Modeling Steel Frame Buildings in Three Dimensions. I: Panel Zone and Plastic Hinge Beam Elements

Transcription

1 Modeing Stee Frame uidings in Three Dimensions. I: Pane Zone and Pastic Hinge eam Eements Swaminathan Krishnan 1 and John F. Ha 2 Abstract: A procedure for efficient three-dimensiona noninear time-history anaysis of stee framed buidings is derived. It incorporates two types of noninear beam eements the pastic hinge type and the eastofiber type and noninear pane zone eements to mode yieding and strain-hardening in moment-frames. Foors and roofs of buidings are modeed using 4-node eastic diaphragm eements. The procedure utiizes an iteration strategy appied to an impicit time-integration scheme to sove the noninear equations of motion at each time step. Geometric noninearity is incuded. An overview of the procedure and the theories for the pane zone and the pastic hinge eements are presented in this paper. The theory for the eastofiber eement aong with iustrative exampes are presented in a companion paper. The pastic hinge beam eement consists of two nodes at which biaxia fexura yieding is permitted, eading to the formation of pastic hinges. Eastic rotationa springs are connected across the pastic hinge ocations to mode strain-hardening. Axia yieding is aso permitted. The pane zone eement consists of two orthogona panes forming a cruciform section. Each pane may yied and strain-harden in shear. DOI: /ASE :4345 E Database subject headings: Framed structures; Stee frames; Noninear anaysis; Pastic hinges; eams; Earthquakes; Threedimensiona anaysis. Introduction 1 Post-Doctora Schoar, Seismoogica Laboratory, M , aifornia Institute of Technoogy, Pasadena, A E-mai: krishnan@catech.edu 2 Professor, ivi Engineering and Appied Mechanics, M , aifornia Institute of Technoogy, Pasadena, A Note. Associate Editor: Francisco Armero. Discussion open unti September 1, Separate discussions must be submitted for individua papers. To extend the cosing date by one month, a written request must be fied with the ASE Managing Editor. The manuscript for this paper was submitted for review and possibe pubication on January 8, 2004; approved on June 30, This paper is part of the Journa of Engineering Mechanics, Vo. 132, No. 4, Apri 1, ASE, ISSN /2006/ /$ A number of noninear anaysis programs incorporating various types of beam and coumn eements have been deveoped by research groups in seismic engineering. These incude OpenSees Mazzoni et a. 2005, ANDERS arson 1999, U-DYNAMIX E-Tawi and Deierein 1996, DRAIN2DX Aahabadi and Powe 1988, DRAIN3DX Powe and ampbe 1994; Prakash et a. 1994, FEDEAS Fiippou and Romero 1998, IDAR Park et a. 1987; Kunnath et a. 1992; Kunnath 1995, IDAR3D Lobo 1994, and ISTAR-ST Lobo et a A summary of their capabiities is provided by arson Of these, the programs that are capabe of three-dimensiona modeing are OpenSees, ANDERS, DRAIN3DX, and IDAR3D. OpenSees is a software framework for simuating the seismic response of structura and geotechnica systems. It incudes beamcoumn eements with distributed pasticity integrated aong the ength of the eement. A corotationa geometric transformation enabes the modeing of arge dispacements. ANDERS modes buidings using panar frames arranged in an orthogona pattern. Two-dimensiona mutisegment fiber eements are empoyed except for coumns at the intersections of the frames where threedimensiona fiber eements are used to mode biaxia bending behavior. The orthogona frames are constrained to move together at the foor eves using a master-save constraint. P effects on frame coumns in the panes of the frames are automaticay accounted for through geometric updating. P effects due to outof-pane dispacements are accounted for by appying appropriate force coupes on orthogona frames. DRAIN3DX incudes a fuy discretized fiber eement with a triinear axia stress-strain aw for the fibers. P effects are incuded by adding geometric stiffness to the tangent stiffness matrix for each eement and accounting for second order effects in the resisting force computation. IDAR3D is a program deveoped for the modeing of reinforced concrete structures in which beams and coumns are modeed as ineastic singe component eements with distributed fexibiity. eamcoumn eements are modeed using fexura springs with a mutiinear moment-curvature reationship. Shear deformations are incuded. Axia deformation is incuded in the coumns but its interaction with bending strength is not. During the time-history anaysis, updating of stiffness matrices is carried out ony in the event of a stiffness change and a singe-step force-equiibrium correction procedure is used. P effects are accounted for by a simpe equivaent force method. The computationa chaenges posed by three-dimensiona seismic anaysis of ta buidings are addressed here by the deveopment of a comprehensive noninear finite eement anaysis program, FRAME3D The focus is on stee moment-frames, athough braced frames can aso be modeed with sight modifications. There are two distinguishing features of the program. First, a geometric updating feature is provided to accommodate arge transations and rotations of the beam eements. This automaticay accounts for P effects and aows the anaysis to foow a buiding s response we into coapse. onsiderabe detai is JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006 / 345

