Damage-plasticity model for mixed hinges in steel frames

Journal of Constructional Steel Research 67 (2011) 1272–1281

Contents lists available at ScienceDirect

Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr

Damage-plasticity model for mixed hinges in steel frames M.T. Kazemi ∗ , M. Hoseinzadeh Asl Department of Civil Engineering, Sharif University of Technology, P.O. Box: 11155-9313, Iran

article

info

Article history: Received 25 August 2010 Accepted 6 March 2011 Keywords: Mixed link element Shear-flexural interaction Multi-surface plasticity Deformation space Damage-plasticity Softening

abstract A damage-plasticity based mixed axial-shear-flexural (PVM) link element for the inelastic analysis of frames is introduced in this paper. The multi-surfaces yield concept is utilized in the definition of the element. The yield surfaces are defined in deformation space and interaction of axial-shearflexural deformations is considered by defining non-rectangular yield surfaces. The element is capable of considering damage and post-peak softening behavior. The analytical results of introduced element are verified by existing experimental results of steel beam to column connection and it is indicated that the analytical results have reasonable agreement with the test results. Nonlinear dynamic analyses are performed on moment frames with the verified element in two cases: with and without consideration of damage. The results exhibit that damage consideration considerably changes the results. The effect of consideration of the interaction between shear deformation and flexural deformation at the ends of beams of frames under lateral and gravity loads is also investigated. The results for the long beams show that, although in the stage of small plastic deformations the shear-flexural interaction is negligible, for the extensive plastic deformation near the failure point, it could be decisive. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The common approach of modeling the inelastic behavior of the frame elements is to adopt inelastic lumped hinge formation in the special locations of the members, where nonlinear behavior is expected. One way to increase the accuracy of the predictions of the plasticity hinge models is to use more than one yield surface with the concept of multi-surface plasticity. The multi-surface plasticity models allow the use of additional parameters by adding extra yield surfaces by which a close fit of the model to the real material behavior is possible. Mroz [1] and Iwan [2] initiated theories of so-called multisurface plasticity. Most of the experimental studies of the yield behavior has been conventionally conducted in stress space (opposed to strain space), and the yield surfaces were initially defined in stress space. However, the shortcoming of stress space driven loading criteria to include softening behavior led the researchers to propose strain space plasticity where the yield surfaces are defined in strain space [3–5]. The yield surfaces in the stress space can be transformed into the strain space, which has been recognized as more convenient for modeling the softening behavior in materials. The concept of multi-yield surfaces can be generalized to force or deformation space. The difference is that the stress or strain

∗

Corresponding author. Tel.: +98 21 66164237; fax: +98 21 66014828. E-mail addresses: [email protected] (M.T. Kazemi), [email protected] (M.H. Asl). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.03.010

space plasticity is intended to model the ‘‘material’’ nonlinearity, while the force or deformation space plasticity describes the structural ‘‘member’’ nonlinearity. In the most of the works done with the latter case, the yield surfaces are defined in force space and it is common to define a lumped plastic hinge in the locations which may include internal axial, shear, flexural or torsional nonlinearity. A generalized plastic hinge element accounting for interaction for axial, torsional and biaxial bending moments and based on a multi-surfaces plasticity concept was presented by Powell and Chen [6]. Krenk et al. [7] by utilizing a piecewise linearized yield surface and linear kinematic hardening rule, developed a formulation for displacement discontinuities with extension and rotation components. A method for modeling the members with yielding under combined flexure and axial force in steel frames subjected to earthquake ground motions was also presented by Kim and Engelhardt [8]. Their method had the capability of modeling plastic axial deformation and changes in axial stiffness based on isotropic and kinematic strain-hardening defined in axial-flexural space. Liew and Tang [9] applied the two surface plasticity concept for considering the inelastic interaction between axial force and bending moment as well. Considering 2D frames and forces, the yield surfaces are more commonly defined in: 1-VM (shear-moment) space for beam elements; 2-PM (axial force-moment) space for beam–column elements; 3-PVM (axial-shear-moment) space for beam column element. The VM space is used for the cases where combined shear-flexural yielding of the beam is the governing behavioral mode. Ricles and Popov [10] developed a formulation for modeling link elements in eccentrically braced frames (EBFs) based on a

M.T. Kazemi, M.H. Asl / Journal of Constructional Steel Research 67 (2011) 1272–1281

1273

Fig. 1. (a) The element’s end forces and displacements, (b) The element’s internal forces and deformations.

