A Ductility-Centred Analytical Model for Axially Restrained Double-span Steel Beam Systems Subjected to Sudden Columns Loss

    A Ductility-Centred Analytical Model for Axially Restrained Double-span Steel Beam Systems Subjected to Sudden Columns Loss Haoran Fu, Jinfan Zhang, Jianqun Jiang, Zhenyu Wang PII: DOI: Reference:

S2352-0124(16)30081-9 doi: 10.1016/j.istruc.2016.09.013 ISTRUC 156

To appear in: Received date: Revised date: Accepted date:

24 March 2016 20 September 2016 22 September 2016

Please cite this article as: Fu Haoran, Zhang Jinfan, Jiang Jianqun, Wang Zhenyu, A Ductility-Centred Analytical Model for Axially Restrained Double-span Steel Beam Systems Subjected to Sudden Columns Loss, (2016), doi: 10.1016/j.istruc.2016.09.013

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ACCEPTED MANUSCRIPT A ductility-centred analytical model for axially restrained double-span

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steel beam systems subjected to sudden columns loss

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Haoran Fu, Jinfan Zhang, Jianqun Jiang, Zhenyu Wang*

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(College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, Zhejiang 310058,China）

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Date of Submission: March 2016.

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Corresponding author: Prof. Zhenyu Wang

Address: College of Civil Engineering and Architecture, Zhejiang University,

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Yuhangtang Road 866#, Hangzhou 310058, People’s Republic of China

Fax: +86-571-87952261

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E-mail: [email protected]

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Telephone: +86-13777466589

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ACCEPTED MANUSCRIPT Abstract: This paper presents a ductility-centred analytical model for robustness assessment of axially restrained double-span steel beam systems further to sudden columns loss. The main benefit of the model is its ability of linking the response of semi-rigid beam-to-column

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joints to the load carrying capacity of the system, and in addition, the model supports the use of codified methods for joint response. The analytical model is first validated through

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comparisons against detailed double-span FE models, where it is found that the load-joint rotation responses of the axially restrained systems can be accurately captured by the

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analytical model, especially at catenary stages. Comparisons are also made against the FE results excluding the axial restraints. The axial restraints are shown to play a crucial role in

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the development of catenary action. Following the verification study, the analytical predictions of the double-span beam systems with varying parameters were further presented

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and discussed, and an illustrative design example was finally given to clearly demonstrate the

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practical use of the analytical model for robustness assessment of building frames. The

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analytical model is design effective and hand-calculable, and thus it brings convenience to robustness assessment of structures in practice. The proposed method may be considered as a feasible alternative method to detailed nonlinear dynamic analysis.

ductility.

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Keywords: robustness; progressive collapse; analytical model; sudden column loss; joint

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ACCEPTED MANUSCRIPT 1. Introduction Under extreme loading conditions, the ability of a structure to resist the progression of

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local damage to global collapse is referred to as ‘robustness’. The failure of the London

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Ronan Point apartment block in 1968 raised an initial concern on structural robustness

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against progressive collapse. After the incident, this collapse mechanism started to receive increasing attention among the community of structural engineers and researchers, and early design rules against such type of massive structural collapse were proposed and documented

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in McGuire (1974), Burnett (1975), and Breen and Siess (1979). A global attention on this

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issue was triggered by the World Trade Centre terrorist attack which shocked the general public. Since then, further numerous efforts have been devoted to this research theme, and the

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research progress was particularly fast during the last decade. A series of experimental

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investigations have been launched, which shed considerable light on the underlying progressive collapse mechanism of building structures (Astaneh-Asl et al., 2002; Jaspart et al.,

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2008; Song et al., 2014; Wang et al., 2016). Apart from the physical tests, numerical models with varied levels of modelling idealisations were established to offer complementary

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investigations, which enabled further understanding of the progressive collapse mechanism (Liu et al., 2005; Sadek et al., 2008; Yu et al., 2009; Alashker et al., 2010; Fang et al., 2011; Fang et al., 2012). In parallel to the experimental and numerical investigations, analytical solutions were also attempted by researchers, and practicing engineers were encouraged to employ these simplified analytical models to perform quick and reliable robustness evaluations on structures subjected to potential extreme loadings (Izzuddin, 2005; Masoero et al., 2013; Gerasimidis, 2014). The continuously emerging research output significantly contributed to the stipulation of design guidelines and recommendations against progressive collapse (GSA, 2003; DoD, 2009). The current design approaches are normally categorized into two types, namely, indirect design approach and direct design approach. The indirect design approach focuses on 3

ACCEPTED MANUSCRIPT increasing the robustness of structures by prescriptively enhancing the continuity, redundancy and ductility. This can be realized by design procedure control, including structural type selection, member size selection, structural layout arrangement, and tying resistance of joints.

