Hysteretic Models for Deteriorating Inelastic Structures

The modeling of deteriorating hysteretic behavior is becoming increasingly important, especially in the context of seismic analysis and design. This paper presents the development of a versatile smooth hysteretic model based on internal variables, with stiffness and strength deterioration and with pinching characteristics. The theoretical background, development, and implementation of the model are discussed. Examples are shown to illustrate the features of the model. Many inelastic constitutive models in popular use have been developed independently of each other based on different behavioral, physical, or mathematical motivations. This paper attempts to unify the concepts underlying such models. Such a holistic understanding is essential to realize limitations in application of inelastic models and to extend 1D models to 3D models featuring interaction between various stress resultants. INTRODUCTION Hysteresis is a highly nonlinear phenomenon occurring in many disciplines involving systems that possess memory, including inelasticity, electricity, magnetism etc. Structures subjected to strong earthquake excitation are designed to dissipate energy by inelastic material behavior, interface friction, etc. However, under repeated cyclic deformation, there is invariably deterioration in the characteristics of such hysteretic behavior. Such deterioration must be taken into account in the modeling and design of seismic-resistant structural systems. The basic requirement to perform such analyses is the availability of accurate constitutive models capable of representing deteriorating structural behavior. Several hysteretic models have been developed. These can be broadly classified into two types, polygonal hysteretic model (PHM) and smooth hysteretic model (SHM). Models based on piecewise linear behavior are PHMs. Such models are most often motivated by actual behavioral stages of an element or structure, such as initial or elastic, cracking, yielding, stiffness and strength degrading stages, and crack and gap closures. Examples include Clough’s model (Clough 1966), Takeda’s model (Takeda et al. 1970), and the Park’s ‘‘threeparameters’’ model (Park et al. 1987). Sivaselvan and Reinhorn (1999) presented a detailed description of a general framework for PHMs. On the other hand, SHMs refer to models with continuous change of stiffness due to yielding but sharp changes due to unloading and deteriorating behavior. The Wen-Bouc model (Bouc 1967; Wen 1976) and Ozdemir’s model (Ozdemir 1976) are some examples of SHMs. Thyagarajan (1989) discussed a discrete element model for hysteretic behavior based on the concept proposed by Iwan (1966). This is a polygonal model that becomes smooth in the limit of infinite elements. Mostaghel (1999) presented a differential equation description of a class of PHMs with deteriorating characteristics. Many of these models that are in popular use have been developed independently of each other based on different behavioral, physical, or mathematical motivations. However, a closer examination presented herein would show that they share several features and stem from a common theoretical Grad. Res. Asst., Dept. of Civ., Struct. and Envir. Engrg., State Univ. of New York at Buffalo, Buffalo, NY 14260. Prof., Dept. of Civ., Struct. and Envir. Engrg., State Univ. of New York at Buffalo, Buffalo, NY. E-mail: [email protected] Note. Associate Editor: George Z. Voyiadjis. Discussion open until November 1, 2000. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on July 6, 1999. This paper is part of the Journal of Engineering Mechanics, Vol. 126, No. 6, June, 2000. qASCE, ISSN 0733-9399/00/0006-0633– 0640/$8.00 1 $.50 per page. Paper No. 21348. base. Such an understanding is important in developing new models and in recognizing physical limitations of many existing models. The objectives of the work reported here are twofold: (1) To present versatile SHM, with stiffness and strength deterioration and pinching characteristics, derived from inelastic material behavior; and (2) to present a holistic picture of the modeling of 1D inelastic material behavior and justify the use of such models to represent the relationships between stressresultants and strains. The SHM