multi-surfaces plasticity concept. In their work, the link beam has a nonlinear hinge at each end and each hinge consists of uncoupled shear and flexural nonlinear sub-hinges. Ramadan and Ghobarah [11] utilized the separate shear and flexural nonlinear hinges and investigated the link element of EBFs as well. In their work, kinematic hardening for flexural and isotropic hardening for shear was employed. The PM space is typically used for modeling of beam–column elements in well-known nonlinear static and dynamic analysis programs such as SAP2000 [12]. A beam–column element for modeling the inelastic cyclic behavior of steel braces is introduced by Jin and EI-Tawil [13], in which bounding surface plasticity model in PM space has been used to represent inelastic brace buckling. The PVM space is also studied by some researchers. Elmandooh and Ghobarah [14] applied the multi-surfaces hinge concept and proposed an element to model the biaxial flexural and shear behaviors of RC columns subjected to varying axial load. Liu et al. [15] derived a failure surface accounting for the combined action of bending moment, shear force, and axial force based on the principle of maximum plastic strain energy. In the present paper a general PVM link element, considering the interaction between axial-shear and flexural deformations, is introduced for steel frames. The multi-surfaces concept with dissimilar yield surfaces is utilized. The yield surfaces are defined in deformation space. With this definition the modeling of softening behavior is also easily achieved by defining negative values for the flexibility matrix. The softening behavior in steel frames is mainly due to local buckling of structural members, when they are subjected to overloads. The proposed link element has also the capability of modeling combined axial-flexural-shear yielding and damage (stiffness degradation). The proposed element is an extension of a previously developed element by Kazemi and Erfani [16], which was defined in force space without damage or softening consideration. A summary of the previous element and some of its applications is also presented in [17,18].

displacements are described as column vectors of Ui = [ui , vi , θi ]T and Uj = [uj , vj , θj ]T in the element’s local coordinate system as presented in Fig. 1(a). Fig. 1(b) shows the element’s internal forces and deformations in the inner inelastic hinge node. The element’s internal forces and deformations in the inner inelastic hinge node are defined as column vectors of Ph and Uh , respectively. Subsequently, one could write [19]:

2. Definition of PVM link element

If the rate of deformations and forces in the inner hinge of the ˙ h and P˙ h , respectively, one could write: element are shown as U

The PVM link element includes one inner inelastic combined axial-shear-rotational hinge with zero length and two rigid parts with nonzero lengths in two sides of it. Geometrical presentation of this element has been shown in Fig. 1, where ‘‘i′′ and ‘‘j′′ are the outer nodes and ‘‘h′′ is the inner hinge. The inner hinge has an arbitrary location. The lengths of two rigid parts are Li and Lj , and L = Li + Lj is the total length of the element. The element will be more representative of inelastic zone if we take L equal to the length of inelastic zone. The properties of the PVM link element are defined such that relative deformations between two ends of it to be equal to the relative deformations between two ends of inelastic zone in real. The element’s end forces are described as column vectors of Pi = [Ni , Vi , Mi ]T and Pj = [Nj , Vj , Mj ]T and the element’s end

˙ h = Fh P˙ h . U

[ ]T [ ] Uh =

Ai Aj

Ui , Uj

[ ] Pi Pj

[ ] =

Ai Ph Aj

(1)

where A is the transformation matrix and its components depend only on lengths of rigid parts of the element and are as follows:

−1 [ ] 0 0 Ai A= = 1 Aj  0 0

0 −1 −Li 0 1 −L j

0 0  −1  . 0   0 1



(2)

In order to relate the flexibility matrix of the inner hinge to the flexibility matrix of the element assuming a fixed end condition for one of the end joints of the element and using Eq. (1), one could write:

[ ]T [ ] Uh =

Ai Aj

0 Uj

= ATj Fj Aj Ph

= ATj Uj = ATj Fj Pj (3)

where Fj is the element flexibility matrix corresponding to node j. It is assumed that no loads and masses are assigned to the hinge and the rigid parts except at the end nodes. Considering Eq. (3), the inner hinge tangential flexibility matrix, Fh , is as follows: Fh = ATj Fj Aj .

(4)

(5)

The components of tangential flexibility matrix of inner hinge, Fh , depend on applied deformations and the loading history. 3. Yield surfaces To consider the interaction between axial, shear and flexural deformations the multi-yield surfaces concept in deformation space is utilized. The axial-shear-rotational deformation space has been shown in Fig. 2. It should be noted that the yield surfaces are traditionally reported and formulated in stress or force space and there is an extensive literature on the experimental deformation of yield surfaces in stress or force space [6–8,10,18,20].

1274

M.T. Kazemi, M.H. Asl / Journal of Constructional Steel Research 67 (2011) 1272–1281

Fig. 3. Typical yield surfaces in the shear-rotation deformation space, (a) Elastic behavior, (b) Elasto-plastic behavior.

Fig. 2. Typical yield surfaces in the axial-shear-rotation deformation space.

