Masing Rules Research Articles

지진시 지반의 반복경화/연화 현상에 의한 부지응답 특성 영향 연구

본 논문에서는 반복하중에 의한 반복 경화 및 연화 현상을 나타낼 수 있는 수정 IWAN 모델을 이용하여, 1차원 비선형 부지응답 해석프로그램(이하 KODSAP; Kaist One Dimensional Site-response Analysis Program)을 개발하였다. 개발된 프로그램은 지진하중 재하에 따른 지반의 반복경화 및 연화현상에 의한 부지응답 특성 변화를 재현할 수 있다. KODSAP을 이용하여 기반암 상부 40m인 모형지반의 반복경화 및 연화 정도, 지진가속도의 크기에 따른 부지응답특성 변화를 살펴 보았으며, 현재 실무에서 널리 적용되고 있는 등가선형, 비선형해석과 KODSAP 해석결과(지반의 반복경화 및 연화현상을 고려한)과의 차이점을 살펴 보았다. A one-dimensional site response analysis program (KODSAP) was developed using cyclic soil behavior model by using the modified parallel IWAN model. The model is able to predict the cyclic hardening and degradation of soil through the adjustment of the internal slip stresses of its elements beyond the cyclic threshold, and satisfies Bauschinger's effect and the Masing rule in terms of its own behavior characteristics. The program (KODSAP) used the direct integration method in the time domain. The elasticity of the base rock was considered as a viscous damper boundary condition. The effects of cyclic hardening or degradation of soil on site response analysis were evaluated through parametric studies. Three types of analyses were performed to compare the effect of analysis and cyclic parameter on site response. The first type was equivalent linear analysis, the second was nonlinear analysis, and a third was nonlinear analysis using the cyclic hardening or degradation model.

Nonlinear attenuation and rock damage during strong seismic ground motions

Strong seismic waves cause nonlinear behavior in the shallow subsurface in fractured rocks. Seismologists use low‐amplitude signals from small repeating earthquakes to measure S wave velocity decrease after strong motion. The 2004 Parkfield, California, earthquake provides examples of such velocity changes in fractured sandstone with an S wave velocity of ∼300 m s−1. This nonlinear behavior occurred around the wave number depth of the incident waves, ∼30 m, for the ∼10 s−1 dominant angular frequency on a velocity seismogram. The low‐amplitude S wave velocity gradually recovered with the logarithm of time. The attenuation of strong waves in general depends nonlinearly on their amplitude. High dynamic stress triggered small, very shallow earthquakes, at sites including Parkfield. The theoretical frictional behavior of a fractured medium with heterogeneous prestress relates these phenomena. Failure occurs in the highly prestressed domains causing small earthquakes and opening‐mode cracks. The energy to dilate the cracks dissipates a significant fraction of the incoming seismic energy. The local high‐porosity domains close with the logarithm of time, as expected from the aging law of rate and state friction, increasing the S wave velocity. The domain model indicates that nonlinear effects increase gradually over a range of dynamic Coulomb stresses as observed and as included in the widely used Masing rules. The Linker and Dieterich (1992) relationship provides the maximum sustainable dynamic coefficient of friction needed to utilize the Masing rules. This parameter is the coefficient of friction at a laboratory normal traction plus a constant ∼0.15 times the logarithm of the ratio of field normal traction to the laboratory normal traction. It is helpful to relate S wave velocity to starting frictional strength, as coefficient of friction near the quarter‐wavelength depth determines nonlinear behavior. Then the dynamic coefficient of friction and equivalently the maximum sustainable acceleration at the dominant frequency depend weakly on S wave velocity. For example, the coefficient of friction at an angular frequency of 10 s−1 is less than 1.5 for rocks ranging from Parkfield sandstone to intact granite.