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On the other hand, the direct design approach, e.g. the alternative path method, normally requires a series of analysis including failure predictions, e.g. the location of damage, and

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then it needs the assessment of structural robustness following an extreme loading event that can cause local failure which potentially initiates progressive collapse. One of the most

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widely used alternative path methods for robustness assessment is the ‘sudden column loss’ scenario, which assumes an abrupt removal of the targeted column to represent the

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consequence of extreme loading (e.g. blast). Izzuddin (2009) comprehensively reviewed the current specialist design codes and recent developments in performance-based assessments

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for structural robustness. It was highlighted that although column removal scenarios in the

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newly developed design guidelines are more performance-based and event-independent, they

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are computationally expensive for design-oriented practice when addressed using nonlinear dynamic analysis. On the other hand, the simplified consideration of dynamic effects using dynamic increase factors (DIFs) can sometimes be overly conservative for some types of

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structures, or unsafe for others. To address this, a more rational ‘Ductility-Centred’ Approach with a practical and multi-level assessment framework was proposed (Izzuddin et al., 2008; Vlassis et al., 2008). In this framework, only the nonlinear static analysis is required, while the dynamic response can be obtained using a simplified dynamic assessment approach considering the principle of energy balance. The system is considered to fail when the joint ductility demand at the maximum dynamic response exceeds the joint ductility supply. Compared with the nonlinear dynamic analysis, the ductility-centred framework with only static analysis required could significantly decrease the computational effort, and furthermore, it supports the use of simplified models for determining the load-deflection responses at various levels of structural idealisation. In the current design practice, however, 4

ACCEPTED MANUSCRIPT computer-aided analysis still needs to be considerably relied on due to the insufficiency of effective analytical tools that are capable of capturing the key behavioural characteristics (e.g. joint responses) of frame or sub-frame systems subjected to member removal. With the aim

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of addressing this issue and thus to bridge the gap between the newly proposed ductility-centred framework and the existing design practice, a practical analytical model for

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double span systems subjected to sudden column loss is proposed in this study, with particular emphasis given to the ductility issue of semi-rigid joints. The analytical model

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enables the prediction of the load-joint rotation responses of double-span systems with varying beam-to-column joint flexibilities, and by this way the ductility demand of the joints

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at any load level can be directly reflected. This potentially overcomes the conventional load-deflection presentations where the connection behaviour is

not interpreted

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straightforwardly. Employing the proposed analytical model, the robustness of the double

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span system can be evaluated through comparing the connection ductility demand with the

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corresponding ductility supply.

In the following discussions, the basic assumptions made for the analytical model are presented first, which is followed by a detailed elaboration of equation derivation. The

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analytical model is then validated through comparing the analytical load-joint rotation responses with those predicted by detailed 3D finite element analysis. The accuracy of the analytical model is discussed, and its potential limitations are also identified. Following the verification study, more results are presented to show the influences of various parameters on the load-joint rotation response. An illustrative design example is finally given to clearly demonstrate the design steps using the proposed analytical model. 2. Analytical model 2.1 General An axially restrained double-span beam system subjected to sudden column loss has 5

ACCEPTED MANUSCRIPT now been widely recognized as a basic and reasonable scenario for discussing the progressive collapse resisting mechanisms of building structures. Fig. 1 shows a typical double-span subassembly with semi-rigid beam-to-column joints. The analytical model proposed in this

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study is intended to predict both static and dynamic responses of this system upon column loss, where the dynamic response is obtained from the static response using the energy

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balance approach (Izzuddin et al., 2008; Vlassis et al., 2008). Before elaborating the derivation process of the proposed model, the underlying assumptions are outlined first as

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follows:



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Fig. 1 Typical double-span beam system with semi-rigid joints A consistent symmetric deflected shape (shape function) of the double-span beam



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system is assumed, regardless of the level of the applied load. The plastic deformation is concentrated in the joint zone, while the steel beams always deflect with the predefined deflected shape according to the shape function. 