However, when the member shows strain softening beyond the peak or initial yield surface, the use of deformation space will ease the definition of the loading and unloading criteria [3,5,21]. On the other hand, when the member deforms at perfect plastic deformation, defining the yield surface in deformation space may be more efficient, since the yield surfaces in force space remain unaffected during plastic deformation. It is assumed that the yield surfaces are convex and could be translated and changed in size. If the action point is internal to the initial yield surface, the behavior is elastic as in Fig. 3(a); when it is on each of the surfaces, the behavior will be elasto-plastic as in Fig. 3(b). In tangency of several yield surfaces, the outer surface properties will govern the element behavior. The yield surfaces may be dissimilar to each other and consequently, they will intersect as the inner surface moves toward the outer surface as shown in Fig. 4(a). The similarity assumption has been utilized in the most works in order to have parallel normals for the corresponding intersecting points on the yield surfaces. In that case, they will not intersect and asymptotically will be tangential to each other. If it is assumed that in the PVM space the yield surface i is similar to the yield surface j we will have: uihy j

=

uhy

i vhy j vhy

=

i θhy

(6)

j θhy

i i where uihy , vhy and θhy , are pure axial, shear and rotational deformation points on yield surface i (see Fig. 2). In general, this may not be a reasonable assumption. Ricles and Popov [10] did not use the similarity assumption of the yield surfaces by choosing rectangular yield surfaces in shear-flexural space with different ratios of length to width. By utilization of rectangular yield surfaces, the interaction of shear–flexure is ignored, practically. In the present research, the surfaces can be dissimilar. When the inner surface i interacts with the yield surface j, it is assumed that the border of inner surface is cut by the outer surface in the interaction region (Fig. 4(a)). As the inner surface i moves back inwards, it changes to its initial shape as shown in Fig. 4(b). A full discussion of intersecting yield surfaces in stress space is given in [22]. The yield function of surface ‘‘i’’ is assumed to be as:

  i  u Iu h

 uihy Iui , Iθi , Iui θ

 + and

θh i θhy

Iθi Iui θ  

Ivi ≥ 1

 +

vh i vhy

Ivi =1

Fig. 4. Interaction of the yield surfaces, (a) Inner surface i interacts with outer surface j, (b) The inner surface regains its initial shape.

where Iui , Iθi , Iui θ , and Ivi are the interaction parameters which are member dependant parameters. For each yield surface ‘‘ i ’’ we i i have 7 parameters, including Iui , Iθi , Iui θ , Ivi , uihy , vhy , and θhy , to be determined. The procedure for estimating these parameters will be explained in Section 8. A larger value for interaction parameters means less interaction between the deformation components of the hinge. Different values of interaction parameter can be assumed for different yield surfaces. The inner hinge flexibility matrix, between the two consecutive yield surfaces can be decomposed as: p

Fh = Feh + Fh Feh

(8) FPh

where and are the elastic and plastic tangential flexibility matrices of the inner hinge, respectively. For definition of FPh , the flow rule proposed by Kazemi and Erfani [16] is used: FPh = m2u FPhu + m2v FPhv + m2θ FPhθ

(9)

where mu , mv , and mθ are the components of vector m in PVM space which is the unit location vector of action point with respect to the center of active yield surface (Fig. 3(b)). FPhu , FPhv , and FPhθ are the plastic flexibility matrices related to the pure axial, pure shear and pure rotational deformations, respectively. 4. Hardening rule The hardening rule defines the manner by which the yield surface changes. For all surfaces, translation (kinematic hardening) and also change in size (isotropic hardening) are permitted. On this premise, the ith yield function is written as follows:

φi (Uh − αi , Hi ) = 0

(10)

where φi is the ith yield function of the inner hinge, Uh is the action vector of inner hinge, αi is the ith yield function center (column vector), and Hi is the expansion matrix of the ith yield surface for the consideration of isotropic hardening and is defined as:

(7)

hu 0 0

 Hi =

0 hv 0

0 0 hθ

 (11)

M.T. Kazemi, M.H. Asl / Journal of Constructional Steel Research 67 (2011) 1272–1281

1275

Fig. 5. Action point and corresponding points on two adjacent yield surfaces.

where hu , hv , and hθ are expansion parameters for axial, shear and flexural yield deformations, respectively. In the present paper the element formulation is based on combined kinematic and isotropic hardening for the shear deformation and only kinematic hardening for flexure and axial deformations (in Eq. (11) hu = hθ = 1, and hv ≥ 1) The shear isotropic hardening is adopted from Ramadan and Ghobarah’s work [11]:

vyie = hv vyi

(12)

where vyi and vyie are the initial and developed values of the shear deformation at the ith yielding surface, respectively. hv is written as: hv = 1 + C1 (1 − exp(C2 pv ))

(13)

where C1 and C2 are material constants and pv represent the accumulated plastic deformation of the hinge associated to shear  deformation (i.e. pv = |dupv |). Various kinematic hardening rules are utilized in the research which are generally defined in force space [8,10,11]. In this paper, the kinematic hardening rule, similar to the one introduced by Kazemi and Erfani [16] is introduced and applied. As is observed in Fig. 5, assume that the action location, Uh , is on the ith yield surface and plastic loading is occurring. The rate of the ith yield surface translation, α˙ i , is assumed as being:

α˙ i = (Uhi+1 − Uhi )µ ˙

(14)

where Uhi+1 is the intersection point between the direction of the action rate U˙ h , and the (i + 1)th yield surface, and Uhi is the conjugate point of Uhi+1 on the ith yield surface. By this definition when the action point Uh , approaches closer to the (i + 1)th yield surface, the ith yield surface moves such that the two points on the ith yield surface, Uhi and Uh approach closer to each other and coincide with Uhi+1 asymptotically. With this assumption the probability and severity of surfaces’ intersection reduces. To calculate µ ˙ , the plastic loading condition is employed:

φ˙ i = 0.

