Inelastic analysis types

Based on how material nonlinearity is accounted for, elastic-plastic analysis types can be further divided into:

 Elastic-Plastic Hinge Analysis

 Refined-Plastic Hinge Analysis

 Reduced Tangent Modulus Analysis

 Plastic-Zone Analysis

(1) Elastic-plastic hinge analysis

In an elastic-plastic hinge analysis, spread of yielding along the member length is concentrated in discrete regions (zero length) where plastic hinges form. Fully yielded sections are modeled using plastic hinges that enforce a yield surface criterion and allow for elastic unloading. Plastic hinge methods rely on

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simplified force interaction expressions to approximate the yield surface. It neglects residual stresses and typically one element is used to model each frame member. This analysis may be adequate for frames with slender members whose failure is governed by elastic instability (Chen, 2000; Chen and Lui, 2017). But it significantly overestimates the ultimate strength of multistory frames whose failure is governed by material yielding (White et al, 1991; White and Chen, 1993), since material yielding is concentrated in the zero- length plastic hinges and the associated instability are not considered.

(2) Refined-plastic hinge analysis

Based on modifications of the elastic-plastic hinge analysis, the refined-plastic hinge analysis captures partial yielding along member and across cross-section due to residual stresses and large axial force. The influence of spread of plasticity is accounted for by using a tangent modulus (Et). It provides a more accurate estimate of member capacities in an inelastic analysis than a typical elastic-plastic hinge analysis (Ziemian, 2010). Great efforts have been contributed to this approach for the stability design of in-plane frames (Abdel-Ghaffar et al.,1991; Al-Mashary and Chen,1991; Clarke et al., 1992; King, et al.,1991; Liew and Chen,1991; Liew et al.,1993a,1993b; White,1993; Kim,1996; Kim and Chen,1996a, 1996b,1997,1998; Kim et al,2000; Ziemian and McGuire, 2002; Ziemian et al., 2008) and three-dimensional frames (Liew and Tang,1998; Kim et.al, 2001; Kim and Choi, 2001). Avery (1998) extended this approach to account for the effect of local buckling. A program (MASTAN2, 2000) that can perform refined-plastic hinge analysis, has been developed by Ziemian and McGuire (Ziemian and McGuire, 2000; McGuire et al., 2000). This program is based on MATLAB.

(3) Reduced tangent modulus analysis

This types of analysis captures the effects of spread-of plasticity by means of reducing the stiffness of members (Cheong-Siat-Moy,1977; Orbison et al., 1982; White and Chen, 1993; Ziemian et al., 1992a, 2008; Ziemian and McGuire, 2002, Surovek-Maleck and White, 2004a, 2004b; Zubydan, 2010; Kucukler et al., 2014, 2015a, 2015b, 2016; Kucukler and Gardner, 2018, 2019a,2019b; White et al., 2016). Some researchers use stiffness reduction factor derived from column flexural buckling curves to approximate stiffness reductions in steel members with high axial load (Ziemian et al., 1992a, 1992b; Ziemian and McGuire, 2002; White and Chen, 1993; Orbison et al., 1982). Since this tangent-modulus adjustment does not consider the combined action of axial compression and bending, it produces considerable errors for the

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members whose plastification under combined axial compression and bending are significant (Ziemian 2010). To accurately capture plastification effects in a second-order analysis, the reduced stiffness must consider the combined effects of axial compression and bending, residual stresses, and shape factor for different cross-sections (Surovek-Maleck and White, 2004a, 2004b; Ziemian et al., 2008; White et al.,2016; Kucukler et al., 2014, 2016; Kucukler and Gardner, 2019a).

Surovek-Maleck and White (2004a, 2004b) proposed a general flexural stiffness reduction factor 0.9τb for strong axis bending and 0.8τb weak axis bending. The factor τb, derived from Column Research Council

(CRC) column strength curve (Johnston, 1966), is intended to mainly account for the influence of partial yielding accentuated by the presence of residual stresses. The factor 0.8 or 0.9, accounts for additional stiffness reduction under combined axial loading and bending moment. The flexural stiffness reduction factor 0.8τb has been adopted in the Direct Analysis Method (DM) for the design of steel frames provided in AISC 360-16 (2016). The definition for τb in AISC 360-16 (2016) has been modified to account for the effects of local buckling of slender elements in compression members.