All the joints undergo the same joint rotation θj at the same loading level, i.e. θj = θ0 = θL, as shown in Fig. 1. This assumption is reasonable if the same joint type/size is employed over the considered double-span beam system.

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The axial deformation in the joints is neglected. This assumption can be acceptable if applied to H-shaped beams. These beams have large bending stiffness to tension stiffness ratios, i.e. (EI/EA), such that the axial strain in the joint caused by the bending moment can be relatively small compared with that caused by axial force. But for other beams with small bending stiffness to tension stiffness ratios, such axial 6

ACCEPTED MANUSCRIPT deformation can be very significant, since the centre of joint rotation is typically at one of the beam flanges. 

The beam is regarded as Euler-Bernoulli Beam such that the shear deformation can be

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

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neglected.

Bilinear moment-rotation responses are considered for the joints, as shown in Fig. 1.

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At the elastic range, i.e. 0 ≤ θ ≤ θp, the initial stiffness of the joint is k1; beyond the elastic range, i.e. θ > θp, the post-elastic stiffness is k2. The simplified bi-linear

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moment-rotation characteristic is compatible to the elastic-plastic global analysis design model recommended in major codes, e.g. Eurocode 3 (2005), although in these

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codes the strain hardening effect of the joints is often ignored by taking k2 as zero. Only catenary action is reflected in the analytical model, whereas the possible

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compressive arching effect is not considered. This assumption can be acceptable if the

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main interest is given to the large deformation behaviour of the system beyond the

analysis). 

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compressive arching stage (which would be the case for progressive collapse

The sudden column loss scenario is treated as the load being abruptly applied over the

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double-span beam system bridging over the missed column. Again, when interests are given to the large deformation behaviour of the system, this assumption can sufficiently reflect the actual structural behaviour. 

The considered double span steel beam system is axially restrained, such that a full catenary action can be formed. This assumption is reasonable when the neighbouring structures are relatively ‘rigid’, but for more flexible neighbouring structures, where the catenary action may not be developed due to the absence of axial restraints, the progressive collapse resistance of the system may be overestimated by this assumption. However, it was argued in Wang et al. (2011) that for most typical steel-composite frame, considerable axial restraints are normally offered by the 7

ACCEPTED MANUSCRIPT neighbouring bays due to the tying action of the adjacent beams as well as the diaphragm action of the adjacent floor slabs. The influence of axial restraining conditions is further discussed in Section 3.4.

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If the deformation of the double-span beam system is dominated by a single mode

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

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with discrete hinges, as would be the case when excessive deformations are expected in the scope of progressive collapse study, the system response becomes almost independent of the load distribution (Izzuddin et al., 2008). Therefore, the nonlinear

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static response of the model could be predicted with sufficient accuracy when a concentrated load instead of a uniformly distributed load (UDL) is considered.

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Therefore, it is assumed in the analytical model that a concentrated load is applied at the mid-span of the double-span beam system. It was suggested by Izzuddin et al.

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(2008) that for progressive collapse analysis, the UDL can be converted to a

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concentrated load by considering a non-dimensional weighting factor a (concentrated

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load = a × the total gravity load within the affected bay) depending on the incremental floor deformation mode. For a typical 1D double-span beam system, the value of a was taken as 0.5; for a typical 2D double-span individual floor system, the value of a

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was taken as 0.25.

2.2 Deflected shape configuration Considering half of the double-span system as shown Fig. 1 (i.e. 0 ≤ x ≤ L), the deflected shape in the elastic state can be derived from:

 d 4w  EI dx 4  0  3  EI d w  Q  dx3 and the corresponding boundary conditions could be expressed as:

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(1)

ACCEPTED MANUSCRIPT  wL = 0   M 0 = kθ0  M =  kθ  L L

(2)

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where E = modulus of elasticity, I = secondary moment of area of the beam section, w =

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beam deflection (w0 and wL denote the deflections at the locations of x = 0 and x = L,

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respectively), M = moment (M0 and ML denote the moments at the locations of x = 0 and x = L, respectively), Q = shear force, θ0 = θL = joint rotation, and k = generalized joint rotational

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stiffness. Since the direction of moment at x=L is different from that at x=0, M0 and ML have the different signs.