(15)

Hence:

∂ϕi ˙ ∂ϕi ∂ϕi ˙ · Uh + · α˙ i + · Hi = 0 ∂ Uh ∂αi ∂ Hi

(16)

Fig. 6. Loading/unloading criterion.

5. Loading-unloading criterion Assuming that the action point of deformation is on the yield surface, the loading-unloading criterion is expressed as:

 λ = n · U˙ h > 0 ⇒ plastic loading λ = n · U˙ h = 0 ⇒ neutral loading  λ = n · U˙ h < 0 ⇒ elastic unloading

where n is normal vector of the active yield surface on the action point as in Fig. 6. Since the yield surfaces are defined in deformation space, the loading-unloading criteria of Eq. (20) holds in general, irrespective of whether the element has a hardening or softening behavior. This helps to mitigate the numerical problems of cyclic severe loading analyses, where on the final cycles the steel members may show softening behavior, especially due to local bucking and damage of their components. 6. Damage and stiffness degradation It is well known from experimental evidence that any material deteriorates as a function of the loading history. Every inelastic excursion causes damage and the damage accumulates as the number of excursions increases. Therefore, it is necessary to include degradation effects in modeling hysteretic behavior. The PVM element is capable of considering stiffness degradation. Stiffness degradation refers to the increase of flexibility (decrease of stiffness) as a function of the deformation history. Considering Eq. (4), the inner hinge flexibility matrices between the two consecutive yield surfaces for the pure shear and pure flexural deformations are assumed as follows: pure

Fh

(18)

By pre-multiplying both sides of Eq. (14) by ∂ϕi /∂ Uh , and using the Eq. (18), one reaches the following:

µ ˙ =

∂ϕi ∂ Uh ∂ϕi ∂ Uh

· U˙ h +

∂ϕi ∂ Hi

· H˙ i

· (Uhi+1 − Uhi )

.

pure Fj

f11

 k1 (1 − duf )2   0 =    0

Eq. (16) can be written as:

(19)

(21) pure



(17)

∂ϕi ˙ ∂ϕi ˙ ∂ϕi · Uh + · Hi = · α˙ i . ∂ Uh ∂ Hi ∂ Uh

Aj = ATj Fpure j

where Aj is defined as Eq. (2). Fj is the element’s flexibility matrix corresponding to node j for the pure shear or pure flexural deformation. In order to consider the increase in the element’s pure flexibility matrix, Fj is defined as [23]:

where (.) presents inner product of two vectors. And since one has:

∂ϕi ∂ϕi =− . ∂ Uh ∂αi

(20)

0 f22 k2 (1 − dv f )2 f32 k3 (1 − dv f ) (1 − dθ f )

0



    (22) k3 (1 − dv f ) (1 − dθ f )    f33 f23

k4 (1 − dθ f )2

where f11 , f22 , f23 = f32 , and f33 are the elastic flexibility parameters of the element which may be obtained from classic analyses or formulas or finite element analysis results. k1 , k2 , k3 , and k4 are the hardening coefficients, which represent the stiffness ratio between each two yield surfaces. For the elastic cases (Fig. 3(a)), k1 = k2 = k3 = k4 = 1 is utilized. duf , dv f and dθ f are damage parameters associated to the stiffness degradation due to the axial deformation, shear deformation and flexural

1276

M.T. Kazemi, M.H. Asl / Journal of Constructional Steel Research 67 (2011) 1272–1281

Fig. 7. Stiffness degradation, (a) Yield surfaces are defined in deformation space, (b) Yield surfaces are defined in force space.

deformation respectively. These variables can take values between 0 (no damage) and 1 (total damage) and are equal to: duf = 1 − exp(−su pu ) 0 ≤ duf < 1 dv f = 1 − exp(−sv pv ) 0 ≤ dv f < 1 dθ f = 1 − exp(−sθ pθ ) 0 ≤ dθ f < 1

Fig. 8. The connection test setup carried out by Ricles et al. [26], dimensions are in mm.



