# Rigid-Plastic Analysis of Seismic Resistant T-Frame considering Moment-Shear Interaction

Authors: Ghader Bagheri; Payam Ashtari; Farhad Behnamfar
Journal: Shock and Vibration
 (2021)

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## Abstract

To select a seismic resistant system, in addition to strength and stiffness, ductility and energy dissipation are important to be considered. Structures have nonlinear behavior under the influence of moderate and strong earthquakes. One of the primary aims in designing seismic resistant structures is to prevent the formation of undesirable collapse mechanisms such as the collapse in only a few storeys of the structure that leads to low energy dissipation. In order to achieve a global collapse mechanism, modern seismic codes provide simple rules for design, which is called the hierarchy criteria. Although these simple criteria could prevent the formation of a soft storey mechanism, they could not lead to an optimal global collapse mechanism. In these mechanisms, the energy dissipation zones include all the yielding zones such as beams, while all other parts of the structure have remained in the elastic range. TRF (T-resisting frame) is an innovative lateral resistant system introduced for architectural reasons and to provide more energy dissipating capability. This system has several collapse mechanisms due to the moment, shear, or moment-shear behavior of its members. In this paper, within the framework of the theory of plastic mechanism control, the rigid-plastic analysis of the TRF system to achieve the desired collapse mechanism is used by considering the moment-shear interaction. According to these analyses, which are performed on a single storey frame, simple hierarchy criteria are developed to create the desired collapse mechanism. Also, these criteria prevent undesired collapse mechanisms in order to have more energy dissipation and more ductility. Finally, the validity of the proposed criteria has been verified by the pushover analysis.


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## Body

## 1. Introduction

T-resisting frame is a new lateral force resisting system, which consists of two structural parts, T-part as the main component and the second part of side columns. In steel structures, T-part is composed of a vertical plate girder (VPG) and a pair of horizontal plate girders (HPGs) to provide signiﬁcant lateral stiffness and high energy dissipation capability under severe earthquakes. TRF dissipates energy predominantly through stable shear yielding of the web shear panels of HPGs. Also, side columns in TRF, such as columns in conventional braced frames, withstand compression or tension to resist the overturning moments (Figure1) [1].Figure 1
Proposed basic structural configurations of TRF and alternatives [1]. (a) Single-T configuration of TRF. (b) Double-T configuration of TRF. (c) Multi-T configuration of TRF. (d) Stylish single-T configuration of TRF. (e) Stylish single-T configuration of TRF. (f) Single-T configuration of TRF using vertical truss. (g) Single-T configuration of TRF using composite side columns. (h) Single-T configuration of reinforced concrete TRF.Ashtari proposed the configuration of TRF in 2008 and Bandehzadeh in the M.Sc thesis demonstrated the seismic performance advantages of the optimized single-T TRFs [1]. Ashtari (2008) to provide and improve building’s behavior and performance presented the initial idea of this lateral seismic system. Initial numerical and experimental studies have been conducted on the behavior and performance of the TRF system by Ashtari et al. [1, 2–5]. The results of nonlinear finite element analyses in this system demonstrate that TRF has high initial stiffness, high ductility, excellent energy dissipation capacity, and stable hysteretic behavior at large inelastic deformations under numerous cycles of loading. According to the results from a series of nonlinear analyses, a properly designed TRF exhibited a great efficiency as the lateral resistant system of a building in a high seismic zone due to its potential for extremely high ductility and energy dissipating capability. In this building, the observed drifts and rotation demands were significantly lower than those commonly associated with the buildings using common moment-resisting frames [1]. Various numerical studies and laboratory tests have been performed on different configurations of this system. The results show that this system has good seismic behavior. The results show that this system has high initial stiffness, high ductility, excellent energy dissipation capacity, and stable hysteretic behavior at large inelastic deformations under numerous cycles of loading [4, 5].In the capacity design, the energy dissipation zones should be designed according to the internal forces resulting from the load combinations provided by the seismic codes. The other parts of the structure have to be designed proportionally to the force of the energy dissipation zones. It means that these parts do not yield under the maximum internal reactions transmitted by the energy dissipation zones in the fully yielded state [6, 7].In the eccentrically braced frames, dissipative zones are constituted by the members called link elements [8–10]. The use of members as energy dissipation zones, which they are yielding in shear, was originally proposed for eccentrically braced frames. This issue has been the starting point for the formation of new structural types such as knee braces [11], aluminum shear links [12], and shear links made of low-flow point steel [13]. Shear yielding is separated from the energy dissipation due to moment yielding, so the moment-shear interaction could not be neglected.In this paper, the TRF-H system has been studied, in which the task of energy dissipation is assigned to the T-part members as link elements, which are affected by moment, shear, or moment-shear interaction. The problem of moment-shear interaction is considered within the framework of rigid-plastic analysis and the solution for the TRF-H system is presented. Solving the interaction problem is affected by compatibility conditions, which depend on the configuration of the structure. Such issue is currently being studied for eccentrically braced frames with D-scheme, K-scheme, V-scheme, inverted V-scheme [14], and Y EB-Frames [15], and here for the T-resisting frame system. The purpose of developing the hierarchy criteria is to control the yielding pattern. Investigations are focused on the one-storey TRF-H system to determine which members and how they yield. This theory is based on the rigid-plastic analysis with the purpose of achieving the global collapse mechanism, in which all the dissipation zones are yielding, while the other members stay in the elastic range. This theory has been presented for frames with horizontal link beam [16], knee braces (KB-frame) [17], dual MRF-CBF systems [18], and truss moment frame (DTMFs) [19]. Also, the solution of the theory of plastic mechanism control (TPMC) currently being studied in the case of moment-resisting frames: eccentrically braced frames (MRF-EBF) dual systems [20] and eccentrically braced frames with inverted Y-scheme [21]. These studies provide a complete and exhaustive design procedure for MRF-EBF dual systems and EB-frames with inverted Y-scheme, considering all the brace configurations commonly adopted and with the goal of assuring the development of a collapse mechanism of global type [20, 21].In Section2, the study of moment-shear interaction in the TRF-H system and the development of the hierarchy criteria at the storey level have been studied. To assess the accuracy of the proposed hierarchy criteria for the TRF-H system has been examined by means of static pushover analyses with the SAP2000 computer program. In addition, the theory of controlling plastic mechanisms for the multistorey TRF system and fixed multistorey TRF system would be provided in subsequent papers.