2 Fig. 2. Pane zone shear caused by beam and coumn moments at the joint. Fig. 1. Eement arrangement in frame mode, showing joint nodes, attachment points, oca beam nodes, and coordinate systems. provided here to document the geometric updating capabiity. Second is the use of computationay efficient eements which incorporate the important types of noninear behavior. Fu threedimensiona anayses can be made of arge, compex structures. At the present time, the eement ibrary of FRAME3D incudes the foowing finite eements: 1 three-dimensiona pastic hinge eement to mode beams and coumns, 2 three-dimensiona eastofiber eement to mode beams and coumns, 3 threedimensiona pane zone eement to mode the beam-to-coumn joints in frames, and 4 eastic diaphragm eement to mode the in-pane action of foors and roofs that tie the frames together. The first three eement types contain materia noninearity, and the pastic hinge and eastofiber eements incude geometric stiffness. An overview of the anaysis procedure which is the basis for this program is presented in this paper aong with the theories for the pane zone eement and the pastic hinge eement. The theory for the eastofiber eement and iustrative exampes are given in a companion paper. Description of Mode The three-dimensiona structura mode of framed buidings considered here consists of grids of beams and coumns. The setup of the mode is comprised of three eement casses: pane zone eements, beam eements for beams and coumns, and diaphragm eements for foors and roofs. The arrangement of these eements in a typica structura mode is iustrated in Fig. 1. The two beam eement types can be used for either beams or coumns. Fu geometric updating is incuded in both static and dynamic anayses. This invoves updating the ocations of the joint nodes, attachment points, and the oca beam nodes shown in Fig. 1 as we as the orientations of the oca eement coordinate systems. Pane Zone Eement This eement modes noninear shear deformation in the region of the joint where the beams and coumns intersect. The joint region consists of a ength of coumn within the depth of the connecting beams. The shear deformation is due primariy to opposing moments from the beams and coumns at the joint caused by the frame being subjected to atera oads Fig. 2. Each pane zone eement is associated with a joint node J, K, etc. at the center of the joint where the goba degrees of freedom DOF are defined. Each pane zone eement consists of two orthogona panes and which aways remain panar and orthogona. Edges of these panes contain attachment points a, b, c, and d, where beams attach and e and f on the top and bottom where coumns attach. The theory of the pane zone eement is presented ater in this paper. eam Eement This eement is used to mode beams and coumns. Two types of beam eements have been deveoped: pastic hinge type and eastofiber type. The pastic hinge eement has two nodes with oca node numbers 1 and 2, whie the eastofiber eement consists of three segments with four nodes. The exterior nodes at the ends of the eastofiber eement are numbered 1 and 2, whie the interior nodes are numbered 3 and 4. oth types of beam eements consider noninear behavior for fexura and axia deformations. Geometric stiffness, i.e., the effect of axia oad on fexura stiffness, is incuded. The theory of the pastic hinge eement is presented ater in this paper and that of the eastofiber eement is presented in the companion paper. Diaphragm Eement This eement is used to mode the in-pane stiffness of foor sabs. It is essentiay a 4-node pane-stress eement connecting to joint nodes J, K, L, and M. Detais of the diaphragm eement are given by Krishnan Its behavior is ineary eastic. oordinate Systems Various coordinate systems are used in the structura mode; each set of axes is right-handed and orthogona. XYZ is the goba, fixed coordinate system that is used to define the structure in space. The coordinates of the joint nodes, attachment points, and oca beam nodes are with respect to this coordinate system. The Z axis is oriented verticay. 346 / JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006

3 The indicates that the rotating pane edges are connected to beams; and 8. : rotation of the ine c-d about the Z axis. It corresponds to deformation of pane into the shape of a paraeogram. The indicates that the rotating pane edges are connected to coumns. Note that and together accommodate a rigid rotation of pane about Z pus its shear deformation. Pane zone eements contribute stiffness directy to the four goba DOF: JȲ, JȲ,, and. eam eements are formuated in terms of three transationa and three rotationa DOF at each of the oca nodes 1 and 2 for pastic hinge eement and 1, 2, 3, and 4 for eastofiber eement. However, none of these DOF appear in the goba equations. Appropriate stiffness terms from the beam eements are assembed into the goba DOF through transformation matrices based on the geometry of the deformed pane zone eements. Fig. 3. Four rotationa degrees of freedom of a pane zone eement. Other coordinate systems are oca and are associated with the eements. Separate X ȲZ systems are used for each pane zone eement, and separate XYZ systems are used for each pastic hinge eement and each segment of each eastofiber eement. These oca coordinate systems transate and rotate with their associated eements and segments. The orientations of the oca axes are defined ater, except that it is now noted that Ȳ is perpendicuar to pane of a pane zone eement, Z is perpendicuar to pane, and X is aong the pane intersection ine e-f. Structure Degrees of Freedom The goba degrees of freedom are associated ony with the nodes J, K, etc. at the joints. The structura mode consists of eight goba DOF at each node, J. These DOF are isted beow. Note that the transationa DOF are with respect to the goba axes X, Y, Z, and the rotationa DOF are with respect to the oca sets of pane zone axes, X,Ȳ,Z. The four and DOF are shown in Fig U JX : transation in the X direction; 2. U JY : transation in the Y direction; 3. U JZ : transation in the Z direction; 4. JX : rotation of pane zone eement as a rigid body about the X axis; 5. JȲ : rotation of the ine e-f about the Ȳ axis. It corresponds to deformation of pane into the shape of a paraeogram. The indicates that the rotating pane edges are connected to beams; 6. JȲ : rotation of the ine a-b about the Ȳ axis. It corresponds to deformation of pane into the shape of a paraeogram. The indicates that the rotating pane edges are connected to coumns. Note that JȲ and JȲ together accommodate a rigid rotation of pane about Ȳ pus its shear deformation; 7. : rotation of the ine e-f about the Z axis. It corresponds to deformation of pane into the shape of a paraeogram. ase ondition Additiona modeing considerations, not discussed here in detai, can be empoyed at the base of the buiding or at basement eves. Stiffnesses representing foundation fexibiity can be assembed into transationa DOF in contact with the ground, or those DOF are fixed if the foundation can be assumed to be rigid. Fexibe basement was on the perimeter can be modeed with wa eements, or, again, appropriate DOF are fixed if such was can be assumed to be rigid. Ha 1995 contains exampes where foundation fexibiity and basement was are considered. Equations of Motion and Soution Process The matrix equation of motion of the buiding ook et a. 1989; hopra 1995 as a function of time, t, is MÜt + U t + Rt = f g MrÜ g t 1 In the above, 1. Ut=vector of goba dispacements at time, t, comprising the eight goba transations and rotations at each node, J, K, etc., not incuding the fixed support DOF; 2. U t, Üt=vector of noda veocities and acceerations, respectivey, corresponding to the goba transations and rotations at each node; 3. M=structure mass matrix. The mass in the structura system is umped at the joint nodes, rendering the mass matrix diagona. Further, the rotary inertia of the umped noda masses is negected, so the ony nonzero terms of the mass matrix are the diagona terms corresponding to the transationa degrees of freedom; 4. =structure damping matrix. Damping is assumed to be of the Rayeigh type stiffness and mass proportiona. Thus the damping matrix is computed as = a 0 M + a 1 K 2 where a 0 and a 1 =user defined proportionaity constants. The initia eastic stiffness matrix is used in the above computation; 5. Rt=vector of stiffness forces corresponding to the configuration Ut. It is computed considering a materia and geometric noninear effects; JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006 / 347