(23)

where pu , pv and pθ represent the accumulated plastic deformation of the hinge associated to axial, shear and flexural deformations,  p p respectively. (i.e. pu = |duu |, pv = |dupv | and pθ = |duθ |). su , sv and sθ are the stiffness degradation parameters associated to axial, shear and flexural deformations respectively and can be determined based on the experimental or finite element analysis results. One should note that since the yield surfaces are defined in deformation space, an increase in flexibility values will result in a yield strength decrease as in Fig. 7(a). In another words, consideration of stiffness degradation in deformation space implicitly means the consideration of strength degradation. Fig. 7(b) shows the consideration of stiffness degradation when the yield surfaces are defined in force space where there is no strength p p p degradation. After calculating Feh , Fhu , Fhv , and Fhθ using Eq. (21), the total flexibility matrix, Fh , is obtained from Eqs. (8) and (9).

Fig. 9. Loading protocol [26].

arranged in beams or special elements with 2D deformations, which means 2D version will be adequate for modeling of the most steel structures.

7. Computer modeling 8. Verification of the link element with existing test results Based on the element model just described, a computer program has been developed. The computer code is written and complied with in Visual C++ programming language [24]. The program is capable of running static and dynamic nonlinear analyses on planner frame structures with elastic and inelastic elements. The flexibility matrix of nonlinear element, which is obtained based on experimental or refined FEM analysis, contains local large deformation effects like local buckling. The tangent stiffness matrix of the structure is constructed based on updated nodal coordinate to consider large deformation effects. For elastic elements, the geometric stiffness matrix is also considered. The resulting response of the frame was calculated using a timestepping technique employing the Newmark-beta method [25] to solve the governing equations of dynamic equilibrium coupled with Arc-Length iteration method for carrying out the nonlinear analyses. The program also considers the classical Rayleigh damping matrix. It should be noted that the introduced 2D element in shearflexural-axial space can be extended to a 3D element with six deformational components of one axial, two shears, two flexural rotations and one twist. The transformation matrix of Eq. (2) will change to a 12 by 6 matrix. A yield surface, similar to the one given in Eq. (7) but including 6 deformational components, should be also considered. Increasing the deformational components of the element will increase the required number of tests or finite element analyses to capture the interaction parameters of the different deformational components. In new practical design of ductile structures, the potential inelastic hinge locations are

Here, test results from a previous investigation by Ricles et al. [26] comprising an improved pre-Northridge welded-flangebolted-web, beam to column connection is utilized for verification. Experimental studies were performed at Lehigh University to verify the effect of the weld access hole modifications and the continuity plate effect, which show that the improved weld access hole detail significantly increased the ductility of the connection. The results of the two of these tests, specimens C3 and C4 in their research have been used here. The configuration of specimen C3 is the same as specimen C4 with the continuity plates removed in specimen C3. Test setup for the connection is shown in Fig. 8 where an interior connection exists between two beams and a column. The test was conducted by applying cyclic variable amplitude displacement at the top of the column. The cyclic loading protocol is shown in Fig. 9. The model has been analyzed by means of the proposed link element. As in the Fig. 10 the beams and columns are divided into two segments of elastic beam and inelastic link element. The inelastic elements are considered as PVM and VM link elements for columns and beams, respectively. The link element model of the beams includes the panel zone. The VM link element of the beam only considers the shear-flexural interaction, while the PVM link element of the column considers the axial, shear and flexural interaction. An experimental or numerical approach is required to obtain complete and precise hysteretic behavior and yield surfaces of the link elements. In the present study, the yield surfaces properties

M.T. Kazemi, M.H. Asl / Journal of Constructional Steel Research 67 (2011) 1272–1281

1277

Fig. 10. Modeling of the existing test configuration with PVM link element (Dimensions in mm).

of the elements are captured by running inelastic finite element analyses. To accomplish this purpose, the link element of the beam has been modeled by a general purpose finite element program and is analyzed separately for pure shear deformation, pure flexural deformation, and a combination of shear-flexural deformations with different ratios of moment to shear deformation. In pure shear deformation the rotation at inner hinge, θh , is zero. For pure flexural deformation the shear deformation at inner hinge, vh , is zero (see Fig. 11). Then the interaction parameters (Iui , Iθi , Iui θ , Ivi ) i i and the yield points (uihy , vhy , θhy ) introduced in Eq. (7), and also flexibility parameters (f11 , f22 , f23 = f32 , f33 ) and the hardening coefficients (k1 to k4 ) of Eq. (22) will be determined based on the iteration and comparison of the curves obtained by the finite element analyses with those obtained by the proposed model. The parameters of an element can be used to predict the parameters of similar elements, approximately. The stiffness degradation parameters (su , sv , sθ ), of Eq. (23), can be extracted with a cyclic loading test or finite element analysis. In this paper, to simplify the approach, only pure shear and pure flexural loading are considered, separately. In the present form, yield surfaces are symmetric about the axial, shear, and flexural axes and the tension–compression behavior of an element is assumed to be similar. This means that the proposed nonlinear element cannot model the post buckling behavior of column or bracing members, when they suffer both cyclic compression and tension deformations. For the columns of moment resisting frames, which they yield only in compression, the post buckling behavior can be considered. The resulting yield surfaces for inner hinge of the VM link elements of beams are simplified as in Figs. 12 and 13, respectively. Fig. 14 shows the yield surfaces for PVM link element of column, which have been extracted in the same way as for VM element of beams, with the difference that the axial deformation interaction is also involved. The values of the elastic flexibility parameters, defined in Eq. (22), for link elements of specimens C3 and C4 are presented

Fig. 12. Yield surfaces of VM link element for specimen C3.