## 2. The Plastic Mechanisms in the TRF-H System

Independent from the configuration of the structure, the main feature of the links is due to the interaction between the bending moment and the shear force, which could not be ignored in predicting the ultimate behavior of the structural system. Generally, the classification of the link beams depends on the length of the member. In the following, general classification is proposed [7–10]:(i)
Short link fore≤1.6MP/VP(ii)
Intermediate link for1.6MP/VP≤e≤3MP/VP(iii)
Long links fore≥3MP/VPVp and Mp, respectively, are plastic shear strength and bending strength of the cross section of the link beam. The long link beam is considered to be in frequent revisions as 2.6MP/VP [22–24]. Moment-shear interaction diagram and the short, medium, and long links’ range are shown in Figure 2.Figure 2
Moment-shear interaction diagram (p=Mf/Mp).The plastic behavior of the link beam, for a bending moment less thanMf, could be considered as a pure shear, where Mf is the flange contribution of the plastic moment of the link beam; thus, for moments greater than Mf, the shear strength of the element is reduced due to the interaction between moment and shear. According to Neal research, the plastic zone is calculated using the following relationships [14, 25]:(1)M−MfMp−Mf2+VVp2=1,forMf≤M≤MP,(2)V=Vp,forM≤Mf.In capacity design, an important issue that has been reported in previous experiments by Hjelmstad and Popov [26], and Malley and Popov [27] is the overstrength developed by the link beam. Overstrength of the link beam is the ratio between the maximum shear forces developed by the link beam to its plastic shear strength. The evaluation of the results of experiments is usually based on the inclusion of the plastic shear strength as VP=0.6fyd−tftw, where fy is the yield stress, d is the depth of the cross section of the beam, and tf and tw are the thickness of the flange and the web, respectively. However, the link overstrength occurs due to the strain hardening and the contribution of link flanges to the shear strength. Therefore, according to European codes [28], plastic shear strength is calculated using the web surface shear strength with the contribution of the flange.According to the capacity design, the link overstrength should be calculated in the assessment of the maximum internal forces transferred to the nonplastic elements. The link overstrength factor is generally recommended as 1.50, based on previous experimental studies [29]. In the 2005 AISC seismic provisions [23], the link overstrength factor is proposed equal to 1.25 for designing the diagonal braces and equal to 1.10 for designing columns and the beam outside the link. A similar approach has been adopted in the eurocode [6]. Recent experimental results and investigations are indicated that overstrength factors can be exceeding the code suggested values. For very short links, with compact cross-sections and perfect axial restraints, values of shear overstrength up to 2 have been obtained. For built-up links with a very compact shape and short length, even larger values could be obtained, especially if larger rotation capacity is achievable [30]. Barecchia et al., by means of ﬁnite element analyses on standard European rolled proﬁles, have found overstrength factors between 2.00 and 2.50 for a link shear deformation equal to 0.10 rad [31]. Nevertheless, in using the above results, some caution should be adopted because they come from ﬁnite element analyses where the stiffeners’ inﬂuence cannot be calculated accurately.Recently, Okazaki et al., based on several experimental tests on shear links with properly detailed web stiffener, have obtained the overstrength factor between 1.34 and 1.48 for short links and in the range of 1.12–1.28 for longer links [24]. These experimental results of the overstrength factor have been computed considering the measured values of the material properties. Therefore, based on these experimental results, it was concluded that the overstrength factor of 1.50, which forms worldwide the basis of capacity design provisions for EB-frames and appears reasonable for links made from rolled profiles. A simple method for calculating the overstrength of the hierarchy criteria obtained by the rigid-plastic analysis has recently been presented [14]. It is based on the use of the interaction domain referring to the ultimate conditions obtained from the homothetic or nonhomothetic expansion of the plastic domain (Figure 2).The topologies of the plastic collapse mechanisms of the one-storey TRF-H are shown in Figure3. In the type “A” mechanism, ends of the both horizontal links and the bottom of the central column yield. In mechanism “B,” two ends of the central column yield. Type of collapse mechanism is determined by the elements and structure specification. The aim of the research presented here is to define the hierarchy criteria, which includes the effect of the moment-shear interaction, to ensure that the desired mechanism would be developed.Figure 3
Modeling TRF-H system by considering the moment and shear interaction. (a) Mechanism “A.” (b) Mechanism “B.”
(a)(b)In fact, in order to calculate the moment and shear interaction in the framework of rigid-plastic analysis, the end-link yield is modeled with a combination of a pair of plastic double pendulum and a plastic hinge that can calculate both types of plastic shear deformationνP and moment plastic rotation ϕp, respectively.Use of the rigid-plastic analysis to calculate the plastic strength of possible mechanisms needs to determine the plastic deformations of the linksνP and ϕp. Also, the values of shear force V and corresponding moment M for a given value of the θ parameter is needed. Problem solving could be done by minimizing the internal work of the link and applying the yielding conditions and considering of the normal plastic flow rule. Additionally, compatibility conditions should be satisfied.


## 3. Moment-Shear Interaction in TRF-H System Elements

In the equations, the index of parameters is based on the member’s names, meaning theBL and BR indexes are related to the left and right horizontal link (beam), respectively, and the AB is related to the vertical member (central column).