4 6. f g =vector of static gravity oads for which a static anaysis is performed first; 7. Ü g t=vector consisting of two horizonta components X and Y and one vertica component Z of the free-fied ground acceeration at time t. The ground motion is assumed spatiay uniform; and 8. r=three-coumn matrix of zeroes except for ones in the first, second, and third coumns in the positions corresponding to the X, Y, and Z transationa DOF, respectivey. The noninear effects contained in Rt can be inearized over the time interva between t and t+t as Rt + t = K T U + Rt 3 where K T =tangent stiffness matrix corresponding to the configuration Ut. Eq. 3 is substituted into Eq. 1 written at time t+t aong with the foowing reations representing a constant average acceeration over the time step t U t + t = U t + Üt + Üt + t t 2 Üt + t = 4 t 2Ut + t Ut 4 t U t Üt The resut is 4 t 2M + 2 t + K TU = f g Rt MrÜ g t 4a 4b + M 4 t U t + Üt + U t 5 The soution of Eq. 5 for U eads to the new dispacements via Ut + t = Ut + U. Once K T and R are updated to time t+t and U t+t and Üt+t are determined from Eqs. 4a and 4b, the next time step can commence. This process continues forward in time step by step. However, since it is unikey that the inearization in Eq. 3 wi hod throughout the entire time step without K T changing, iterations are used within each time step. In iteration, dispacement configuration U has been reached, and the equation to be soved uses K T and R corresponding to this configuration 4 t 2M + 2 t + K T U = f g R MrÜ g t + M 4 t 2Ut + 4 U t + Üt t + 2 t Ut + U t 4 t 2M + 2 t U The soution of Eq. 7 eads to U which adds to U to give U +1. The next iteration uses the updated vaues of K T +1 and R +1 corresponding to configuration U +1. Iterations continue unti convergence, and then the next time step begins. 6 7 For noninear probems, the basic computationa task is to update K T and R using U. The updating is done at the eement eve and then assemby is used to construct K +1 T and R +1, i.e., contributions to K +1 T are the eement matrices K +1 pz, K +1 ph, K +1 ef, and K +1 d, and contributions to R +1 are the eement vectors R +1 pz, R +1 ph, R +1 ef, and R +1 d. The subscripts stand for pane zone eement, pastic hinge beam eement, eastofiber beam eement, and diaphragm eement, respectivey. Take, for exampe, computation of the updated K +1 ph and R +1 ph for a pastic hinge eement. The foowing steps are performed: 1. The 16 goba dispacement increments from U computed by Eq. 7 for the nodes J and K associated with the pastic hinge eement are extracted and paced in the vector U ph. 2. Using a matrix T ph, vector U ph is transformed into U ph L which contains dispacement increments for the eement s 12 degrees of freedom at nodes 1 and 2 in the XYZ coordinate system U ph L = T ph U ph 8 The subscript L indicates that the vector contains terms for the eement s 12 DOF at its oca nodes 1 and The eement s dispacement increments U ph L are used to first compute K +1 ph L and R +1 ph L, which are in terms of the eement s 12 DOF. 4. Then, K +1 ph L and R +1 ph L are transformed to the 16 goba DOF at nodes J and K using the matrix T +1 ph updated to the +1 configuration K +1 ph = T +1 ph T K +1 ph L T +1 ph 9 R +1 ph = T +1 ph T R +1 ph L 10 The updating process is a itte different for the other eement types. Since the and degrees of freedom for the pane zone eement are aso goba degrees of freedom, no transformation with a T matrix is necessary. Some additiona steps are necessary for the eastofiber eement because the presence of interior nodes means that a noninear structura anaysis probem must be soved for each eement, using iterations. These iterations are within those performed during the goba soution process. Detais for the pane zone and pastic hinge beam eements are provided in the next few sections. Parts deaing with the eement tangent stiffness matrices use a differentia notation since tangent is a imit concept. Pane Zone Eement Genera Description Each joint is modeed by a pane zone eement consisting of a ength of coumn within the depth of the connecting beams. This impies that one coumn, the associated coumn of the joint, runs continuousy through the height of the joint. The pane zone eement is an ideaization of the joint region of this coumn. It consists of two rectanguar panes which are perpendicuar to each other, pane in the X -Z pane and pane in the X -Ȳ pane, forming a cruciform section Figs. 1 and 4. The thicknesses of a web pates and web douber pates of the associated coumn are combined to form the thickness of pane, t p, whie the thicknesses of a fange pates of the associated coumn are combined to form the thickness of pane, t p. The depth D of the joint is 348 / JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006

5 Fig. 5. Shear stress-strain backbone curve for a pane zone eement. Fig. 4. Joint region of the associated coumn eft and representation as a pane zone eement right. taken to be equa to the depth dimension aong the minor axis Z of the associated coumn. The width W of the joint is taken to be equa to the width dimension aong the major axis Y. The height H of the joint is taken to be equa to the depth of the deepest beam framing into that joint. eams and coumns modeed using beam eements connect to the pane zone eement at the midpoints of the edges of the two panes. These connection points are referred to as attachment points. There are six attachment points, a through f. Attachment points a, b, c, and d are reserved for beams, whie attachment points e and f are reserved for coumns. The panes are assumed to deform ony in shear as a resut of the end moments and shears of the attached beams and coumns. However, they remain panar and perpendicuar to each other at a times. The reation between the shear stress and the shear strain in each pane can be a noninear reation. Detaied experimenta ertero et a. 1972; Popov and Petersson 1978; Kato 1982; Krawinker and Popov 1982 and anaytica Fieding and hen 1973; Pinkney 1973; Krawinker et a. 1975; Popov and Petersson 1978; Kato 1982 studies have been carried out on stee-frame joints and have resuted in a much better understanding of their hysteretic behavior. A hysteretic mode Ha and haa 1995 is impemented here as described in the next section. Each pane has two degrees of freedom as shown in Fig. 3: JȲ and JȲ for pane and and for pane, where J=goba node at the center of the joint. These DOF are aso goba DOF. Strain or rotation in one of the panes causes a rigid body rotation but no strain in the orthogona pane. The joint coordinate system X ȲZ with origin at the center of the joint at the node is defined as foows: 1. X axis is oriented from attachment point f to attachment point e of the joint. As the pane zone eement moves and deforms, X foows the f-to-e orientation. 2. Ȳ axis is in the pane of pane and is initiay in the d-to-c direction Fig. 4. As the pane zone eement moves and deforms, Ȳ remains in the pane of pane but it may rotate off the d-to-c direction since it must remain perpendicuar to X. 3. Z axis is in the pane of pane and is initiay in the b-to-a direction Fig. 4. As the pane zone eement moves and deforms, Z remains in the pane of pane but it may rotate off the b-to-a direction since it must remain perpendicuar to X. Materia Mode for Pane Shear The hysteresis mode for shear stress-strain behavior in a pane proposed by Ha and haa 1995 defines a backbone curve as shown in Fig. 5. This curve is assumed to be inear unti a stress of 0.8 y is reached corresponding to a strain of 0.8 y, where y and y =yied shear stress and strain, respectivey. The utimate shear stress u of the pane zone is assumed to be equa to 2.35 y, with an utimate shear strain u of 100 y. These contro points are seected based on monotonic test data on pane zones in beam-coumn assembages Kato The post-yied behavior is infuenced by the resistance of the pane boundary eements coumn fanges and continuity pates, and the restraint to defor- Fig. 6. Panes top and bottom showing dimensions and degrees of freedom eft and edge shear forces right. JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006 / 349