Fig. 13. Yield surfaces of VM link element for specimen C4.

in Table 1 and the values of hardening coefficients and stiffness degradation parameters, introduced in Eqs. (22), and (23) are presented in Tables 2 and 3. The experimental and PVM link results for column top displacement of specimens C3 and C4 are compared in Figs. 15 and 16, respectively. As shown, the results for the PVM link element have a satisfactory agreement with the experimental results.

Fig. 11. Yield surfaces extraction by finite element modeling.

1278

M.T. Kazemi, M.H. Asl / Journal of Constructional Steel Research 67 (2011) 1272–1281

Fig. 15. Comparison between PVM link element results and experimental results for specimen C3.

Fig. 14. Yield surfaces of PVM link element for modeling of column link element.

Table 1 Elastic flexibility parameters. Element

f11 (m/N)

f22 (m/N)

f23 = f32 (1/N)

f33 (1/mN)

VM element of specimen C3 VM element of specimen C4 PVM element of specimens C3&C4

3.190E−10

3.651E−09

1.833E−09

2.439E−09

3.090E−10

3.519E−09

1.790E−09

2.381E−09

9.091E−11

9.848E−10

4.545E−10

9.091E−10

Fig. 16. Comparison between PVM link element results and experimental results for specimen C4.

Table 2 Hardening coefficient and damage parameters of VM element of specimen C3. Yield Surf.

Specimen C3

Specimen C4

0–1 1–2 2–3 3–4 4–5 0–1 1–2 2–3 3–4 4–5 5–6

Shear deformation

sv = sθ

Flexural deformation

k2

k3 = k4

k2

k3 = k4

0.4204 0.0858 0.0347 −0.0317 −0.0005 0.3942 0.0802 0.0344 0.0034 −0.0476 −0.0024

0.6829 0.1707 0.0634 −0.0317 −0.0005 0.5952 0.1429 0.0714 0.0014 −0.0476 −0.0024

0.6822 0.1442 0.0429 −0.0317 −0.0005 0.6292 0.1320 0.0444 0.0068 −0.0476 −0.0024

0.6829 0.1707 0.0317 −0.0317 −0.0005 0.5952 0.1429 0.0357 0.0029 −0.0476 −0.0024

0 0.1 0.35 0.4 0.5 0 0 0.1 0.3 0.4 0.4

Fig. 17. Comparison between the results of PVM link element with no damage and experimental results for specimen C3.

In order to study the effect of damage consideration on the results, the connections have been remodeled with PVM link elements without consideration of damage in which the degradation parameters are set to zero. The analytical results are compared with experimental results in Figs. 17 and 18 for specimens C3 and C4, respectively. As can be seen in Figs. 16 and 17, at the initial cycles there is good agreement between analytical and experimental results, while at the last cycles with larger inelastic deformations there is considerable difference between analytical and experimental

Fig. 18. Comparison between the results of PVM link element with no damage and experimental results for specimen C4.

Table 3 Hardening coefficients and damage parameters of PVM element of both specimens. Surf.

0–1 1–2 2–3 3–4

Axial def.

Shear def.

sv = sθ

Flexural def.

k1

k2

k3 = k4

k1

k2

k3 = k4

k1

k2

k3 = k4

0.500 0.036 0.030 0.018

0.139 0.045 0.037 0.019

0.545 0.118 0.055 0.018

0.750 0.065 0.054 0.033

0.142 0.047 0.041 0.015

0.818 0.236 0.109 0.009

0.750 0.065 0.054 0.033

0.139 0.045 0.037 0.019

0.545 0.118 0.055 0.018

0 0 0.1 0.3

M.T. Kazemi, M.H. Asl / Journal of Constructional Steel Research 67 (2011) 1272–1281

1279

Fig. 19. (a) Plan of the building, (b) Elevation of the frames A and D.