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Through a successive integration of Eq. (1) and considering the boundary conditions given in Eq. (2), the following equation for beam deflection can be derived: (3)

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Q 3 LQ 2 LQ L2Q( Lk  6EI ) x  x  x 6 EI 4 EI 2k 12 EIk

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w x  

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the vertical displacement w0 at the mid-span can be derived by considering x = 0 in Eq. (3):

L2Q( Lk  6 EI ) w0   12 EIk

(4)

12 EIkw0 L ( Lk  6 EI )

(5)

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Eq. (4) can be rearranged to:

Q

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and replacing Q in Eq.(3) by Eq.(5), one can obtain:

w x 

2kw0 3kw0 6 EIw0 x3  x2  x  w0 L ( Lk  6 EI ) L( Lk  6 EI ) L( Lk  6 EI ) 2

(6)

Eq. (6) presents a deflected shape configuration when the beams behave elastically, which is not related to the load applied on it. In this paper, it is assumed that the deformed shape (shape function) is the same at both elastic and plastic stages of joint response; therefore, Eq. (6) can be also expressed by:

w  f ( x, k1 , w0 )w0  wp    w  f ( x, k1 , wp )  f ( x, k2 , w0  wp )w0  wp 9

(7)

ACCEPTED MANUSCRIPT where wp is the mid-span vertical deflection when the joint first experiences plastic rotation. 2.3 Static load-joint rotation response

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Employing the above shape function of deflection, the static load-joint rotation response

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of a double-span steel beam system with semi-rigid connections can be derived. The axial

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strain ε in the longitudinal axis of the beam at the catenary stage can be given as: n





L

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L g L L 2 1  w '  L 2  i 1  w '( 2 zi  2 )  i 1 1 L L 2

(8)

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in which zi [1,1] are the Gauss integration points, ωi is the corresponding weighting factor

n

n

g 1 g 1 L L 2 1 L L  (1  w '( z  ) )  1  i w '( zi  )2   i i 2 i 1 2 2 2 2 2 i 1 4

(9)

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

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and ng is the number of integration points. As w '    1 , Eq. (8) can be re-written as:



w2p 2

2L



k12 w2p

10( Lk1  6 EI )

2

w0  wp



k1k2 ( w0  wp ) wp

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

w02 k12 w02  2 L2 10( Lk1  6 EI ) 2

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

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5 5 8 Substituting the Gauss points ( ,- 0.6) , ( , 0.6) , and ( , 0) for (ωi, zi) in Eq.(9): 9 9 9

5( Lk2  6 EI )( Lk1  6 EI )

( w0  wp ) 2 2

2L



k22 ( w0  wp ) 2 10( Lk2  6 EI )

2



( w0  wp ) wp L2

(10)

w0  wp

w0 in Eq. (10) then needs to be expressed by θ0. Since the chord rotation can be derived by derivation of w (Eq. (6)) with respect to x:

  x 

6kw0 6kw0 6 EIw0 x2  x L ( Lk  6 EI ) L( Lk  6 EI ) L( Lk  6 EI ) 2

(11)

Taking x = 0 in Eq. (11), w0 in Eq. (10) can be expressed by:

L( Lk1  6 EI ) 0 0   p 6 EI L( Lk1  6 EI ) L( Lk2  6 EI ) w0   p  ( 0   p ) 6 EI 6 EI w0 =-

0   p (12)

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ACCEPTED MANUSCRIPT With the axial strain ε derived, the equations regarding the joint moment M and axial force N (N=εEA, where ε is expressed by Eq. (10)) can be given by:

A[5( Lk1  6 EI )2  k12 L2 ] 2 0 360 EI 2

θ0  θ p

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M 0  k1θ0 , N 

A[5( Lk1  6 EI ) 2  k12 L2 ] 2 A[5( Lk2  6 EI ) 2  k22 L2 ] p  (θ0  θ p ) 2 2 2 360 EI 360 EI 2 A[5( Lk2  6 EI )( Lk1  6 EI )  k1k2 L ]  θ0   p 180 EI 2 (13)

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M 0  M P  k2 (θ0  θ p ), N 

The next step is to derive the relationship between the applied concentrated force P and

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joint rotation θ0. The governing equations of the beam in equilibrium can be expressed as:

d 2M d 2w  N 0 dx 2 dx 2

(14)