A modified parallel IWAN model for cyclic hardening behavior of sand

A modified parallel IWAN model, which includes a cyclic hardening function, is proposed and verified. The proposed model consists of elasto-perfect plastic and isotropic hardening elements. The model is able to predict cyclic hardening behavior through the adjustment of the internal slip stresses of its elements beyond the cyclic threshold, and satisfies Bauschinger's effect and the Masing rule with its own behavior characteristics. The cyclic hardening function is developed based on the irrecoverable plastic strain (accumulated shear strain) of dry sand during shearing, which is assumed to be a summation of shear strain beyond the cyclic threshold. Symmetric-limit cyclic loading and irregular loading tests were performed to determine model parameters and to verify the behavior of the proposed model. Finally, a one-dimensional site response analysis program (KODSAP) is developed by using the proposed model. The effects of cyclic hardening behavior on site response are evaluated using KODSAP.

Application of hysteresis functions in vibration problems

In the paper, the notion of the hysteresis function introduced by the author is developed further and applied in some vibration problems for hysteretic materials. The stress–strain relation is built on the basis of well-known extended Masing's rules in which the scaled backbone curve used in the second rule is replaced by a more general function called the hysteresis function. This modification allows us to regulate the dependence of the damping ratio on the strain amplitude in the process of cycle deforming of a material with further extending to arbitrary deformation processes. The three cases of hysteretic systems are considered: the system with limited stress when strain increases without bound, the system with linear backbone curve, and the system with the backbone curve having increasing stiffness for increasing strain. For a number of dynamic examples, a comparison of different hysteresis functions is carried out.

Generalized Masing Approach to Modeling Hysteretic Deteriorating Behavior

The modeling of hysteretic behavior is of significant importance in several areas, including civil engineering and mechanics. This paper focuses on finding a method for modeling hysteretic behavior that is simple and efficient in terms of capturing the response and retaining the memory, if any, and at the same time is proper for use in physically meaningful modeling and identification of the system with few parameters. A distributed-element model (DEM) capable of capturing deterioration is used as a starting point, and its characteristics are studied, with a particular focus on the way memory is stored in the model. It is observed that keeping track of the response at a few of the past extremes of input displacement, called the Sequence of Dominant Alternating Extremes, is enough for representing the effect of history. The relation of this behavior to a generalized Masing model is studied. A set of rules is proposed which is a generalization of the Masing rules and can capture the deteriorating (or nondeteriorating) response of DEMs with any distribution of element yield displacement thresholds to any arbitrary loading. The presented formulation provides a framework for efficient modeling and identification of dynamic models of very different characteristics with only a few physically meaningful parameters.

Mathematical modelling of a "post-Northridge" steel connection

A robust mathematical model to capture the behaviour of an attractive post-Northridge type, Partially Restrained (PR) steel connection is proposed. The proposed procedure captures the loading behaviour for this type of connection reasonably well. It is suggested that the unloading and reloading parts of M ? θ curves required for seismic analysis can be theoretically generated by using the Masing Rule. This study confirms some of the important desirable features of the connection. Although it is a PR connection, its initial stiffness is similar to that of bolted, a single web angle with top and seat connection, generally considered to be a Fully Restrained (FR) connection. Thus, a steel frame with the connection studied is not expected to show weakness in satisfying serviceability requirements. Furthermore, the beam develops full plastic moment and the ductility of the connection and its seismic energy dissipation characteristics are expected to be significantly improved.

Modeling cyclic behavior of geosynthetics using mathematical functions combined with Masing rule and bounding surface plasticity

In the numerical analysis of geosynthetic-reinforced soil structures subjected to earthquake loadings, the constitutive model for the geosynthetic reinforcement is important, because it affects the accuracy of numerical simulations. Two different approaches for modeling the monotonic and cyclic behavior of geosynthetic reinforcements are discussed in this paper: a nonlinear mathematical function combined with the modified Masing rule, and bounding surface plasticity. The Masing rule was modified so that it can simulate the hysteretic behavior of geosynthetics under cyclic loading, including the decrease in the residual strain rate with the number of loading cycles (cyclic strain-hardening). For the bounding surface model, different bounding lines were defined for primary loading, unloading, and reloading. An exponential function was used as the bounding line for primary loading in order to capture the stiffening behavior of some geosynthetics. Constitutive models were proposed based on the two approaches for different types of geosynthetic. Experimental results for six geogrids were used to verify the proposed models, and it was shown that they were able to capture the nonlinear and hysteretic behavior of the corresponding geosynthetics. It was shown that the approach utilizing a mathematical function combined with the modified Masing rule is simple and straightforward. Bounding surface plasticity provides an approach that is more rational for simulating the cyclic behavior of geosynthetics, especially for the case of irregular earthquake loading.