6 Fig. 7. DOF of the pastic hinge beam eement showing noda transations/rotations and noda forces/moments. mation of joint-pane by the adjacent beam and coumn webs, in addition to the strain-hardening of the pane pate. The postutimate behavior is assumed to be perfecty pastic. The curve between the joint shear strain imits of 0.8 y and 100 y is defined by a quadratic eipse. Hysteresis oops are defined by inear segments and cubic eipses, and the hysteresis rues to define the cycic response of each pane are given by haa These rues are the same as those used for each fiber in an eastofiber eement and are iustrated in Fig. 3 of the companion paper. Theory of Pane Zone Eement Fig. 6 shows panes and with their dimensions, degrees of freedom, and shear forces acting on the pane edges. The shear forces come from the end moments and shears from the connected beam eements modeing beams and coumns. Each pane is under a uniform state of shear stress: for pane and for pane. onsider pane with its degrees of freedom, JȲ and JȲ. The four edge forces form a doube coupe of ampitude t p HD. The haves of this doube coupe are the pane moments which correspond to the two rotationa DOF and can be written as M = MJȲ JȲ = t p HD 11 The shear stress is reated to the shear strain by the incrementa reation d = G T d 12 where G T =tangent shear moduus. The incrementa shear strain is reated to the degrees of freedom by d = d JȲ djȳ 13 Fig. 8. Sign convention for interna forces and moments in a pastic hinge beam eement. A simiar treatment for pane resuts in the foowing tangent reation: dm dm = G T t p HW 1 1 JZ d d ombining Eqs. 14 and 15 eads to dr pz = K T,pz du pz where dr pz =incrementa version of R pz = M JȲ M JȲ M M K T,pz =44 tangent stiffness matrix for the pane zone eement, and du pz = d JȲ d JȲ d d 18 Updating Process In goba iteration, U is computed from Eq. 7. The four rotation increments for a pane zone eement, JȲ, JȲ,, and, are extracted from U and paced in U pz. The coordinates of node J of the pane zone eement are updated using U JX, U JY, and U JZ, aso obtained from U. Using the new ocation of node J and the incrementa rotations U pz the updated coordinates for attachment points a through f and the updated orientations for the X, Ȳ, and Z axes are found. The procedure for this geometry updating is given by Krishnan Aso updated are anges and used in the pastic hinge eement formuation described ater. Shear strain increments in the two panes are computed as = JȲ JȲ 19a ombining Eqs. 12 and 13 with the incrementa version of Eq. 11 eads to dm JȲ = G T t p dm JȲ HD 1 1 JY d d JȲ which is the tangent reation for pane. 350 / JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006 = from which the updated shear strains are,+1 =, +,+1 =, + 19b 20a 20b

7 Fig. 9. P-M py -M pz interaction diagrams used for the pastic hinge eement. Fig. 10. Pane zone deformation geometry for construction of T 2 for a coumn. The updated shear stresses,+1 and,+1 can now be found using the materia mode described previousy. From Eq. 11, the moments for pane are,+1,+1 M = MJȲ JȲ =,+1 t p HD Simiary, the moments for pane are given by Then, as in Eq. 17,,+1,+1 M = M =,+1 t p HW R +1,+1 pz = M JȲ,+1 M JȲ,+1 M,+1 M The eement tangent stiffness matrix can be updated once the tangent shear modui G,+1 T and G,+1 T are found for each pane. This information is avaiabe as a resut of computing R +1 pz, after which K +1 T,pz can be found from the formuas in the previous section. Since the rotationa DOF of the pane zone eement are aso goba DOF, no transformations of R +1 pz and K +1 T,pz are necessary before they are assembed into the goba equations. Three-Dimensiona Pastic Hinge eam Eement Genera Description The foowing assumptions are made in the formuation of this eement: 1. The cross section is uniform aong the ength of the eement; 2. The cross section is douby symmetric with shear center at the centroid; 3. Pane sections remain pane; however, they do not have to remain norma to the beam axis. Thus shear deformations are incuded; 4. Strains in the eement are sma; 5. Latera defections reative to the chord are sma. This means that the effect of bowing Oran 1973a,b; Kassimai 1983 on axia stiffness is negected. This effect is important ony for very sender members or during postbucking of compression members such as braces in braced frames; 6. Warping restraint under twisting is negected; and 7. The eement is not oaded aong its span. The pastic hinge beam eement has two nodes which connect to the attachment points a through f of the pane zone eement. It can mode beams and coumns in framed structures. oumns connect to attachment points e and f whie beams connect to attachment points a through d. The beam eement aong with its degrees of freedom is shown in Fig. 7. Origina ength of the eement is. The pastic hinge eement oca coordinate system XYZ is defined as foows: 1. X axis runs aong the ongitudina axis of the eement at the centroid of the cross section. It is defined as a vector from node 1 to node 2 which are aso ocated at the centroid; 2. Y axis is orthogona to X and is the major principa axis of the cross section; and 3. Z axis is the minor principa axis of the cross section. The major and minor principa axes Y and Z of the cross section are oriented using a user-defined orientation ange or Krishnan Degrees of Freedom, and Noda Forces and Moments The degrees of freedom Fig. 7 of the pastic hinge eement are 1. U 1, U 2 =X transations of nodes 1 and 2, respectivey; 2. V 1Y, V 2Y =Y transations of nodes 1 and 2, respectivey; 3. V 1Z, V 2Z =Z transations of nodes 1 and 2, respectivey; 4. 1, 2 =rotations about X at nodes 1 and 2, respectivey; 5. 1Y, 2Y =rotations about Y at nodes 1 and 2, respectivey; and 6. 1Z, 2Z =rotations about Z at nodes 1 and 2, respectivey. orresponding to these DOF are noda forces and moments Fig. 7: 1. P 1, P 2 =forces in X direction at nodes 1 and 2, respectivey; 2. Q 1Y, Q 2Y =forces in Y direction at nodes 1 and 2, respectivey; 3. Q 1Z, Q 2Z =forces in Z direction at nodes 1 and 2, respectivey; 4. T 1, T 2 =moments about X at nodes 1 and 2, respectivey; 5. M 1Y, M 2Y =moments about Y at nodes 1 and 2, respectivey; and 6. M 1Z, M 2Z =moments about Z at nodes 1 and 2, respectivey. Interna Forces and Moments The interna forces and moments in the pastic hinge beam eement are the axia force, P, the shear forces in the Y and Z directions, Q Y and Q Z, respectivey, the twisting moment torque, T, and the bending moments about the Y and Z axes, M Y and M Z, respectivey. The sign convention positive directions for these forces and moments is shown in Fig. 8. JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006 / 351