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by successive integrations of Eq. (14), and considering the boundary condition of M = M0 when x = 0 as well as Eq. (6), the moment M over the beam length can be expressed by:

2kw0 3kw0 12 EIM 0  2 LM 0 k  LNkw0 x3  N x2  x  M 0 (15) L ( Lk  6 EI ) L( Lk  6EI ) L( Lk  6EI )

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M  x  N

2

Now considering the vertical equilibrium at mid-span (at x = 0), the vertical load P is

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resisted by the vertical components of axial force N and shear force Q:

P  Q cos(0 )  N sin(0 ) 2

(16)

and assuming sin(0 )  0 , cos(0 )  1 , Eq.(16) can be re-expressed by:

P dM  |x 0  N0 2 dx

(17)

Considering Eq. (11) through Eq. (15), and replacing the previously used symbol θ0 by a consistent symbol θj (θj = -θ0 , θj = -θL) in the final expression, the relationship between the load P and the joint rotation θj can be derived from Eq. (17), as given by:

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ACCEPTED MANUSCRIPT 4k1  j  C1 j3  j   p L 4k1 4k  p  2 ( j   p )  C1 p3  C 2 ( j   p )3  C 3 p2 ( j   p )  C 4 p ( j   p )2  j   p L L (18)

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  P   P  

where:

A[5( Lk1  6 EI )2  k12 L2 ](6 EI  Lk1 ) A[5( Lk2  6 EI )2  k22 L2 ](6 EI  Lk2 ) C  , 2 1080 E 2 I 3 1080 E 2 I 3

C3 

A{[5( Lk1  6EI )2  k12 L2 ](6EI  Lk2 )  2[5( Lk1  6EI )( Lk2  6EI )  k1 k2 L2 ](6 EI  Lk1 )} 1080 E 2 I 3

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A{[5( Lk2  6 EI )2  k22 L2 ](6 EI  Lk1 )  2[5( Lk1  6EI )( Lk2  6 EI )  k1 k2 L2 ](6 EI  Lk2 )} 1080 E 2 I 3

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C1 

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2.4 Dynamic load-joint rotation response

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The energy-based robustness assessment approach (Izzuddin et al. 2008) is employed

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herein to convert the obtained static load-joint rotation response to the dynamic (pseudo-static) response. An important benefit of this approach its convenience of conducting simplified dynamic assessment using the principle of energy balance instead of directly undertaking

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complex nonlinear dynamic analysis. For a given nonlinear static response, the maximum dynamic deflection wd of the system subjected to a certain level of load Pi can be obtained from the energy balance between the work done by the external load and the internal energy dissipated in the deformed system, as illustrated in Fig. 2. Initially at small deflections, the load Pi exceeds the static structural resistance, and the differential work done over the incremental deformations is transformed into additional kinetic energy, thus leading to increasing velocities. As the deformation increases further, the static resistance would eventually exceed the load Pi, and the differential energy absorbed leads to a reduction in the kinetic energy, thus leading to decreasing velocities.

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Fig. 2 Illustration of the principle of energy balance

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With the assumption that the deflected floor system is dominated by a single deformation mode, which is likely to be the case for a sudden column loss scenario, the maximum dynamic deflection wd is achieved for a gravity load level Pi when the kinetic

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energy is reduced back to zero. At this point of equilibrium, the external work done by the suddenly applied gravity load Pi is identical to the internal strain energy dissipated by the

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structure, i.e. the equivalence between external work and internal energy is achieved when the two hatched areas shown in Fig. 2 become identical. The state of equilibrium can be

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expressed by:

Pw i d  

wd

0

 Pdw  P 

wd

0

i

Pdw

wd

(19)

Considering the relationship between θ0 and w0 as given in Eq. (12), and then incorporating Eq. (18) into Eq. (19), the dynamic load-joint response can be obtained in Eq. (20), which can be directly used to obtain the dynamic ductility demand of joint θj at any load level Pi (or to obtain the allowable load level with known ductility supply of joint θj):

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2k1 C1 3  P  L  j  4  j  j   p  2 2 4 4 2 2 3  2k1  p 2k2 ( j   p ) C1  p C 2 ( j   p ) C 3  p ( j   p ) C 4  p ( j   p )      P  L j L j 4 j 4 j 2 j 3 j     p  j  (20)

2.5 Design procedure

For typical steel framed structures employing semi-rigid joints, the sudden loss of a