Implicit plane stress algorithm for multilayer J2‐plasticity using the Prager–Ziegler translation rule

AbstractA totally implicit algorithm for plane stress multilayer plasticity is presented. The algorithm presents the plane stress version of the 3D/plane strain model for multilayer plasticity presented in a previous work. As in the 3D/plane strain case, the model is consistent with the principle of maximum dissipation and it may be considered as an extension of classical J2‐plasticity for anisotropic non‐linear kinematic behaviour preserving Masing's rules. In order to obtain the asymptotic second‐order convergence of the Newton algorithm, the consistent tangent moduli are also given. Copyright © 2003 John Wiley & Sons, Ltd.

Implicit J2‐bounding surface plasticity using Prager's translation rule

AbstractA bounding surface J2‐plasticity model that uses Prager's translation rule is presented. The model preserves Masing's rules and is developed from the same ideas as classical infinitesimal J2‐plasticity, resulting in the same formulation with the exception of the algorithm for the computation of the hardening function. Instead of utilizing a loading surface as in a previous formulation, hardening surfaces are introduced; the formulation is similar to that of multilayer plasticity using Prager's rule, presented in previous work. An implicit algorithm based on the radial return concept is used, and the consistent elastoplastic tangent is also developed in closed form. Examples illustrating anisotropic behaviour are presented and compared to that predicted by a multilayer J2‐plasticity model. The model is also applied to a soil dynamics problem to show the robustness of the algorithm and its applicability to complex loading. Copyright © 2002 John Wiley & Sons, Ltd.

Seismic reliability of non-linear frames with PR connections using systematic RSM

An effective, efficient, and robust reliability analysis algorithm is proposed for non-linear structures, where seismic loading can be applied in the time domain. The method is developed specifically for steel frame structures considering all major sources of non-linearity, including geometry, material, and partially restrained (PR) connections. The non-linearity due to PR connections is modeled by moment-relative rotation curves using the four-parameter Richard model. For seismic excitation, the loading, unloading, and reloading behavior at PR connections is modeled using moment-relative rotation curves and the Masing rule. The proposed algorithm intelligently integrates the response surface method, the finite element method, the first-order reliability method, and an iterative linear interpolation scheme. The uncertainties in all the random variables including the four parameters of Richard model are considered. Two unique features of the proposed algorithm are that (1) actual earthquake time histories can be used to excite structures in the presence of major sources of non-linearity and uncertainty and (2) it is possible to estimate the risk corresponding to both the serviceability and strength limit states. The algorithm is verified using the Monte Carlo simulation technique. The verified algorithm is first used to study the reliability of a frame structure in the presence of PR connections with different degrees of flexibility. Then the algorithm is used to estimate the reliability of a frame structure excited by 13 actual recorded earthquake time histories, 12 of them recorded during the Northridge earthquake of 1994. As expected, the reliabilities of the frame are found to be quite different, when excited by several time histories of the Northridge earthquake.

Application of experimental results on cyclic deforming of soils to seismic response analysis

It is shown in this paper that experimental results on stiffness and damping corresponding to cyclic deforming of soils can be well approached theoretically on an interval of strain variation from zero to arbitrary large values of strains. For this purpose, the following two methods are recommended: a modification of the hysteretic model obeying Masing's rule by adding elastic stresses to pure hysteretic stresses that do not change the corresponding backbone curve; and, as the second method, a non-linear model with frequency independent loss of energy in periodic processes. It is shown that in the area close to the free surface of soil, normalized strains resulting from propagation of seismic waves can be very large, even in the case of an input motion of relatively low intensity. Therefore, a using analytical approximations for experimental stress–strain relationships, which are suitable only for an interval of small and medium values of normalized strains, is insufficient in seismic response analysis.