### 3.1. Mechanism Type “A”

Within the framework of rigid-plastic analysis, moment-shear interaction occurring in the link of EB-frames is provided in the prior studies [15]. The moment and shear interaction in the elements of the TRF-H frame (Figure 3(a)), within the framework of the rigid-plastic analysis, is obtained by the following relationships (equations (3)–(10)):(1)
Minimizing the internal workWi according to the collapse mechanism under the influence of the plastic rotation θ at the columns, for a hinge state, is(3)Min.:WiA=MBLAϕp,BLA+ϕp,ABA+VBLAvp,BLA+MBRAϕp,BRA+ϕp,ABA+VBRAvp,BRA+MABAϕp,ABA+VABAvp,ABA.(2)
Kinematic compatibility conditions are(4)vp,BLA+ϕp,BLAeL=0,(5)vp,BRA+ϕp,BRAeR=0,(6)vp,ABA+ϕp,ABAh=θh.(3)
Elements’ yield condition leads to these three functions:(7)FMBLA,VBLA=MBLA−Mf,bMp,b−Mf,b2+VBLAVp,b2−1,(8)FMBRA,VBRA=MBRA−Mf,bMp,b−Mf,b2+VBRAVp,b2−1,(9)FMABA,VABA=MABA−Mf,cMp,c−Mf,c2+VABAVp,c2−1.(4)
Normal plastic flow law gives(10)vpA=λ∂F∂VA,ϕpA=λ∂F∂MA.Here,MAand VA are the moment and the shear formed at the end of TRF-H elements, respectively. ϕPA and νPA are the plastic rotation and plastic shear deformation occurring in the mechanism “A,” respectively. e is length of the link, h is height of the storey, and θ is the plastic rotation at the end of the beam and columns. The “b” and “c” index are for the TRF-H horizontal link beam and central column, respectively. For each member of the TRF-H system, the values of the shear and the moment would be determined using the above equations. These calculations are made for the left horizontal link (BL), the right horizontal link (BR), and the central column of the TRF-H (AB) system. For the BL member, by substituting equations (4)–(6) into equation (3), an equation will be obtained that expresses the internal work as a function of the plastic rotation of the T-part’s members (ϕp). So, minimizing the internal work of each member could be expressed as(11)dWiAdϕp,BLA=0⟶VBLA=MBLAeL,(12)dWiAdϕp,BRA=0⟶VBRA=MBRAeR,(13)dWiAdϕp,ABA=0⟶VABA=MBLA+MBRA+MABAh.By defining the plastic moment of the web asMw (Mw=Mp+Mf) and by combining equations (7)–(9) with equations (11)–(13), for all T-part members, the value of MA and VA could be determined.The value ofMA and VA for the left horizontal link (BL) is determined as follows:(14)MBLA=Mf,b+Mw,b1−Mf,b2−Mw,b2/Vp,b2eL21+Mw,b2/Vp,b2eL2,(15)VBLA=Mf,b+Mw,b1−Mf,b2−Mw,b2/Vp,b2eL2eL1+Mw,b2/Vp,b2eL2.By defining the parameterpb=Mf,b/Mp,b, the moments can be rewritten based on the beam’s plastic moment. Accordingly, by substituting the values of in Mf,b=pbMp,b and Mw,b=1−pbMp,b and Vp,b=41−pb/3hw,bMp,b into equation (14), the BL moment is defined as the beam plastic moment coefficient as follows:(16)MBLA=xBLA⋅Mp,b,(17)VBLA=xBLAeL⋅Mp,b.In these equations, the multiplierxBLA is equal to(18)xBLA=pb+1−pb1−3/16hw,b/eL22pb−1/1−pb21+3/16hw,b/eL2,pb≤xBLA≤1.Using the above equations, with the definition of the shear length in the form ofev=Mf,b/Vp,b, for different shear values VBLA=sv,b.Vp,b (Figure 1), link length and moment factor (xBLA) can be obtained as follows:(19)eL=pb+1−pb1−sb2pbsbev,(20)xBLA=pb+1−pb1−sb2.According to the above equations, forsv,b=1, equations (19) and (20) conclude that eL=ev and xBLA=pb, respectively. Consequently, using equation (16), we have MBLA=Mf,b, which indicates a short (shear) link. In addition, where eL⟶∞, equation (18) concludes that xBLA=1. So, according to equation (16), we would have MBLA=Mp,b and equation (17) gives VBLA=Mp,b/eL, which this state indicates the long (moment) link. Also, the values of MBLA and VBLA could be determined simply by combining the equilibrium conditions in the link with the yielding condition. In other words, the solution obtained is also reliable in a static state so that it provides the actual internal reactions under the yielding conditions in moment-shear interactions.In addition, using these equations and choosing the value ofsv,b, the length of the long link could also be determined. According to AISC99 and AASHTO, if this value is considered 0.6, then we have [32, 33](21)em=pb+43pbev,xBLA=0.2pb+0.8,MBLA=0.2pb+0.8Mp,b⟶0<pb<1MBLA=0.8∼1.0Mp,blong link.In this relation,em is defined as the length of moment (long) link.Also, for the right beam (BR), it can be determined as follows:(22)MBRA=Mf,b+Mw,b1−Mf,b2−Mw,b2/Vp,b2eR21+Mw,b2/Vp,b2eR2,(23)VBRA=Mf,b+Mw,b1−Mf,b2−Mw,b2/Vp,b2eR2eR1+Mw,b2/Vp,b2eR2,(24)xBRA=pb+1−pb1−3/16hw,b/eR22pb−1/1−pb21+3/16hw,b/eR2,(25)MBRA=xBRA⋅Mp,b,(26)VBRA=xBRAeR⋅Mp,b.In this case, for a link with long lengtheR⟶∞, according to equation (24) xBRA=1. Consequently, by using equations (25) and (26), we have MBRA=Mp,b and VBRA=Mp,b/eR, respectively, which indicates the long link. All the relations and conditions examined for the left link (BL) are confirmed for the right link (BR).Finally, for the central column (AB) by defining the parameter pc=Mf,c/Mp,c and by substituting the values Mp,c=z.Mp,b, Mf,c=pcMp,c, Mw,c=1−pcMp,c, and Vp,c=41−pc/3hw,cMp,c, we have(27)MABA=xABA⋅Mp,c,(28)VABA=xBLA+xBRA+z⋅xABAz⋅h⋅Mp,c,(29)xABA=pc−xBLA+xBRA/z3/16hw,c/h21+3/16hw,c/h2+1−pc1−2pc+xBLA+xBRA/zxBLA+xBRA/z⋅1−pc2+2pc−1/1−pc23/16hw,c/h21+3/16hw,c/h2.According to equation (27), moment of central column can be defined as a coefficient of the column plastic moment (pc≤xABA≤1) in collapse mechanism. In this column, by substituting VABA=sv,c.Vp,c into the above equations, the minimum moment link length (hm) can be obtained as follows:(30)hm=xBLA+xBRA+z⋅pc+1−pc1−sv,c23hw,c4z⋅1−pc⋅sv,c.By usingsv,c=1 in equation (30), maximum shear link length hv can be obtained as(31)hv=xBLA+xBRA+z⋅pc3hw,c4z⋅1−pc.Forh=hv, based on equation (29), xABA=pc, and consequently, MABA=Mf,c; this state represents a short or shear link. Also, for sv,c⟶0, equation (30) gives hm⟶∞, and according to equation (29), we have xABA=1, and consequently, MABA=Mp,c, which indicates long link. This solution is valid within the framework of the static approach so that it provides actual internal reactions under yielding conditions in moment-shear interaction, and also, it can evaluate the corresponding plastic shear deformation νPA and plastic rotation ϕPA. According to the normal plastic flow rule (equation (10)), the plastic shear deformation, νP,ABA and the plastic rotation ϕP,ABA for the central column can be obtained as follows:(32)vp,ABA=2λVABAVp,c2,ϕp,ABA=2λMABA−Mf,cMw,c2.where, the parameter λ, which is controlling the large plastic flow, can be eliminated by dividing ϕPA and νPA. By combining resultant equation with the kinematic compatibility conditions (equation (6)), the plastic shear deformation of the central column νP,ABA and its plastic rotation ϕP,ABA can be obtained as follows:(33)vp,ABA=θh11+MABA−Mf,b/Mw,b2Vp,b2/VABAh,(34)ϕp,ABA=θ11+Mw,b2/MABA−Mf,bVABA/Vp,b2h.Finally, regarding this issue, the internal workWiA can be obtained by substituting equations (11)–(13) in equation (3) as follows:(35)WiA=MBLAθ+MBRAθ+MABAθ.Equations (35) shows that, in this type of mechanism, internal work in the beams and the central column of the TRF-H system can simply be expressed as Wlink=MAθ=Mp⋅eqθ, where Mp⋅eq is yielding moment considering shear-moment interaction, that is, between Mf and Mp. This result means that MA can be interpreted as an equivalent plastic moment, allowing to write the internal work simply on the basis of the equivalent plastic moment and the equivalent plastic rotation, even in the case of moment-shear interaction [14].