8 Materia Noninearity Two types of materia noninearity are considered. 1. Axia yieding when axia force P in the eement reaches the yied axia force P y which is given by P y = y A 24 where y =yied stress of the materia and A=area of the cross section. 2. Fexura yieding when the bending moment M Y or M Z at the ends of the eement reaches the pastic moment capacity M py or M pz which depend on the axia force P as shown in Fig. 9. The pastic moment capacities when P=0, denoted by 0 0 M py, and M pz, are given by 0 M py = y Z Y 25a 0 M pz = y Z Z 25b where Z Y and Z Z =pastic modui of the cross section of the eement about its major and minor axes, respectivey. Once M Y or M Z reaches M py or M pz at node 1 or node 2, further oading causes a kink, termed a pastic hinge, to form in the beam at that node. etween these pastic hinge ocations, the beam behaves easticay. To approximate the effect of strain hardening, eastic rotationa springs are mounted across the pastic hinges to exert moments proportiona to the kink anges about the Y and Z axes. Transformation Matrix, T ph T ph =transformation matrix between the 16 goba DOF at the nodes J and K of the joints, U ph, and the 12 oca degrees of freedom at nodes 1 and 2, U ph L. For notationa purposes, it is assumed that node 1 of the eement connects to one of the attachment points at node J and that node 2 of the eement connects to one of the attachment points at node K. The transformation is carried out in four steps for dispacement increments 1 U ph 2 Ū ph 3 Ū ph L 4 U ph L U ph L 26 The goba dispacement increments in U ph are isted as U ph = U JX U JY U JZ JX JȲ JȲ U KX U KY U KZ KX KȲ KȲ KZ KZ 27 In U ph, transationa DOF are with respect to XYZ; rotationa DOF are with respect to the respective X ȲZ s at the nodes J and K. The first transformation produces Ū ph = T 1 U ph 28a Ū ph = U JX U JȲ U JX JȲ JȲ U KX U KȲ U KZ KX KȲ KȲ KZ KZ 28b where the transationa DOF are aso now with respect to the X ȲZ s at nodes J and K. The second transformation produces Ū ph L = T 2 Ū ph 29a Ū ph L = U 1X U 1Ȳ U 1Z 1X 1Ȳ 1Z U 2X U 2Ȳ U 2Z 2X 2Ȳ 2Z 29b where the eight DOF at node J have been transformed to the six oca beam DOF at node 1 sti in node J s X ȲZ and the eight DOF at node K have been transformed to the six oca beam DOF at node 2 sti in node K s X ȲZ. The subscript L denotes the presence of the terms for the 12 DOF at oca nodes 1 and 2. The third transformation produces U ph L = T 3 Ū ph L U ph L = U 1X U 1Y U 1Z 1X 1Y 1Z U 2X U 2Y U 2Z 2X 2Y 2Z 30b where a DOF are now in XYZ. Finay, the fourth transformation U ph L = T 4 U ph L 31a produces U ph L = U 1 V 1Y V 1Z 1 1Y 1Z U 2 V 2Y V 2Z 2 2Y 2Z 31b where a DOF are now in the beam eement s oca XYZ. 352 / JOURNAL OF ENGINEERING MEHANIS ASE / APRIL a

9 ombining the above eads to U ph L = T ph U ph 32a where T ph = T 4 T 3 T 2 T 1 32b The components T 1, T 3, and T 4 are defined beow. J T 1 = I K I where I=55 identity matrix and J=matrix consisting of the direction cosines of the pane zone coordinate system X ȲZ at joint node J with respect to the goba coordinate system XYZ, simiar for K at joint node K. J T 3 = T T J T K T K T 4 = where =matrix consisting of the direction cosines of the beam oca coordinate system XYZ with respect to the goba coordinate system XYZ. T 2 is different for beams and coumns. It aso depends on the attachment points to which the eement oca nodes 1 and 2 are connected at nodes J and K. As an exampe, the foowing matrix is derived for a coumn whose oca node 1 connects to attachment point f at node J and oca node 2 connects to attachment point e at node K. The deformed pane zone geometry which is the source of some of the terms of T 2 for a coumn is shown in Fig. 10. For instance, an incrementa twist in the joint about the joint X axis eads to an equa amount of incrementa twist in the coumns that the joint is part of. Hence term 4,4 of the T 2 matrix in Eq. 36 is H J H J T co f e = H K H K The deformed pane zone geometry for cacuating some of the terms of T 2 for a beam is shown in Figs. 11 and 12. As another exampe, the foowing matrix is derived for a beam whose oca node 1 connects to attachment point a at node J and oca node 2 connects to attachment point b at node K. For instance, an incrementa twist in the joint eads to a transation of a the beam attachment points of the joint as shown in Fig. 12. Hence term 2,4 of the T 2 matrix in Eq. 37 is 0.5D J cos J. JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006 / 353