Effects of the constitutive relationship on seismic response of soils. Part I. Constitutive modeling of cyclic behavior of soils

This paper is the first part of the general paper dealing with effects of constitutive modeling of cyclic stress–strain behavior of soils on site amplification. The paper concentrates on modeling of pseudo-static cyclic soil behavior in small to medium strain range. In order to fit the small strain data accurately, the chosen analytical stress–strain relationship should satisfy the specific small strain condition formulated for soils using the small strain data from the pseudo-static cyclic tests. Analysis of conventional relationships, in particular the Ramberg–Osgood (R–O) relationship, indicated that a failure to satisfy this condition lead to low accuracy of prediction of both tangent stiffness and damping ratio at small and medium strains. The logarithmic function originally proposed to describe static monotonic stress–strain behavior is applied to fit experimental cyclic backbone curves. Constructed to satisfy the formulated small strain condition for soils, this function has proven to be free from the limitations of the R–O and other relationships. When applied in combination with the Masing rules to predict damping ratios, it gives a good prediction in the small to medium strain range, where the Masing hypothesis is supported by experimental evidence.

Modèle de comportement dynamique cyclique de sols en grandes déformations

A two-parameter cyclic empirical law, as used traditionnally by the engineer, is not able to fit the dynamic behaviour of soils under large strains. When simulating the classical laboratory results, a Hardin-Drnevich law associated with Masing's rule, although able to fit satisfactorily the average strain modulus with only two parameters (Go, α), is not adapted for fitting the damping factor at moderate and high strains. To overcome this limitation, a modified empirical law with three parameters (Go, α, β) is proposed. According to the initial law, parameter Gocharacterizes low strains and parameter α allows simulating the average modulus. The extra parameter β is used for adjusting the damping factor, as obtained by Masing's rule, to the experimental results at high strain.Key words: soil behaviour, dynamic, cyclic, large deformation, hysteresis, damping.

Bounding surface plasticity model with extended Masing behavior

In this paper, a deviatoric bounding surface plasticity model that preserves Masing's rule and an algorithm that allows for the lack (or existence) of elastic range is formulated. This model inherits the advantages of multilayer models like Mroz's and the simplicity and continuity of the bounding surface plasticity. The material-specific parameters can be obtained directly and automatically from the experimental non-linear stress–strain “backbone” curves. A mapping converts monotonic isotropic hardening functions into anisotropic ones that depend on the previous load history. The cyclic behavior resembles that of the nested yield surface models, the cycles are stabilized from the first loop, the unloading curve preserves a fixed homological ratio (of two) with the initial monotonic one and the process reloads through previous hardening functions when the corresponding stress–strain curves are intersected. Furthermore, it also maintains the advantages of bounding surface models, like the possibility of using a continuous non-linear hardening function that depends solely on stresses and that is interpolated through only two surfaces. Although the formulation in this paper is based on the von Mises criterion, different criteria could be used.

Modèle de comportement dynamique cyclique de sols en grandes déformations

A two-parameter cyclic empirical law, as used traditionnally by the engineer, is not able to fit the dynamic behaviour of soils under large strains. When simulating the classical laboratory results, a Hardin-Drnevich law associated with Masing's rule, although able to fit satisfactorily the average strain modulus with only two parameters (Go, α), is not adapted for fitting the damping factor at moderate and high strains. To overcome this limitation, a modified empirical law with three parameters (Go, α, β) is proposed. According to the initial law, parameter Go characterizes low strains and parameter α allows simulating the average modulus. The extra parameter β is used for adjusting the damping factor, as obtained by Masing's rule, to the experimental results at high strain.Key words: soil behaviour, dynamic, cyclic, large deformation, hysteresis, damping.