### 3.2. Mechanism Type “B”

The relations required to solve the problem of moment-shear interaction in the intermediate links for the Mechanism “B” are as follows.(1)
Condition for minimizing internal work of the “B” type collapse mechanism under the influence of the rotation θ can be expressed as(36)Min⋅WiB=2MABBϕp,ABB+2VABBvp,ABB.(2)
Kinematic compatibility condition is(37)2vp,ABB+ϕp,ABBh=θh.(3)
Yielding condition of the link leads to the following function:(38)FMABB,VABB=MABB−Mf,cMp,c−Mf,c2+VABBVp,c2−1.(4)
Normal plastic flow rule are as follows:(39)vpB=λ∂F∂VB,ϕpB=λ∂F∂MB,where MABB and VABB are, respectively, the bending moment and the shear created on the ends of the central column (AB) of the TRF-H system in the mechanism “B.” ϕPB and νPB, respectively, are plastic rotation and plastic shear deformation created at the two ends of central column. For each member of the TRF-H system, the values of the shear and moment can be determined using the above relations. By substituting equation (37) into equation (36), the following equation is obtained:(40)WiB=2MABBϕp,ABB+VABBhθ−hΦp,ABB.For the certain value ofθ, the internal work relation can be expressed as a function of the plastic rotation of end of the columns ϕPB. So, minimizing the internal work could be expressed by(41)dWiBdϕp,ABB=0⟶VABB=2MABBh.By combining equation (38) with equation (41), the internal reactions in the central column of the TRF-H system can be determined as follows:(42)MABB=Mfc+Mw,c1−4Mf,c2−Mw,c2/Vp,c2h21+4Mw,c2/Vp,c2h2,(43)VABB=2Mf,c+Mw,c1−4Mf,c2−Mw,c2/Vp,c2h2h1+4Mw,c2/Vp,c2h2.Same as mechanism type “A,” by substituting Mf,c, Mw,c, and Vp,c into equations (42) and (43), the moment and shear value would be defined as a coefficient of the beam plastic moment:(44)MABB=xABB⋅Mp,c,(45)VABB=2xABBh⋅Mp,c,where(46)xABB=pc+1−pc1−3/4hw,c/h22pc−1/1−pc21+3/4hw,c/h2,pc≤xABB≤1.Using the above relations, for the different shear valuesV=sv,c⋅Vp,c, the moment multiplier can be obtained as follows:(47)xBLA=pb+1−pb1−sc2,with respect to the above relation for a state, where sc=1; according to equation (47), we have xABB=pc, and consequently, using equation (44), MABB=Mf,c; this state indicates a short link. Also, for state where sc⟶0, equation (47) would conclude that xABB=1; according to equation (44), MABB=Mp,c, which indicates the long link.Same as mechanism “A,” plastic deformations νPB and ϕPB can be calculated at the ends of the link, depending on the kinematic conditions. Using the normal plastic flow rule, the parameter λ, which is controlling large plastic yield, could be eliminated by dividing ϕPB and νPB. Then, by combining resultant equation with the kinematic compatibility, the plastic shear deformation of the central column νP,ABB and its plastic rotation ϕP,ABB can be obtained as follows:(48)ϕp,ABB=θ11+2Mw,c2/MABB−Mf,cVABB/Vp,c2h,(49)vp,ABB=θh12+MABB−Mf,c/Mw,c2Vp,c2h/VABB.