10 DJ cos J T bm a b = D J cos J D J sin J D K cos K D K cos K D K sin K The anges and, the atter needed for a beam which connects to attachment points c and d, are defined in Fig. 11. The detais of how the T 2 matrix is computed in other instances are given in Krishnan Deveopment of Tangent Stiffness Matrix Axia Deformation The axia force P in the eement is a function of the axia strain as shown in Fig. 13. =axia strain at the centroid of the cross section. Increments in P and are reated by dp = E T Ad 38a where E T =tangent moduus, either Young s moduus E or zero as shown in Fig. 13, and A=area of the cross section. The tangent stiffness reation for the noda forces P 1 and P 2 associated with axia deformation is given by dp 1 dp 2 = E TA 1 1 du du 2 38b Twisting Twisting is assumed to be ineary eastic with increments in torque T and twisting strain twist per unit ength reated by dt = GJd 39a where G=shear moduus and J=torsiona constant of the cross section. The tangent stiffness reation for the noda forces T 1 and T 2 associated with twisting deformation is given by dt 1 dt 2 = GJ 1 1 d d 2 39b ending and Shearing in X-Z Pane (ending about the Major Axis of the ross Section) Fig. 14 shows the pastic hinge beam eement with the pastic hinges ocated aong the beam just inside the nodes inside means the side towards the midde of the beam. 1Y and 2Y =Y rotations of the cross section between the pastic hinges and the nodes, and ˆ 1Y and ˆ 2Y are the Y rotations of the cross section just inside the pastic hinges. These rotations are reative to the chord straight ine connecting nodes 1 and 2. Since the behavior between the pastic hinges is eastic, Fig. 11. Pane zone deformation geometry for construction of T 2 for a beam. Fig. 12. Pane zone deformation geometry for construction of T 2 for a beam continued. 354 / JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006

11 Fig. 15. Effective shear areas for wide-fange and box sections Fig. 13. P- reation for pastic hinge eement h s M 1Y = M 1Y + M 1Y 42a M 1Y M 2Y = a b b 1Y cˆ ˆ 2Y 40 where a, b, and c depend on E, the moment of inertia I Y about the major axis, G since shear deformations wi be incuded Przemeniecki 1968, the effective shear area A SZ as shown in Fig. 15, and P since effect of axia oad on stiffness wi be incuded Samon and Johnson For convenience, subscripts denoting the X-Z pane wi be omitted on a, b, and c. Soutions for a, b, and c Ha and haa 1995 come from the foowing differentia equations: d 2 M Y dx 2 + Q Z = Pd2 V Z dx 2 dq Z dx =0 41a 41b dˆ Y M Y = EI Y 41c dx Q Z = GA SZ dv Z Y dx + ˆ 41d subject to V Z =0 at nodes 1 and 2, and ˆ Y =ˆ 1Y at node 1 and ˆ Y =ˆ 2Y at node 2. In these equations, V Z =Z transation aong the beam reative to the chord, and ˆ Y =Y rotation of the cross section aong the beam reative to the chord. Each moment M 1Y and M 2Y consists of a part carried by the pastic hinge and a part carried by the rotationa spring h s M 2Y = M 2Y + M 2Y 42b The hinge contribution is a rigid-pastic function of the kink rotation as shown in Fig. 16 and the spring contribution is a inear function of the kink rotation s M 1Y = k 1 1Y 43a s M 2Y = k 2 2Y 43b where the rotationa spring constants at the two ends of the eement are k 1 and k 2 subscripts to indicate the X-Z pane are omitted for convenience, and 1Y and 2Y =kink anges given by 1Y = 1Y ˆ 1Y 2Y = 2Y ˆ 2Y as shown in Fig. 14. Substituting Eqs. 44 into Eq. 40 M 1Y M 2Y = a b b 2Y 2Y c 1Y 1Y 44a 44b 45 Taking the differentia form, dropping terms containing dp, and with appropriate substitutions from above, this equation can be rewritten as dm 1Y dm 2Y = a T b T b T d T 1Y c d 2Y 46 where a T, b T, and c T =tangent stiffness coefficients which depend on whether pastic hinges are active during the increment Tabe 1. Eq. 46 needs to be transformed and expanded to the noda degrees of freedom. The foowing equations are empoyed: d 1Y d 1Y d 2Y = SdV1Z dv 2Z d 2Y 47a Fig. 14. Geometry of pastic hinge beam eement for bending and shearing in the X-Z pane. dq1z dm 1Y dq 2Z ST dm1y dm dm 2Y + P 0 L 2Y= d 1Y Z dV1Z d 2Y 47b JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006 / 355