A homogenized transversely isotropic model for cyclic behavior of reinforced soil

A constitutive model for cyclic behavior of reinforced soil is proposed that uses the homogenized transversely isotropic concept. The cyclically non‐linear reinforced soil behaviors and the proper boundary conditions are considered in the model. The hyperbolic function and the Masing rule are combined to model cyclic behaviors of reinforced soil. Assessment of the predictive capabilities and limitations of the proposed model are compared with some experimental results.

THERMAL RATCHETTING ANALYSIS OF CYLINDERS SUBJECTED TO AXIAL TRAVELLING OF TEMPERATURE DISTRIBUTION

The paper deals with estimation of thermal ratchetting strain in cylinders subjected to short axial travelling of temperature distribution. The estimation is made by considering partly pressurized cylinders as a model. It is assumed that the pressurized region, which is taken to occupy the temperature travel region, is ruled by the deformation theory with a constant secant modulus so as to estimate the thermal ratchetting strain conservatively. This assumption enables us to utilize a linear elastic solution for thin-walled cylinders. The secant modulus is specified using the analytical solution derived in the long travel case by assuming Masing's rule and no cyclic hardening. It is shown that the model gives good estimates to finite element analysis of the thermal ratchetting strain as well as to experiments on 316FR steel cylinders done by Kitade et al. An extension of the model, in which plastic yielding outside the travel region is taken into account, is also discussed.

高温強度 温度分布の軸方向移動を受ける円筒の熱ラチェット解析 温度分布移動距離の影響

This paper deals with the estimation of thermal ratchetting strain of cylinders subjected to short axial travelling of temperature distribution. The estimation is made by considering partly pressurized cylinders as a model. It is assumed that the pressurized region is ruled by the deformation theory with the uniform secant modulus determined so as to be exact in the long travelling case. This assumption enables us to utilize a linear elastic solution for the problem. The secant modulus is specified using the analytical solution which was derived by the present authors for the long travelling case by assuming Masing's rule. It is shown that the model based on the partly pressurized cylinders gives good estimates comparable to the finite element analysis of the thermal ratchetting strain as well as to the experiments on 316FR steel cylinders done by Kitade et al. An extension of the model, in which plastic yielding outside the travel region is taken into account, is also discussed.

Periodic and Non-Periodic Oscillations of a Class of Hysteretic Two Degree of Freedom Systems

The equations governing the response of hysteretic systems to sinusoidal forces, which are memory dependent in the classical phase space, can be given as a vector field over a suitable phase space with increased dimension. Hence, the stationary response can be studied with the aids of classical tools of nonlinear dynamics, as for example the Poincare map. The particular system studied in the paper, based on hysteretic Masing rules, allows the reduction of the dimension of the phase space and the implementation of efficient algorithms. The paper summarises results on one degree of freedom systems and concentrates on a two degree of freedom system as the prototype of many degree of freedom systems. This system has been chosen to be in 1:3 internal resonance situation. Depending on the energy dissipation of the elements restoring force, the response may be more or less complex. The periodic response, described by frequency response curves for various levels of excitation intensity, is highly complex. The coupling produces a strong modification of the response around the first mode resonance, whereas it is negligible around the second mode. Quasi-periodic motion starts bifurcating for sufficiently high values of the excitation intensity; windows of periodic motions are embedded in the dominion of the quasi-periodic motion, as consequence of a locking frequency phenomenon.

Reduced phase space analysis for hysteretic oscillators of masing type

Hysteretic oscillators can be represented as dynamic systems provided a high dimension phase space is considered. It is shown that for a fairly large class of oscillators, based on Masing rules, the dynamics can be studied over a two-dimensional manifold. Theoretical motivations and algorithms are presented. Developed procedures are tested on two simple but important oscillators, one of them has an unstable branch and is the hysteretic counterpart of the Duffing one-wells two-bumps oscillator. Though the applications mainly have the scope to illustrate the way the procedures of solution work, some aspects of the dynamic response are investigated in depth. In particular it is shown that the well-known sequence: breaking of symmetry-cascade of period doubling, does not appear in practice in the two bumps hysteretic oscillator. Subharmonic oscillations are observed throughout.