## 3.1. Mechanism Type “A”

Within the framework of rigid-plastic analysis, moment-shear interaction occurring in the link of EB-frames is provided in the prior studies [15]. The moment and shear interaction in the elements of the TRF-H frame (Figure 3(a)), within the framework of the rigid-plastic analysis, is obtained by the following relationships (equations (3)–(10)):(1)
Minimizing the internal workWi according to the collapse mechanism under the influence of the plastic rotation θ at the columns, for a hinge state, is(3)Min.:WiA=MBLAϕp,BLA+ϕp,ABA+VBLAvp,BLA+MBRAϕp,BRA+ϕp,ABA+VBRAvp,BRA+MABAϕp,ABA+VABAvp,ABA.(2)
Kinematic compatibility conditions are(4)vp,BLA+ϕp,BLAeL=0,(5)vp,BRA+ϕp,BRAeR=0,(6)vp,ABA+ϕp,ABAh=θh.(3)
Elements’ yield condition leads to these three functions:(7)FMBLA,VBLA=MBLA−Mf,bMp,b−Mf,b2+VBLAVp,b2−1,(8)FMBRA,VBRA=MBRA−Mf,bMp,b−Mf,b2+VBRAVp,b2−1,(9)FMABA,VABA=MABA−Mf,cMp,c−Mf,c2+VABAVp,c2−1.(4)
Normal plastic flow law gives(10)vpA=λ∂F∂VA,ϕpA=λ∂F∂MA.Here,MAand VA are the moment and the shear formed at the end of TRF-H elements, respectively. ϕPA and νPA are the plastic rotation and plastic shear deformation occurring in the mechanism “A,” respectively. e is length of the link, h is height of the storey, and θ is the plastic rotation at the end of the beam and columns. The “b” and “c” index are for the TRF-H horizontal link beam and central column, respectively. For each member of the TRF-H system, the values of the shear and the moment would be determined using the above equations. These calculations are made for the left horizontal link (BL), the right horizontal link (BR), and the central column of the TRF-H (AB) system. For the BL member, by substituting equations (4)–(6) into equation (3), an equation will be obtained that expresses the internal work as a function of the plastic rotation of the T-part’s members (ϕp). So, minimizing the internal work of each member could be expressed as(11)dWiAdϕp,BLA=0⟶VBLA=MBLAeL,(12)dWiAdϕp,BRA=0⟶VBRA=MBRAeR,(13)dWiAdϕp,ABA=0⟶VABA=MBLA+MBRA+MABAh.By defining the plastic moment of the web asMw (Mw=Mp+Mf) and by combining equations (7)–(9) with equations (11)–(13), for all T-part members, the value of MA and VA could be determined.The value ofMA and VA for the left horizontal link (BL) is determined as follows:(14)MBLA=Mf,b+Mw,b1−Mf,b2−Mw,b2/Vp,b2eL21+Mw,b2/Vp,b2eL2,(15)VBLA=Mf,b+Mw,b1−Mf,b2−Mw,b2/Vp,b2eL2eL1+Mw,b2/Vp,b2eL2.By defining the parameterpb=Mf,b/Mp,b, the moments can be rewritten based on the beam’s plastic moment. Accordingly, by substituting the values of in Mf,b=pbMp,b and Mw,b=1−pbMp,b and Vp,b=41−pb/3hw,bMp,b into equation (14), the BL moment is defined as the beam plastic moment coefficient as follows:(16)MBLA=xBLA⋅Mp,b,(17)VBLA=xBLAeL⋅Mp,b.In these equations, the multiplierxBLA is equal to(18)xBLA=pb+1−pb1−3/16hw,b/eL22pb−1/1−pb21+3/16hw,b/eL2,pb≤xBLA≤1.Using the above equations, with the definition of the shear length in the form ofev=Mf,b/Vp,b, for different shear values VBLA=sv,b.Vp,b (Figure 1), link length and moment factor (xBLA) can be obtained as follows:(19)eL=pb+1−pb1−sb2pbsbev,(20)xBLA=pb+1−pb1−sb2.According to the above equations, forsv,b=1, equations (19) and (20) conclude that eL=ev and xBLA=pb, respectively. Consequently, using equation (16), we have MBLA=Mf,b, which indicates a short (shear) link. In addition, where eL⟶∞, equation (18) concludes that xBLA=1. So, according to equation (16), we would have MBLA=Mp,b and equation (17) gives VBLA=Mp,b/eL, which this state indicates the long (moment) link. Also, the values of MBLA and VBLA could be determined simply by combining the equilibrium conditions in the link with the yielding condition. In other words, the solution obtained is also reliable in a static state so that it provides the actual internal reactions under the yielding conditions in moment-shear interactions.In addition, using these equations and choosing the value ofsv,b, the length of the long link could also be determined. According to AISC99 and AASHTO, if this value is considered 0.6, then we have [32, 33](21)em=pb+43pbev,xBLA=0.2pb+0.8,MBLA=0.2pb+0.8Mp,b⟶0<pb<1MBLA=0.8∼1.0Mp,blong link.In this relation,em is defined as the length of moment (long) link.Also, for the right beam (BR), it can be determined as follows:(22)MBRA=Mf,b+Mw,b1−Mf,b2−Mw,b2/Vp,b2eR21+Mw,b2/Vp,b2eR2,(23)VBRA=Mf,b+Mw,b1−Mf,b2−Mw,b2/Vp,b2eR2eR1+Mw,b2/Vp,b2eR2,(24)xBRA=pb+1−pb1−3/16hw,b/eR22pb−1/1−pb21+3/16hw,b/eR2,(25)MBRA=xBRA⋅Mp,b,(26)VBRA=xBRAeR⋅Mp,b.In this case, for a link with long lengtheR⟶∞, according to equation (24) xBRA=1. Consequently, by using equations (25) and (26), we have MBRA=Mp,b and VBRA=Mp,b/eR, respectively, which indicates the long link. All the relations and conditions examined for the left link (BL) are confirmed for the right link (BR).Finally, for the central column (AB) by defining the parameter pc=Mf,c/Mp,c and by substituting the values Mp,c=z.Mp,b, Mf,c=pcMp,c, Mw,c=1−pcMp,c, and Vp,c=41−pc/3hw,cMp,c, we have(27)MABA=xABA⋅Mp,c,(28)VABA=xBLA+xBRA+z⋅xABAz⋅h⋅Mp,c,(29)xABA=pc−xBLA+xBRA/z3/16hw,c/h21+3/16hw,c/h2+1−pc1−2pc+xBLA+xBRA/zxBLA+xBRA/z⋅1−pc2+2pc−1/1−pc23/16hw,c/h21+3/16hw,c/h2.According to equation (27), moment of central column can be defined as a coefficient of the column plastic moment (pc≤xABA≤1) in collapse mechanism. In this column, by substituting VABA=sv,c.Vp,c into the above equations, the minimum moment link length (hm) can be obtained as follows:(30)hm=xBLA+xBRA+z⋅pc+1−pc1−sv,c23hw,c4z⋅1−pc⋅sv,c.By usingsv,c=1 in equation (30), maximum shear link length hv can be obtained as(31)hv=xBLA+xBRA+z⋅pc3hw,c4z⋅1−pc.Forh=hv, based on equation (29), xABA=pc, and consequently, MABA=Mf,c; this state represents a short or shear link. Also, for sv,c⟶0, equation (30) gives hm⟶∞, and according to equation (29), we have xABA=1, and consequently, MABA=Mp,c, which indicates long link. This solution is valid within the framework of the static approach so that it provides actual internal reactions under yielding conditions in moment-shear interaction, and also, it can evaluate the corresponding plastic shear deformation νPA and plastic rotation ϕPA. According to the normal plastic flow rule (equation (10)), the plastic shear deformation, νP,ABA and the plastic rotation ϕP,ABA for the central column can be obtained as follows:(32)vp,ABA=2λVABAVp,c2,ϕp,ABA=2λMABA−Mf,cMw,c2.where, the parameter λ, which is controlling the large plastic flow, can be eliminated by dividing ϕPA and νPA. By combining resultant equation with the kinematic compatibility conditions (equation (6)), the plastic shear deformation of the central column νP,ABA and its plastic rotation ϕP,ABA can be obtained as follows:(33)vp,ABA=θh11+MABA−Mf,b/Mw,b2Vp,b2/VABAh,(34)ϕp,ABA=θ11+Mw,b2/MABA−Mf,bVABA/Vp,b2h.Finally, regarding this issue, the internal workWiA can be obtained by substituting equations (11)–(13) in equation (3) as follows:(35)WiA=MBLAθ+MBRAθ+MABAθ.Equations (35) shows that, in this type of mechanism, internal work in the beams and the central column of the TRF-H system can simply be expressed as Wlink=MAθ=Mp⋅eqθ, where Mp⋅eq is yielding moment considering shear-moment interaction, that is, between Mf and Mp. This result means that MA can be interpreted as an equivalent plastic moment, allowing to write the internal work simply on the basis of the equivalent plastic moment and the equivalent plastic rotation, even in the case of moment-shear interaction [14].