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column often leads to a considerable concentration of deformations in the joint areas within the affected floors. The resistance of the structures against progressive collapse is largely

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determined by the deformation demands on the joints at the maximum dynamic response in relation to their ductility supply. The ductility-centred analytical design approach for

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following three design steps:

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double-span steel beam systems subjected to sudden columns loss is thus comprised of the

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1) Bilinear joint response. With any proposed connection detail, the bilinear moment-joint rotation response needs to be derived first. The joint response is presented by three key parameters: initial rotational stiffness k1; yield rotation θp, and

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post-limit/post-elastic stiffness k2. The three values can be calculated according to relevant codified design methods, such as the component method stipulated in Eurocode 3 (2005). 2) Dynamic load-joint rotation response. The dynamic (pseudo-static) load-joint rotation response of any double-span beam system can be subsequently obtained by incorporating the three key joint parameters and other necessary parameters (e.g. beam dimensions) into the analytical model (Eq. (20)). Based on the constructed the dynamic (pseudo-static) load-joint rotation curve, the maximum rotation that needs to be accommodated by the joints at any load level can be obtained. 3) Ductility assessment. The robustness of the system is assessed via comparing the 14

ACCEPTED MANUSCRIPT obtained ductility demand (i.e. the maximum joint rotation under the required load level) with the ductility supply. The ductility supply of different types of joints (also known as the acceptance criterion) can be based on the current progressive collapse

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design guidelines, e.g. DoD (2009). In these guidelines, the failure criterion is provided as the joint rotation capacity when joint is subject to the loading

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combination of bending moment and axial force. Since no detailed finite element modelling is available regarding the local failure of joint, we only compare the

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ductility demand to the ductility supply from guidelines. It should be pointed out that this guideline underestimate the rotation capacity, such that the ductility supply of the

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structure is conservative in design. On the other hand, the experimental measurement of joint rotation capacity, and the corresponding numerical simulation will be

3.1 General

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3. Model verification

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considered in our future work.

The above analytical model can be used to obtain the dynamic load-joint rotation

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response for double-span steel beam systems. As the rationale behind the simplified dynamic assessment framework using the principle of energy balance (Section 2.4) has already been carefully examined (Izzuddin et al., 2008; Vlassis et al., 2008), and considering the fact that this approach is now widely accepted by the community of progressive collapse researchers (Lee et al., 2009; Yu et al. 2010; Xu and Ellingwood, 2011; Naji and Irani, 2012; Zolghadr Jahromi et al., 2013; Fang et al., 2013a; Ali et al., 2014), focus of the current verification study is only given to the static load-joint rotation analytical response, i.e. Eq. (18). Seven double-span steel beam systems were considered with varying beam lengths, beam cross-section sizes, and material yield strengths. The geometric configurations of the models represent a reasonable range of typical structural layouts adopted in actual steel 15

ACCEPTED MANUSCRIPT frames. For each model, the same material property was considered for the main structural members, i.e. steel beam, steel column, and end-plate. To avoid excessive bolt yielding at large joint rotations, Grade 10.8 M30 high strength bolts were employed. Details of the seven

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models are illustrated in Fig. 3 and Table 1.

Fig. 3 Geometric dimensions of double-span beam models

Column section size D×B×tw×tf

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Beam section size D×B×tw×tf B1 400×300×6×8 B2 450×200×9×14 B3 500×200×10×16 B4 400×300×6×8 B5 400×300×6×8 B6 400×300×6×8 B7 400×300×6×8 Note: unit of length is mm Model

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Table 1 Details of double-span beam models Yielding strength (MPa) 345 345 345 345 345 295 235

be

bi

70 50 50 70 70 70 70

80 50 50 80 80 80 80

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Single beam length L 5000 5000 5000 6000 8000 5000 5000

The following steps were considered for the verification study: 1) The load-joint rotation responses for all the seven double-span beam models are first obtained from the detailed FE analysis, and these results will be later used for comparisons with the analytical results; 2) the pure moment-rotation response of each considered joint is obtained from FE analysis, and then bilinear curves are employed to match the nonlinear moment-rotation joint responses to obtain the values of k1, k2, and θp. It should be noted that in practice, the bilinear moment-rotation joint response k1, k2, and θp can be obtained from either nonlinear FE analysis or codified methods (e.g. the component method stipulated in Eurocode 3 (2005)). For the purpose of verification, however, the former approach was employed because a 16