presented herein was developed in the context of moment-curvature relationships of beam-columns. Therefore the stress variable, or force, is here referred to as ‘‘moment’’ (M ) and the strain variable, or deformation, as ‘‘curvature’’ (f). However, these can be replaced by any other work-conjugate pair, according to the application under consideration. SHM The smooth model discussed here is a variation of the model originally proposed by Bouc (1967) and modified by several others (Wen 1976; Baber and Noori 1985; Casciati 1989; Reinhorn et al. 1995). The derivation of this model from the theory of viscoplasticity and its resemblance to the endochronic constitutive theory are discussed in a subsequent section. Plain Hysteretic Behavior without Degradation Plain hysteretic behavior with postyielding hardening can be modeled using two springs (Springs 1 and 2 of Fig. 1). One of the springs is linear elastic at all deformations, and the second changes stiffness upon yield. The two springs undergo the same deformation under a bending moment (or generalized force M ). The springs share the moment proportionally to their instantaneous stiffnesses. The portion of the moment shared by the hysteretic spring is denoted by M*. The combined stiffness is given by K = K 1 K (1) postyield hysteretic Spring 1: Postyield Spring A linear elastic spring represents the postyielding stiffness K = aK (2) postyield 0 where K0 = total initial stiffness (elastic); and a = ratio of postyielding to initial stiffness ratio. Spring 2: Hysteretic Spring This purely elastoplastic spring has a smooth transition from the elastic to the inelastic range displaying degradation phenomena. The nondegrading stiffness is JOURNAL OF ENGINEERING MECHANICS / JUNE 2000 / 633 FIG. 1. Multiple Spring Representation of Smooth Hysteretic Model N M* ̇ K = (1 2 a)K 1 2 [h sgn(M*f) 1 h ] (3) hysteretic 0 1 2 H U U J M* y where N = power controlling the smoothness of the transition from elastic to inelastic range; h1 and h2 = parameters controlling the shape of the unloading curve (h2 = 1 2 h1 for compatibility with plasticity); M* = portion of the applied moment shared by the hysteretic spring; and = (1 2 a)My, M* y the yield moment of the hysteretic spring. The function in the large parentheses is related to the smooth transition and the signum function governs unloading. When N → `, the model reduces to a bilinear system. The model can accommodate nonsymmetrical yielding if the yield moment is defined ̇ ̇ 1 1 sgn(f) 1 2 sgn(f) 1 2 M* = (1 2 a) M 1 M (4) y y y FS D S D G 2 2 where = positive and negative yield moments, re1 2 M and M y y spectively. The moment can therefore be expressed ̇ ̇ Ṁ = Kf = aK f 0 N M* ̇ ̇ 1 (1 2 a)K 1 2 [h sgn(M*f) 1 h ] f 0 1 2 H U U J M*y (5) The model is similar to the Bouc-Wen model (Wen 1976), but the transition is described explicitly in terms of the ratio between moment and yield moment. Hysteretic Behavior with Degradation Often structures that undergo inelastic deformations and cyclic behavior weaken and lose some of their stiffness and strength. Moreover, often gaps develop due to cracking and the material becomes discontinuous. The hysteretic model developed herein can accommodate changes in the stiffness, strength, and pinching due to gap opening and closing. Stiffness Degradation Stiffness degradation occurs due to geometric effects. The elastic stiffness degrades with increasing ductility. It has been found empirically that the stiffness degradation can be accurately modeled by the pivot rule (Park et al. 1987). According 634 / JOURNAL OF ENGINEERING MECHANICS / JUNE 2000 FIG. 2. Stiffness and Strength Degradation to this rule, the load-reversal branches are assumed to target a pivot point on the initial elastic branch at a distance of aMy on the opposite side, where a is the stiffness degradation parameter. This is shown in Fig. 2(a). From the geometry in Fig. 2(a), the stiffness degradation factor is given by M 1 aM cur y K = R K = K (6) cur K 0 0 K f 1 aM 0 cur y where Mcur = current moment; fcur = current curvature; K0 = initial elastic stiffness; a = stiffness degradation parameter; and My = if (Mcur, fcur) is on the right side of the initial 1 My elastic branch and My = if (Mcur, fcur) is on the left side. 2 My However, since stiffness degradation