results. This indicates the necessity of damage consideration for the inelastic analyses of such connections. 9. Study of seismic behavior of frames employing the mixed PVM link element In order to compare the seismic behavior of connection types C3 and C4 and further to study the effect of consideration of damage in the model, a four-story building was studied. The plan and longitudinal elevation of the building are shown in Fig. 19. The building has a plan dimension of 54 m by 27 m and a story height of 4 m for each floor. In the longitudinal direction, the interior frames B and C are non-moment resisting frames designed to carry only the gravity loads associated with their tributary areas. The perimeter moment resisting frames A and D resist the lateral forces in the longitudinal direction and provide lateral stability for the entire structure in longitudinal direction. The dead and live load for all stories are assumed to be 5.0 kN/m2 and 4.79 kN/m2 , respectively. The seismic dead weight is assumed to be 1.0(5.0) + 0.25(4.79) = 6.198 kN/m2 . As mentioned in the Section 2, for the inelastic elements, it is assumed that no loads or masses are assigned to the element except at the two end nodes. For the elastic beam and column elements distributed or lumped mass can be applied. The program uses a finite element approach with the application of shape functions to transfer and lump the applied loads and masses to the end nodes of elastic elements. The mass matrix includes both translational and rotational components. This study concentrates only on the behavior of the building in the longitudinal direction. Thus, the investigation concentrated on frames A and D. The beams are W36 × 150 girders, and the columns are W27 × 258 in all stories as for specimens C3 and C4. The building was subjected to the El Centro earthquake. The record was scaled up by a factor of 3 to a peak ground acceleration of 1 g. The damping matrix C is assumed to be proportional to the mass matrix M of the building as follows: C = aM

Fig. 20. Comparison of the roof’s lateral displacements.

(24)

where a is determined based on a damping ratio of 5% of critical damping from the first elastic mode of the building. Four models are utilized for the study of the frame. In the first model, indicated by D-C3, the beam to column connections are the same as specimen C3 (without continuity plates) and the PVM link element is employed for modeling the beams and columns. The second model, indicated by D-C4 is the same as D-C3 but the connections are as specimen C4 (with continuity plates). The third model, indicated by ND-C3 is the same as D-C3, with the difference that no damage has been considered in PVM link elements. The third model, indicated by ND-C3 is the same as DC3, with the difference that no damage has been considered in PVM link elements. Fig. 20 compares the time history of roof lateral displacement for analyzed frames. As can be observed, there is significant difference between the displacement response of frames with damage consideration (shown with solid lines in Fig. 20) with those of frames without damage consideration. It is further noted

Fig. 21. Comparison of the deformed shape of frames, (a) Frame D-C3 at 11.4 s of analysis; at failure time, (b) Frame ND-C3; residual configuration, (c) Frame D-C4; residual configuration, (d) Frame ND-C4; residual configuration.

that the D-C3 frame, which has no continuity plates, has been unstable due to excessive damage on the 12 s of analysis. The frame D-C4, which has continuity plates, has suffered less damage in comparison with frame D-C3. Fig. 21 shows the deformed shapes of the end nodes of the elements of frames at the end of the analysis. For the

1280

M.T. Kazemi, M.H. Asl / Journal of Constructional Steel Research 67 (2011) 1272–1281

Fig. 22. Translation history of the yield surfaces of the inner hinge VM1 of frame D-C4, (a) 2.34 s, (b) 5.46 s, (c) 9.91 s, (d) 10.01 s.

deformed shapes of the inelastic link elements the two end nodes’ deformations has been depicted for simplicity. The real translation of nodes of one of the inelastic elements has been shown in Fig. 21(c) (element VM1 in the picture). Comparison of analytical results for damage considered models (D-C3 and D-C4) with models without consideration of damage (ND-C3 and ND-C4) in Figs. 20 and 21 shows that ignoring the damage effect will seriously change the analysis result. It should be noted that this difference depends on the extent of nonlinearity of members under applied loads. As mentioned before the applied record was scaled up to a peak ground acceleration of 1 g which is a relatively high acceleration causing more damage and nonlinearity on the frames. The gravity load applied on the beams reduce their shear capacity and as it can be seen in Fig. 21 with the increase of plastic deformations in some beams they tend to move down due to shear plastic deformation of the VM link elements. This issue is illustrated more explicitly in Fig. 22 which shows the translation history of the yield surfaces of the element VM1, indicated in Fig. 21(c). As can be seen, at the initial seconds of analysis, the outer surfaces have not been passed and the predominant plastic deformation is due to flexural deformations, as expected for beams of that length. In Fig. 22(d), the link element begin to suffer large plastic deformations and shear plastic deformations are increasing. When the outer surfaces of element are passed the element stiffness decreases and with the shear capacity reduction, the gravity loads are strong enough to initiate large shear deformations. The above discussion indicates that, for small plastic deformations, the shear-flexural interaction in inelastic elements of relatively long beams need not to be considered. But it can be important when we are studying the failure behavior of such beams.