12 associated with bending and shearing in the X-Z pane. ending and Shearing in X-Y Pane (ending about Minor Axis of ross Section) A formuation simiar to the preceding section eads to dq1y dm 1Z dq 2Y dm 2Z=ST at bt b T c TS + P Fig. 16. M h - reation at a pastic hinge d 1Z dv1y dv 2Y d 2Z 49 where S = and the P/ term is additiona geometric stiffness. ombining with Eq. 46 eads to dq1z dm 1Y dq 2Z dm 2Y=ST 0 at bt d 1Y dv1z dv 2Z d 2Y b T c TS + P 48 which is the tangent stiffness reation for the noda quantities where S = and a T, b T, and c T in terms of a, b, and c are as described previousy except that a, b, and c use the minor axis moment of inertia I Z and the effective shear area A SY Fig. 15. ombined Resuts The tangent reations in Eqs. 38b, 39b, 48, and 49 are combined into dr ph L = K T,ph L du ph L 50 where L refers to oca nodes 1 and 2, dr ph L =incrementa version of R ph L = P 1 Q 1Y Q 1Z T 1 M 1Y M 1Z P 2 Q 2Y Q 2Z T 2 M 2Y M 2Z 51a K T,ph L =1212 tangent stiffness matrix for the pastic hinge eement, and du ph L = du 1 dv 1Y dv 1Z d 1 d 1Y d 1Z du 2 dv 2Y dv 2Z d 2 d 2Y d 2Z 51b Updating Process In goba iteration 1, U is computed from Eq. 7. The 12 dispacement increments in XYZ at nodes 1 and 2 for a pastic hinge eement are found as U ph L = T ph U ph 52 where U ph contains the 16 terms extracted from U corresponding to joint nodes J and K connected to eement nodes 1 and 2, and T ph =transformation matrix representing configuration. Updating the pane zone eement geometries as discussed previousy wi produce new ocations for the attachment points, from which new X, Ȳ, and Z orientations can be found, eading to +1 J and +1 K and then T +1 1 and T +1 3 by Eqs. 33 and 34, respectivey. Updating and eads to T +1 2 for a beam Eq. 37 for exampe; T +1 2 for a coumn does not need updating Eq. 36 for exampe. The new attachment point ocations aso give the new ocations for nodes 1 and 2, and these wi give the updated direction for the X axis of the pastic hinge eement. Next, the updated vaue for or is found as +1 = or or / JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006

13 Tabe 1. Tangent oefficients, a T, b T, and c T Active pastic hinges a T b T c T None a b c Node 1 ony Node 2 ony Nodes 1 and 2 from which the new directions for Y and Z are found. The updated direction cosine matrix +1 is then used to form T +1 4 by Eq. 35. Then T +1 ph = T +1 4 T +1 3 T +1 2 T Next R +1 ph L is found from R ph L and U ph L. onsider first the noda axia forces P +1 1 and P Once the ocations for nodes 1 and 2 are determined, the new eement ength L +1 can be found. Then and +1 = L+1 = +1 Using and, P +1 can be found from Fig. 13. Then 55a 55b P +1 1 = P +1 2 = P +1 55c Next consider the noda torques T +1 1 and T The increment in twisting strain is and then Noda vaues are a k 1 k 1 + a = 2 1 T +1 = T + GJ 56a 56b T +1 1 = T +1 2 = T +1 56c Next is the X-Z pane bending and shearing. The +1 state wi correspond to one of nine pastic hinge cases which are isted in Tabe 2. The procedure is as foows. The noda rotations are first updated as +1 = 1Y 1Y +1 = 2Y 2Y b2 a k 2 + c b k 1 k 1 + a b k 2 k 2 + c k 1 ak 2 + c b 2 k 1 + ak 2 + c b 2 bk 1 k 2 k 1 + ak 2 + c b 2 + 1Y + V 2Z V 1Z + 2Y + V 2Z V 1Z 57a 57b Second, the pastic moment capacity is updated to M +1 py according to Fig. 9 using P +1. Then each case is examined using Eq. 45 written for state +1 b2 c k 1 + a c k 2 k 2 + c k 2 ck 1 + a b 2 k 1 + ak 2 + c b 2 Tabe 2. Nine Pastic Hinge ases Used for Determining R +1 ph L ase Pastic hinges M None *M p +1 2 Node 1 =M p +1 3 Node 1 = M p +1 4 Node 2 *M p +1 5 Node 2 *M p +1 6 Nodes 1 and 2 =M p +1 7 Nodes 1 and 2 =M p +1 8 Nodes 1 and 2 = M p +1 9 Nodes 1 and 2 = M p +1 M +1 1Y M 2Y +1= a+1 b +1 b +1 M 2 +1 *M p +1 *M p +1 *M p +1 =M p +1 = M p +1 =M p +1 = M p +1 =M p +1 = M p Y 1Y c +1 2Y 2Y 58 where a +1, b +1, and c +1 are computed using P +1. To examine ase 1, +1 1Y = 1Y and +1 2Y = 2Y are substituted into Eq. 58, and M +1 1Y and M +1 2Y are computed. Then, if M +1 1Y M +1 py and M +1 2Y M +1 py, the current state corresponds to ase 1 and the updated moments are accepted. If ase 1 does not check, ase 2 is examined. M +1 1Y =M +1 py and +1 2Y = 2Y are substituted into Eq. 58, and the set of equations is soved for M +1 2Y and +1 1Y.IfM +1 2Y M +1 py and +1 1Y 1Y, the current state corresponds to ase 2 and the updated moments are accepted. If not, ase 3 is examined and so on. Ony one out of the nine cases wi check. Finay Q +1 1Z = Q +1 2Z = M M 1Y 2Y L For X-Y pane bending, Q +1 1Y, M +1 1Z, Q +1 2Y, and M +1 2Z are found by a procedure simiar to the one just described. Then the 12 updated noda forces and moments are assembed into R +1 ph L. The eement tangent stiffness matrix can be updated once the updated axia force P +1 and the appropriate pastic hinge cases for the X-Z and X-Y panes are determined. This information is avaiabe as a resut of computing R +1 ph L, after which K T,ph +1 L can be found from the formuas in the preceding section. The fina step is the transformation to the goba DOF K +1 T,ph = T +1 ph T K T,ph +1 L T +1 ph 60a R +1 ph = T +1 ph T R +1 ph L and then assemby into K +1 T and R b Use of Interior Nodes A mutisegment pastic hinge eement with interior nodes can extend the formuation of the 2-node eement just described. Static gravity oads can be appied to the interior nodes, reieving the assumption of no aong-span oads. With geometric updating, the effects of bowing on axia stiffness are incorporated, and the eement can be used as a brace with post-bucking behavior incuded. The extra DOF associated with the interior nodes can be condensed out so that the number of goba DOF does not increase. This process is described by Ha and haa 1995 where an exampe of a two-segment eement with a singe interior node at midspan acting as a brace is described. This mutisegment approach can aso be used to mode curved members approximatey. = = 1 = = 2 = 2 = JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006 / 357