## 3.2. Mechanism Type “B”

The relations required to solve the problem of moment-shear interaction in the intermediate links for the Mechanism “B” are as follows.(1)
Condition for minimizing internal work of the “B” type collapse mechanism under the influence of the rotation θ can be expressed as(36)Min⋅WiB=2MABBϕp,ABB+2VABBvp,ABB.(2)
Kinematic compatibility condition is(37)2vp,ABB+ϕp,ABBh=θh.(3)
Yielding condition of the link leads to the following function:(38)FMABB,VABB=MABB−Mf,cMp,c−Mf,c2+VABBVp,c2−1.(4)
Normal plastic flow rule are as follows:(39)vpB=λ∂F∂VB,ϕpB=λ∂F∂MB,where MABB and VABB are, respectively, the bending moment and the shear created on the ends of the central column (AB) of the TRF-H system in the mechanism “B.” ϕPB and νPB, respectively, are plastic rotation and plastic shear deformation created at the two ends of central column. For each member of the TRF-H system, the values of the shear and moment can be determined using the above relations. By substituting equation (37) into equation (36), the following equation is obtained:(40)WiB=2MABBϕp,ABB+VABBhθ−hΦp,ABB.For the certain value ofθ, the internal work relation can be expressed as a function of the plastic rotation of end of the columns ϕPB. So, minimizing the internal work could be expressed by(41)dWiBdϕp,ABB=0⟶VABB=2MABBh.By combining equation (38) with equation (41), the internal reactions in the central column of the TRF-H system can be determined as follows:(42)MABB=Mfc+Mw,c1−4Mf,c2−Mw,c2/Vp,c2h21+4Mw,c2/Vp,c2h2,(43)VABB=2Mf,c+Mw,c1−4Mf,c2−Mw,c2/Vp,c2h2h1+4Mw,c2/Vp,c2h2.Same as mechanism type “A,” by substituting Mf,c, Mw,c, and Vp,c into equations (42) and (43), the moment and shear value would be defined as a coefficient of the beam plastic moment:(44)MABB=xABB⋅Mp,c,(45)VABB=2xABBh⋅Mp,c,where(46)xABB=pc+1−pc1−3/4hw,c/h22pc−1/1−pc21+3/4hw,c/h2,pc≤xABB≤1.Using the above relations, for the different shear valuesV=sv,c⋅Vp,c, the moment multiplier can be obtained as follows:(47)xBLA=pb+1−pb1−sc2,with respect to the above relation for a state, where sc=1; according to equation (47), we have xABB=pc, and consequently, using equation (44), MABB=Mf,c; this state indicates a short link. Also, for state where sc⟶0, equation (47) would conclude that xABB=1; according to equation (44), MABB=Mp,c, which indicates the long link.Same as mechanism “A,” plastic deformations νPB and ϕPB can be calculated at the ends of the link, depending on the kinematic conditions. Using the normal plastic flow rule, the parameter λ, which is controlling large plastic yield, could be eliminated by dividing ϕPB and νPB. Then, by combining resultant equation with the kinematic compatibility, the plastic shear deformation of the central column νP,ABB and its plastic rotation ϕP,ABB can be obtained as follows:(48)ϕp,ABB=θ11+2Mw,c2/MABB−Mf,cVABB/Vp,c2h,(49)vp,ABB=θh12+MABB−Mf,c/Mw,c2Vp,c2h/VABB.