10. Conclusion A general PVM link element with the capability of modeling combined axial-shear-flexural yielding and damage is defined. The multi-surfaces concept with dissimilar yield surfaces is utilized. The yield surfaces are defined in deformation space and modeling of softening behavior can be easily achieved by defining negative values for the flexibility matrix. The analytical results of the model were compared with some existing experimental results and close agreements were observed. Nonlinear dynamic time–history analysis is done on a four-story building consisting of perimeter moment resistant frames. The seismic response of the frames analyzed with and without damage consideration is compared which shows the significant effect of damage consideration on seismic response. The seismic response of the frames analyzed with and without continuity plates in the panel zone is further compared. The results show that in the last seconds of analysis, where damage increases due to accumulation of plastic deformations, the frame with continuity plates has better performance. The results also show that at the initial stages of plastic deformations of beams, where just inner yield surfaces have been passed and the element’s stiffness is high enough, the shear plastic deformations are negligible (as expected) and shear-flexural interaction need not to be considered. When the beam elements bear large plastic deformations and the outer yield surfaces of element are passed, the elements stiffness decrease. In these final stages, with the decrease of shear capacity, the element suffers large shear plastic deformations (due to gravity load effect) and the consideration of shear-flexural interaction is needed. References [1] Mroz Z. On the description of anisotropic work-hardening. J Mech Phys Solids 1967;15:163–75.

M.T. Kazemi, M.H. Asl / Journal of Constructional Steel Research 67 (2011) 1272–1281 [2] Iwan WD. On a class of models for the yielding behaviour of continuous and composite systems. J Appl Mech 1967;34:612–7. [3] Casey J, Naghdi PM. On the nonequivalence of the stress space and strain space formulation of plasticity theory. J Appl Mech, ASME 1983;50:350–4. [4] Yoder RJ, Iwan WD. On the formulation of strain–space plasticity with multiple loading surface. J Appl Mech, ASME 1981;48:773–8. [5] Naghdi PM, Trapp JA. The significance of formulating plasticity theory with reference to loading surfaces in strain space. Int J Eng Sci 1975;13:785–97. [6] Powell GH, Chen PFS. 3D beam–column element with generalized plastic hinges. J Eng Mech, ASCE 1986;112:627–41. [7] Krenk S, Vissing S, Vissing -JC. A finite step updating method for elastoplastic analysis of frames. J Eng Mech, ASCE 1993;119:2478–95. [8] Kim KD, Engelhardt MD. Beam–column element for nonlinear seismic analysis of steel frames. J Struct Eng, ASCE 2000;126:916–25. [9] Liew JYR, Tang LK. Advanced plastic hinge analysis for the design of tubular space frames. Eng Struct 2000;22:769–83. [10] Ricles JM, Popov EP. Inelastic link element for EBF seismic analysis. J Struct Eng, ASCE 1994;120:441–63. [11] Ramadan T, Ghobarah A. Analytical model for shear-link behavior. J Struct Eng 1995;121:1574–80. [12] CSI computer & structures inc. SAP2000. Linear and nonlinear static and dynamic analysis of three-dimensional structures. Berkeley (CA). 2009. [13] Jin J, El-Tawil S. Seismic performance of steel frames with reduced beam section connections. J Constr Steel Res 2005;61:453–71. [14] ElMandooh Galal K, Ghobarah A. Flexural and shear hysteretic behaviour of reinforced concrete columns with variable axial load. Eng Struct 2003;25: 1353–67.

1281

[15] Liu Y, Xu L, Grierson DE. Combined MVP failure criterion for steel crosssections. J Constr Steel Res 2008;65:116–24. [16] Kazemi MT, Erfani S. Mixed shear-flexural (VM) hinge element and its applications. Scientia Iranica 2007;14:193–204. [17] Kazemi MT, Erfani S, Hoseinzadeh Asl M VM. shear-flexural, link element for seismic analysis of steel frames. In: Proc. US–Iran seismic workshop on improving earthquake mitigation through innovations and applications in seismic science, engineering, communication, and response, PEER report, 2009/02, p. 229–38. [18] Kazemi MT, Erfani S. Special VM link element for modeling of shear–flexural interaction in frames. Struct Des Tall Special Build 2009;18:119–35. [19] Kazemi MT, Erfani S. Analytical study of special girder moment frames using a mixed shear-flexural link element. Can J Civ Eng 2007;34:1119–30. [20] Mroz Z. An attempt to describe the behavior of metals under cyclic loads using a more general work hardening model. Acta Mech 1969;7:199–212. [21] Khan AS, Huang S. Continuum theory of plasticity. New York: John Wiley & Sons. Inc.; 1995. [22] Puzrin AM, Houlsby GT. On the non-intersection dilemma in multiple surface plasticity. Geotechnique 2001;51:369–72. [23] Florez-Lopez J. Frame analysis and continuum damage mechanics. Eur J Mech A/Solids 1998;17:269–83. [24] Microsoft visual C++ 2005. Microsoft visual studio 2005; Ver. 8.0.50727.42. [25] Newmark NM. A method of computation for structural dynamics. J Eng Mech 1959;85:67–94. [26] Ricles JM, Mao C, Lu LW, Fisher J. Development and evaluation of improved details for ductile welded unreinforced flange connections. Report No. SAC BD 00-24. Sacramento (California). 2000.