14 Treatment of mutisegment eements with interior nodes is aso provided in the companion paper for the eastofiber eement. oncusions A procedure for efficient three-dimensiona noninear time-history anaysis of stee-framed buidings is derived. In this method, four types of eements are empoyed: pane zone, pastic hinge beam, eastofiber beam, and diaphragm. The pane zone eement is intended to mode the three-dimensiona noninear shear behavior of joints in stee moment-frame buidings whie the pastic hinge and eastofiber eements mode the three-dimensiona noninear axia and fexura behavior of beams and coumn. The diaphragm eement is a 4-node eastic pane-stress eement which modes diaphragm action of the foors and roofs of buidings. In this paper, the overa setup of the structura mode, the anaysis procedure, and the theory for the pane zone eement and the pastic hinge beam eement are described. Noninear pane zone shear and beam pastic hinging are incuded. Through geometric updating, the method aows arge noda transations and rotations so as to be abe to foow the response of a buiding into the coapse regime. The theory for the eastofiber beam eement aong with iustrative exampes are given in a companion paper. References Aahabadi, R., and Powe, G. H DRAIN-2DX user guide. Technica Rep. U/EER-88-06, Earthquake Engineering Research enter, Univ. of aifornia, erkeey, aif. ertero, V. V., Popov, E. P., and Krawinker, H eam-coumn subassembages under repeated oading. J. Struct. Div. ASE, 985, arson, A Three-dimensiona noninear ineastic anaysis of stee moment-frame buidings damaged by earthquake excitations. Technica Rep. EERL 99-02, Earthquake Engineering Research Laboratory, aifornia Institute of Technoogy, Pasadena, aif. haa, V. R. M Noninear seismic behavior of stee panar moment-resisting frames. Technica Rep. EERL 92-01, Earthquake Engineering Research Laboratory, aifornia Institute of Technoogy, Pasadena, aif. hopra, A. K Dynamics of structures Theory and appications to earthquake engineering, Prentice-Ha, Engewood iffs, N.J. ook, R. D., Makus, D. S., and Pesha, M. E oncepts and appications of finite eement anaysis, 3rd Ed., Wiey, New York. E-Tawi, S., and Deierein, G. G Ineastic dynamic anaysis of mixed stee-concrete space frames. Technica Rep. Structura Engineering 96-05, orne Univ., Ithaca, New York. Fieding, D. J., and hen, W. F Stee frame anaysis and connection shear deformation. J. Struct. Div. ASE, 991, Fiippou, F., and Romero, M. L Noninear and dynamic anaysis from research to practice. Structura engineering word wide 1998, N. K. Srivatsava, ed., Esevier Science Ltd., New York, T Ha, J. F Parameter study of the response of moment-resisting stee frame buidings to near-source ground motions. Technica Rep. EERL 95-08, Earthquake Engineering Research Laboratory, aifornia Institute of Technoogy, Pasadena, aif. Ha, J. F., and haa, V. R. M eam-coumn modeing. J. Eng. Mech., 12112, Kassimai, A Large deformation anaysis of eastic-pastic frames. J. Struct. Eng., 1098, Kato, eam-to-coumn connection research in Japan. J. Struct. Div. ASE, 1082, Krawinker, H., ertero, V. V., and Popov, E. P Shear behavior of stee frame joints. J. Struct. Div. ASE, 10111, Krawinker, H., and Popov, E. P Seismic behavior of moment connections and joints. J. Struct. Div. ASE, 1082, Krishnan, S Three-dimensiona noninear anaysis of ta irreguar stee buidings subject to strong ground motion. Technica Rep. EERL , Earthquake Engineering Research Laboratory, aifornia Institute of Technoogy, Pasadena, aif. Kunnath, S. K Enhancements to program IDAR: Modeing ineastic behavior of weded connections in stee moment-resisting frames. Technica Rep. NIST GR , uiding and Fire Research Laboratory, Nationa Institute of Standards and Technoogy, Gaithersburg, Md. Kunnath, S. K., Reinhorn, A. M., and Lobo, R IDAR Version 3.0: A program for the ineastic damage anaysis of reinforced concrete structures. Technica Rep. NEER , Nationa enter for Earthquake Engineering Research, Univ. at uffao, State Univ. of New York, uffao, N.Y. Lobo, R IDAR3D: Ineastic damage anaysis of reinforced concrete structures in three dimensions. Technica Rep., Nationa enter for Earthquake Engineering Research, Univ. at uffao, State Univ. of New York, uffao, N.Y. Lobo, R. F., Skokan, M. J., Huang, S.., and Hart, G Threedimensiona anaysis of a 13-story stee buiding with wed connection damage. Structura engineering word wide 1998, N. K. Srivatsava, ed., Esevier Science Ltd., New York, T Mazzoni, S., McKenna, F., and Fenves, G. L Opensees command anguage manua. Technica Rep., opensees.berkeey.edu/, Pacific Earthquake Engineering Research PEER. Oran,. 1973a. Tangent stiffness in pane frames. J. Struct. Div. ASE, 996, Oran,. 1973b. Tangent stiffness in space frames. J. Struct. Div. ASE, 996, Park, Y. J., Reinhorn, A. M., and Kunnath, S. K IDAR: Ineastic damage anaysis of reinforced concrete frame-shearwa structures. Technica Rep. NEER , Nationa enter for Earthquake Engineering Research, State Univ. of New York at uffao. Pinkney, R ycic pastic anaysis of structura stee joints. Technica Rep. U/EER-73-15, Earthquake Engineering Research enter, Univ. of aifornia, erkeey, aif. Popov, E. P., and Petersson, H ycic meta pasticity: Experiments and theory. J. Eng. Mech. Div., Am. Soc. iv. Eng., 1046, Powe, G. H., and ampbe, S DRAIN-3DX eement description and user guide for eement type01, type05, type08, type09, type15, and type17. Technica Rep. U/SEMM-94/08, Structura Engineering Mechanics and Materias, Univ. of aifornia, erkeey, aif. Prakash, V., Powe, G. H., and ampbe, S DRAIN-3DX base program description and user guide, Version Technica Rep. U/SEMM-94/07, Structura Engineering Mechanics and Materias, Univ. of aifornia, erkeey, aif. Przemeniecki, J Theory of matrix structura anaysis, McGraw- Hi, New York. Samon,. G., and Johnson, J. E Stee structures Design and behavior, 2nd Ed., Harper & Row, New York. 358 / JOURNAL OF ENGINEERING MEHANIS ASE / APRIL 2006