occurs only in the hysteretic spring, only the hysteretic stiffness is modified and is given by N M* ̇ K = (R 2 a)K 1 2 [h sgn(M*f) 1 h ] (7) hysteretic K 0 1 2 H U U J M* y Ranges of variation of a indicate that for large values (a > 200), no deterioration occurs, whereas small values (a < 10) produce substantial degradation (Sivaselvan and Reinhorn 1999). Strength Degradation Strength degradation is modeled by reducing the capacity in the backbone curve, as shown schematically in Fig. 2(b). Mathematically, this is equivalent to specifying an evolution equation for the yield moment. The strength degradation rule can be formulated to include an envelope degradation, which occurs when the maximum deformation attained in the past is exceeded, and a continuous energy-based degradation. The rule reads 1/b1 1/2 f b H max 2 1/2 1/2 M = M 1 2 1 2 (8) y y0 F S D G F G 1/2 f 1 2 b H u 2 ult where = positive or negative yield moment; = in1/2 1/2 M M y y0 itial positive or negative yield moment; = maximum pos1/2 fmax itive or negative curvatures; = positive or negative ulti1/2 fu mate curvatures; H = hysteretic energy dissipated, obtained by integrating the hysteretic energy quotient; Hult = hysteretic energy dissipated when loaded monotonically to the ultimate curvature without any degradation; b1 = ductility-based strength degradation parameter; and b2 = energy-based strength degradation parameter. The second term on the right-hand side of (8) represents strength degradation due to increased deformation, and the third term represents strength degradation due to hysteretic energy dissipated. The quotient of the hysteretic energy in incremental form is given by M 1 (M 1 DM ) DM DH = Df 2 (9) F G S D 2 R K K 0 The differential equations governing strength degradation in the SHM can be obtained by differentiating (8) 1/2 dM b H y 2 1/2 = M 1 2 y0 HF G dt 1 2 b H 2 ult 1 1/2 (12b )/b 1/2 1 1 ̇ ? 2 (f ) f max max F G 1/2 1/b1 b (f ) 1 u 1/b1 1/1 f b max 2 ̇ 1 1 2 2 H F S D G F G J 1/2 f (1 2 b )H u 2 ult (10) Eq. (10) requires the hysteretic energy quotient in rate form Ṁ (K 1 R K ) postyield K hysteretic ̇ ̇ Ḣ = M f 2 = Mf 1 2 (11) S D F G R K R K K 0 K 0 The evolution equations for the maximum positive and negative curvatures can be written 1 1 2 2 ̇ ̇ ̇ ̇ ̇ ̇ f = fU(f 2 f )U(f); f = fU(f 2 f)(1 2 U(f)) max max max max (12) where U(x) = heaviside step function. The differential equations [(10)–(12)] govern strength degradation. Pinching or Slip Pinched hysteretic loops usually are a result of crack closure, bolt slip, etc. An additional spring (Spring 3 of Fig. 1) called the slip-lock spring (Baber and Noori 1985; Reinhorn et al. 1995) is added in series to the hysteretic spring to model this effect. The stiffness of the slip-lock spring can be written 2 21 ̄ 2 s 1 M* 2 M* K = exp 2 (13) slip-lock HÎ F S D GJ p M* 2 M* s s where s = slip length = = mea1 2 R (f 2 f ); M* = sM* s max max s y sure of the moment range over which slip occurs; M̄* = = mean moment level on either side about which slip lM*y occurs; Rs, s, and l = parameters of the model; and and 1 fmax = maximum curvatures reached on the positive and neg2 fmax ative sides, respectively, during the response. The variation of the flexibility of the slip-lock element can be chosen as Gaussian or any other distribution provided that (Kslip-lock) 21 dM ` *2` = s the slip length. The stiffness of the combined system is then given by K K hysteretic slip-lock K = K 1 (14) postyield K 1 K slip-lock hysteretic Gap-Closing Behavior Often, hysteretic elements exhibit stiffening under higher deformations. This happens, for example, in metallic dampers (Soong and Dargush 1997) when axial behavior predominates bending behavior and in bridge isolators (Reichman and Reinhorn 1995) due to closing of the expansion joints. Such behavior can be modeled by introducing an additional gap-closing spring in parallel, as shown in Fig. 1. The internal moment and the stiffness of this spring are given by N 21 gap M** = kK N (ufu 2 f ) U(ufu 2 f ) (15) 0 gap gap gap N 21 gap K = kK N (ufu 2 f ) U(ufu 2 f ) (16) gap-closing 0 gap gap gap where M** = moment in the gap-closing spring; Kgap-closing = stiffness of the gap-closing spring; fgap = gap-closing curvature; U = heaviside step function; and k and Ngap = parameters.