Kucukler et al. (2014, 2016) developed a function of beam-column flexural stiffness reduction factor for the stability design of in-plane carbon steel beam-columns and frames with compact I sections (referred to as τMN,Kucukler, note that the symbols presented in this section are not identical to their original symbols). The

main variables of the function include first order maximum axial force, first order maximum bending moment, column flexural stiffness reduction factor (referred to as τN,Kucukler), beam flexural stiffness reduction factor (referred to as τM, Kucukler ), and a moment gradient factor Cm. τN, Kucukler is derived from

column buckling curves given in EN 1993-1-1 (2005), while τM, Kucukler for beams sufficiently restrained against lateral-torsional buckling (LTB) under pure bending, is developed using a similar empirical formulation to the one proposed by Zubydan (2010) for compact I and H cross-sections subjected to combined axial loading and bending moment. For the development of flexural stiffness reduction formulations in Kucukler et al. (2014, 2016) and Zubydan (2010), strain-hardening is not considered and residual stresses pattern recommended by the European Convention for Construction Steelwork (ECCS, 1984) is adopted.

Furthermore, White et al. (2016) proposed a simple interpolation equation to represent beam-column flexural stiffness reduction factor (referred to as τMN,White) for using direct buckling analysis to the stability

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equation is τMN,White = *τa(/90o)+ τM,AISC (1-/90o), where the angle  represents the position of the current force point within a normalized interaction plot of the axial and moment strength ratios for a given cross- section. For the interpolation equation, τa is derived from column buckling curves given in AISC 360-16 (2016) and * accounts for local buckling effects, while τM,AISC is derived from lateral-torsional buckling (LTB) curve of beams given in AISC 360-16 (2016). For using second order refined plastic hinge method to frame stability design, Kim and Chen (1999) extended the column flexural stiffness reduction factor derived from Column Research Council (CRC) column strength curve, to be applicable to beam-columns with compact I cross-sections.

In addition, SEI/ASCE 8-02 (2002) provides the tangent modulus approach to determine the buckling strength of members. The flexural buckling stress (fn) is determined by fn=(π2 Et)/(KL ⁄ r)2. The tangent modulus Et is derived from the nonlinear stress–strain curve determined by the Ramberg–Osgood expression. Et, shown in Fig.2.2, is given by

𝐸𝑡 =

𝑓𝑦𝐸0

𝑓𝑦+0.002𝑛𝐸0(_𝑓𝑦𝑓)

𝑛−1 (2.1)

Where n is Ramberg–Osgood parameter; f is the stress in the member; E0 is initial elastic modulus (Young’s Modulus); fy is 0.2％ proof stress.

εp=0.01% εy=0.2% fp fy E0 Es E0 Et

E0 = initial elastic modulus

Et = tangent modulus

Es = secant modulus

fy = 0.2% proof strength

fp = proportional limit

Fig.2.2 Et derived from Ramberg-Osgood equation-based nonlinear stress–strain

19 reason that:

(1) Many studies have shown that, the single Ramberg-Osgood curve from which the tangent modulus Et is derived, is generally incapable of accurately representing the full stress-strain curve of stainless steel (Mirambell and Real, 2000; Rasmussen, 2003; Gardner and Nethercot, 2004; Arrayago et al. 2015).

(2) The GNA coupled with stiffness reduction method proposed in this paper is aimed to align with AISC 360-16 (2016). Thus, the adopted column buckling strength curve, and beam-column interaction curve are based on AISC provisions, with modifications made where necessary to fit in with test results.

(4) Plastic-zone analysis

A plastic-zone analysis is typically taken as an “exact” solution and is used as a benchmark to verify other simplified analysis. The AISC-LRFD beam-column interaction equations were established based on "exact" beam-column strength curves from the plastic-zone analysis conducted by Kanchanalai (1977). In plastic- zone analysis, members are discretized into finite elements, and furthermore the cross-section of each finite element is subdivided into many fibers. The deflection at each division point along a member is obtained by numerical integration. Spread of plasticity and second-order effects are rigorously captured through the incremental load-deflection response at each loading step.

Over the past decades, the application of this type of analysis was limited to verifying the accuracy of simplified methods, since it was too intensive in computation (Alvarez and Birnstiel, 1967, King et al.,1992; Liew, 1992; Liew et al., 1993a, Clarke et al., 1992; White, 1993; Vogel, 1985; El-Zanaty et al., 1980; Wang, 1988; Chen and Atsuta, 1977). However, with the development computer technology, numerous finite element software, such as ANSYS v19.0 (2018), ABAQUS v2013 (2013), and LS-DYNA SMP R11.0.0 (2019), which can perform plastic-zone analysis, have been developed. These software facilitate the application of plastic-zone analysis in practical engineering.