## 4. Hierarchy Criteria for TRF-H System

### 4.1. Hierarchy Criteria for Intermediate Link Beams

In different mechanisms, three main elements of the TRF-H system could be as short, long, or intermediate links. The purpose of this research is to develop the hierarchy criteria for controlling the frame yield pattern of the one-storey TRF-H system. In the mechanism “A,” all three elements of the TRF-H system have been yielded. In the mechanism “B,” only the yield of the central column of this system has been considered. According to the kinematic theory of plastic collapse, the condition to achieve desired collapse mechanism is that the kinetic external force for a mechanism “A” becomes less than the one for mechanism “B.”The internal virtual work is expressed for the mechanism “A” using equation (35). For the mechanism “B,” WiA can be obtained by substituting equation (41) in equation (40) as follows:(50)B­type mechanism:WiB=2MABBθ.External work is only due to the horizontal inertia force of the earthquake and is obtained for both the mechanisms by the following equation:(51)We,F=Fextθh.The coefficient of horizontal kinematic force is obtained by equating external work and internal work. Therefore, the following relation is obtained:(52)A­type mechanism:FextA=MBLA+MBRA+MABAFh,(53)B­type mechanism:FextB=2MABBFh.Given equations (52) and (53), it is possible to write a design criterion for the formation of the mechanism “A” before the mechanism “B,” for which the following relation must be satisfied:(54)FextA≤FextB⟶MBLA+MBRA+MABA≤2MABB.Substituting equations (13) and (45) in equation (54) gives(55)VABA≤VABB.In addition, substituting equations (17), (26), (27), and (44) in equation (54), the following relation is obtained:(56)xBLA+xBRA+z⋅xABA≤2z⋅xABB.


### 4.2. Hierarchy Criteria for Long Link Beams

In long links, the plastic shear deformation (vp) is negligible and is considered zero. Therefore, the internal work of the kinematic mechanisms discussed is easily obtained. In its mechanism “A” (Figure 2(a)), three members of the TRF-H system including a two beams BL and BR and a column AB would yield. Also, in its mechanism “B,” the central vertical member of the TRF-H (AB) system yields from both ends. In accordance with equations (18), (24), (29), and (46), for elements in different mechanisms, the value is p≤x≤1. In each member, if the value x=1, the member is long and would have a bending behavior. Accordingly, by substituting this value in equation (56) for all the elements, the criteria for the different mechanism links can be obtained as follows:(57)z≥2⟶Mp,c≥2Mp,b.


### 4.3. Hierarchy Criteria for Short Link Beams

In short links, the moment and shear interaction according to Figure2 is negligible, so only the shear plastic deformation is vp and the value of ϕp is zero according to the normal plastic flow rule. In accordance with equations (18), (24), (29), and (46), for each member in different mechanisms, if the value is x=p, the element is a short link and has shear behavior. Therefore, by substituting this value in equation (56), the hierarchy criteria for the short links can be obtained as follows:(58)z⋅pc≥2pb⟶Mf,c≥2Mf,b.


## 4.1. Hierarchy Criteria for Intermediate Link Beams

In different mechanisms, three main elements of the TRF-H system could be as short, long, or intermediate links. The purpose of this research is to develop the hierarchy criteria for controlling the frame yield pattern of the one-storey TRF-H system. In the mechanism “A,” all three elements of the TRF-H system have been yielded. In the mechanism “B,” only the yield of the central column of this system has been considered. According to the kinematic theory of plastic collapse, the condition to achieve desired collapse mechanism is that the kinetic external force for a mechanism “A” becomes less than the one for mechanism “B.”The internal virtual work is expressed for the mechanism “A” using equation (35). For the mechanism “B,” WiA can be obtained by substituting equation (41) in equation (40) as follows:(50)B­type mechanism:WiB=2MABBθ.External work is only due to the horizontal inertia force of the earthquake and is obtained for both the mechanisms by the following equation:(51)We,F=Fextθh.The coefficient of horizontal kinematic force is obtained by equating external work and internal work. Therefore, the following relation is obtained:(52)A­type mechanism:FextA=MBLA+MBRA+MABAFh,(53)B­type mechanism:FextB=2MABBFh.Given equations (52) and (53), it is possible to write a design criterion for the formation of the mechanism “A” before the mechanism “B,” for which the following relation must be satisfied:(54)FextA≤FextB⟶MBLA+MBRA+MABA≤2MABB.Substituting equations (13) and (45) in equation (54) gives(55)VABA≤VABB.In addition, substituting equations (17), (26), (27), and (44) in equation (54), the following relation is obtained:(56)xBLA+xBRA+z⋅xABA≤2z⋅xABB.


## 4.2. Hierarchy Criteria for Long Link Beams

In long links, the plastic shear deformation (vp) is negligible and is considered zero. Therefore, the internal work of the kinematic mechanisms discussed is easily obtained. In its mechanism “A” (Figure 2(a)), three members of the TRF-H system including a two beams BL and BR and a column AB would yield. Also, in its mechanism “B,” the central vertical member of the TRF-H (AB) system yields from both ends. In accordance with equations (18), (24), (29), and (46), for elements in different mechanisms, the value is p≤x≤1. In each member, if the value x=1, the member is long and would have a bending behavior. Accordingly, by substituting this value in equation (56) for all the elements, the criteria for the different mechanism links can be obtained as follows:(57)z≥2⟶Mp,c≥2Mp,b.


## 4.3. Hierarchy Criteria for Short Link Beams

In short links, the moment and shear interaction according to Figure2 is negligible, so only the shear plastic deformation is vp and the value of ϕp is zero according to the normal plastic flow rule. In accordance with equations (18), (24), (29), and (46), for each member in different mechanisms, if the value is x=p, the element is a short link and has shear behavior. Therefore, by substituting this value in equation (56), the hierarchy criteria for the short links can be obtained as follows:(58)z⋅pc≥2pb⟶Mf,c≥2Mf,b.


