shaded area with a negative sign. It is noted that W is not equal to the dissipated energy [18]. If it were, the model would violate the second law of thermodynamics [19].
In case f = y the unloading branch is covered by Eq. (8) and hence Eqs. (13) and (14) are modified, respectively, as follows: Regarding the Bouc-Wen model, the work of the elastic spring over a closed strain loop is zero. Thus, in violation of Ilyushin’s postulate, the total work W over the transition A > B— C of Fig. 3b is attributed to the hysteretic spring only and it is expressed by the shaded area with a negative sign. It is noted that W is not equal to the dissipated energy [18]. If it were, the model would violate the second law of thermodynamics [19].
where the underlined expression is the modification, R,(u,z) € [0,1] is a stiffening factor and H(-) is the Heaviside function defined herein as: Note that definition (23) differs from the one in [20] at x = 0. Due to the Heaviside function, the unloading branches of the modified model remain identical to those of the original model. When loading or reloading, factor R,(u, z) controls the transition between loading (reduced) stiffness and unloading (increased) stiffness. For R, = 0, Eq. (22) reduces to Eq. (2) and the proposed modified model is identical to the original one. For R, = 1, the loading stiffness becomes equal to that of unloading at the same point. When n = 1, the total work is expressed as:
where the underlined expression is the modification, R,(u,z) € [0,1] is a stiffening factor and H(-) is the Heaviside function defined herein as: Note that definition (23) differs from the one in [20] at x = 0. Due to the Heaviside function, the unloading branches of the modified model remain identical to those of the original model. When loading or reloading, factor R,(u, z) controls the transition between loading (reduced) stiffness and unloading (increased) stiffness. For R, = 0, Eq. (22) reduces to Eq. (2) and the proposed modified model is identical to the original one. For R, = 1, the loading stiffness becomes equal to that of unloading at the same point. When n = 1, the total work is expressed as:
Fig. 4. Formulation of stiffening factor R,. A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968 Next, a consistent formulation of stiffening factor R, is considered. By virtue of Eqs. (5) and (8), the ful unloading path from a reversal point is known a priori in analytical form. For 6¥y this path is curved. whereas for f = y it is a straight line. Setting R, equal to unity along this path has the desired effect that partial unloading followed by reloading will guide the hysteretic response exactly on the unloading path up to the reversal point. Upon there, factor R, should revert to zero to allow for further loading with normal (reduced) stiffness. Finally, factor R, should diminish in regions away from the unloading path so that normal behavior of Bouc—Wen model remains unaffected.
Fig. 4. Formulation of stiffening factor R,. A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968 Next, a consistent formulation of stiffening factor R, is considered. By virtue of Eqs. (5) and (8), the ful unloading path from a reversal point is known a priori in analytical form. For 6¥y this path is curved. whereas for f = y it is a straight line. Setting R, equal to unity along this path has the desired effect that partial unloading followed by reloading will guide the hysteretic response exactly on the unloading path up to the reversal point. Upon there, factor R, should revert to zero to allow for further loading with normal (reduced) stiffness. Finally, factor R, should diminish in regions away from the unloading path so that normal behavior of Bouc—Wen model remains unaffected.
Fig. 5. Stiffening effect in u-z space: original model (left), modified model (right) (7 = 2, 6 = 0.1, y = 0.9, uf = L.5u,, z* * 0.905, p = 2). A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 5. Stiffening effect in u-z space: original model (left), modified model (right) (7 = 2, 6 = 0.1, y = 0.9, uf = L.5u,, z* * 0.905, p = 2). A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 6. Contour plot of R, with n = 2, B= 0.1, y= 0.9, u = 1.5uy, zt © 0.905 and (a) p = 1.0, (b) p = 2.0. ut A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 6. Contour plot of R, with n = 2, B= 0.1, y= 0.9, u = 1.5uy, zt © 0.905 and (a) p = 1.0, (b) p = 2.0. ut A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 7. Response under the Northridge TAR090 [21] excitation using last observed reversal points. A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 7. Response under the Northridge TAR090 [21] excitation using last observed reversal points. A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 8. Response under the Northridge TAR090 [21] excitation using maximum displacement reversal points
Fig. 8. Response under the Northridge TAR090 [21] excitation using maximum displacement reversal points
Fig. 9. “Active” zone of reversal point P* (n = 2, B = 0.1, y = 0.9, ut = A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 9. “Active” zone of reversal point P* (n = 2, B = 0.1, y = 0.9, ut = A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 10. Response under the Northridge TAR090 [21] excitation using multiple reversal points.
Fig. 10. Response under the Northridge TAR090 [21] excitation using multiple reversal points.
Strong motion recordings taken from PEER [21]. Table 2
Strong motion recordings taken from PEER [21]. Table 2
Fig. 11. Relative peak displacement error.
Fig. 11. Relative peak displacement error.
Fig. 12. Relative peak hysteretic energy error. A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
Fig. 12. Relative peak hysteretic energy error. A.E. Charalampakis, V.K. Koumousis / Journal of Sound and Vibration 322 (2009) 954-968
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