## 5. Numerical Examples and Verification

In this section, in order to verify the hierarchy criteria obtained, these criteria are considered for the one-storey TRF-H frame, with particular attention to intermediate links in which the moment-shear interaction is important. The inelastic behavior of the designed structure has been examined with the pushover analysis to achieve the design goals, namely, the desired mechanism.The profile of the T-part elements is shown in Table1. The steel used in these members is S235 fyk=235MPa type. In order to validate the hierarchy criteria provided with intermediate links, two models are analyzed named model (I) and model (II).Table 1
Characteristics of used profile sections.
ModelElement sectionFlange plateWeb platebf (mm)tf (mm)hw (mm)tw (mm)(I)Beam (BL and BR)20010206Column (AB)200202008(II)Beam (BL and BR)200152006Column (AB)200152008In according to Eurocode 8 [1], links are classified as short links that have the following length of link:(59)e≤1.6Mp,linkVp,link,where Mp,link=Mf=bftffyd−tf and Vp,link=Vp=d−2tftwfy/3 are plastic moment and plastic shear in the link, respectively. It should be noted that, in equation (59), due to strain hardening, the flexural overstrength factor is about 1.2 (Ωm) and the shear overstrength is equal to 1.5 (Ωv). Therefore, using the equilibrium equation in the link, the maximum shear length in the ultimate state is as follows [34]:(60)e=2.0MuVu=2.01.20Mp,link1.50Vp,link=1.6Mp,linkVp,link,where Mu and Vu are the ultimate moment and ultimate shear, respectively. Equation (60) is established for the links having two end rigid connections. In the TRF system with hinge connections, this equation can be used for the central column, but for the beams, the multiplier 1.6 is replaced by 0.8. Obviously, these results could be easily generalized to the ultimate state using shear and flexural overstrength, as shown in Figure 2. With such an expansion of the plastic domain to reach the ultimate domain, all the relationships obtained in Sections 3 and 4 remain unchanged for the ultimate values of M and V. Limitation provided by equation (60) for the elements of the TRF-H system with mentioned sections in Table 1, in model (I) for beam would be equal to 0.48 meters in the ultimate state and 0.61 meters in the plastic state and for the central column of the TRF-H system would be equal to 1.51 meters in the ultimate state and 0.94 meters in the plastic state. In Model (II) for beam would be equal to 0.74 meters in the ultimate state and 0.93 meters in the plastic state and for the central column of the TRF-H system would be equal to 0.69 meters in the ultimate state and 1.11 meters in the plastic state.In models, (I) and (II), the span length is equal toL = 5.0 m, the height of the storey is equal to h = 3.0 m, and the length of the left beam is considered in the TRF-H system as an average link of 1.0 m eL=1.0m. The moment-shear interaction diagram for cross section of the TRF-H system members is shown in Figure 4. In this figure, BL and BR correspond to the yielding of the left and right beams of the TRF-H system and AB corresponds to the yielding of the central column of the TRF-H system.Figure 4
The moment-shear interaction diagram for T-part elements’ sections. For the models studied, the results of the relationships given in Section4 for determining the hierarchy criteria are shown in Table 2.
(a)(b)(c)(d)Table 2
The results of the moment-shear interaction and determining the type of mechanism.
Model (I)Model (II)xBLA0.96770.9398xBRA0.99810.9972xABA0.97810.9653xABB0.97590.9826xBLA+xBRA+z⋅xABA4.11933.03482z⋅xABB4.29702.2351Mechanism(A)(B)It should be noted that, in design terms, plastic moments are reduced due to the effects of axial forces. This axial force could be easily calculated from the shear reaction results transmitted between the beam and the column of the TRF-H system and balanced by the axial forces of these elements.To evaluate the validity of the hierarchy criteria previously described, various types of structures examined according to Table1 were analyzed using pushover analyzes with the SAP2000 computer program [35]. It should be noted that, in the SAP2000 software, the moment-shear interaction could not be defined, and the elements of the TRF-H system are modeled as a beam and column element with a plastic hinge at their ends, in which the plastic moment is equal to the plastic moment, with the use of relationships in Section 4, has been determined. The mechanisms developed for the models examined are shown in Figure 5, where structural deformation is shown that these deformations are for the displacement value, which is arranged at the top of the storey to form a kinematic mechanism for all models examined, and there is a complete assurance.Figure 5
Results of analysis of models (I) and (II).
(a)(b)Model (I) shown in Figure5(a), according to the results of Table 2 and analysis, the collapse of the structure is in accordance with the Mechanism “A.” In model (II) shown in Figure 5(b), according to the results of Table 2 and analysis, the collapse of the structure is in accordance with the Mechanism “B.” The patterns obtained confirm the validity of the proposed design method.


## 6. Conclusion

In this paper, an analytical technique for considering the results of moment and shear interaction on the TRF-H system’s intermediate links has been presented. The whole analysis was carried out in the framework of the rigid-plastic design using plastic domain diagram, normal flow rule, kinematic compatibility, and kinematic theory of plastic collapse to have a strong theoretical basis. In particular, the analytical relationships for the assessment of internal forces and the plastic deformation of intermediate links have been provided and the appropriate hierarchy criteria for the formation of the desired collapse mechanism have been obtained.From the design point of view, the effect of link’s strain hardening can be considered by applying the moment and shear overstrength factor in the plastic domain. The presented relationships are valid in the homothetic and nonhomothetic expansion of the plastic domain to determine the desired ultimate domain. The results obtained are required as a strong and applied theory in the capacity design principles for seismic design of TRF-H systems. In addition, the resultant hierarchy criteria lead to the development of a design method for failure mode control in multistorey TRF-H systems. This means that the final aim of the presented research is to ensure that a specific collapse mechanism has an expected yield hierarchy for the TRF-H system.



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*Source: 8844039-2021-07-06.xml*