ACCEPTED MANUSCRIPT consistent joint response is necessary. Details of the FE modelling strategy is discussed in the following section; and 3) incorporating k1, k2, and θp and other necessary parameters of the steel beams into the analytical model (i.e. Eq. (18)), the static load-joint rotation responses of

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the double-span beam systems can be obtained. These analytical results are finally compared

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against the FE results (from the first step) for the purpose of verification. 3.2 FE modelling strategy

The general FE program ABAQUS (2010) was employed in this study. All the structural

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components of the double-span beam system were simulated with 8-node linear brick elements with reduced integration and hourglass control (C3D8R in ABAQUS nomenclature).

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A representative finite element model and its meshing scheme are shown Fig. 4. A ‘symmetry’ condition was applied to the vertical symmetric surface cutting the beam section

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in half. ‘Hard contact’ behaviour with no penetration in the normal direction was assumed for all contact faces. A Coulomb friction model was used with a coefficient of friction of 0.2,

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which corresponds to the Class D slip factor for untreated hot roll steel (Eurocode 3, 2005). All fillet welds were considered as ‘tie’ interactions. Static displacement-controlled load was

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applied onto the top surface of the mid-column. It is recalled that the analytical model takes account of the catenary action by assuming full axial restraints applied at the two ends of the double-span beam systems, but in actual cases, a reduced level of axial restraints may be provided by the neighbouring structures. In order to investigate the influence of axial restraints on the load-joint rotation responses of the double-span beam systems, two axial restraining conditions, namely, ‘full axial restraints’ and ‘no axial restraints’, were considered in the FE models. For the latter case, the longitudinal degree of freedom was released such that the end columns are free to move axially but fully constrained in the vertical and out-of-plane directions. The two considered boundary conditions are illustrated in Fig. 5. This modelling strategy is proved robust since it has already been validated by the experiment (Wang et al., 2016). 17

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Fig. 4 Typical meshing scheme for double-span beam system models

Fig. 5 Illustration of axial restraints

3.3 Moment-rotation response of joints In order to obtain the load-joint rotation response of the double-span beam system, the analytical model requires a predetermined bilinear moment-rotation joint response (i.e. k1, k2, and θp). These were obtained via curve fitting of the nonlinear responses predicted by the cantilever beam models which were modified from the double-span beam models. For the cantilever beam models, the joint bending moment was obtained from multiplying the beam tip load by the lever arm (i.e. beam length L plus half of the column section depth D). The rotation of the joint was obtained via dividing the difference of the longitudinal 18

ACCEPTED MANUSCRIPT displacements of two reference points by their vertical distance, as shown in Fig. 6. Using this method, the nonlinear pure moment-rotation responses of the joints were obtained, and the key parameters for depicting the joint response were obtained as shown in Fig. 6. The key

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parameters were then incorporated into Eq. (18) for obtaining the static load-joint rotation

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responses of the double-span beam systems.

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Joint rotation angle

Fig. 6 Pure moment-rotation responses of joints

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ACCEPTED MANUSCRIPT 3.4 Verification of analytical model The analytical model was validated through comparing the predicted static load-joint rotation responses against those obtained from the FE analysis. Employing the key

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parameters of the joints given in Fig. 6, the analytical beam deflection for Beam 1 can be derived, as shown in Fig. 7 (half of the double span beam, 0
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at elastic stage (θj=0.005), and that post the elastic stange (θj=0.02,), both of them agree well with the FE results. The analytical load-joint rotation responses for the seven double-span

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beam systems are shown in Fig. 8, and the responses from the FE models considering the two extreme cases, i.e. full axial restraint and no axial restraint, are also included for comparison.

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It is clearly found from Fig. 8 that the axially restrained double-span beam systems exhibit larger sustained loads than the axially unrestrained cases. This indicates that the presence of

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axial restraints benefits the loading carry capacity.

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Good agreements are generally observed between the analytical and FE results. For each

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load-rotation response analytical curve, two distinguishable segments are shown, i.e. the elastic joint rotation range and the plastic joint rotation range. At initial loading stages (prior to the development of plastic joint rotation), the analytical predictions seem to agree well

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with the axially unrestrained curves, whereas higher load carrying capacities are exhibited for the FE results of the axially restrained models. The difference between the axially restrained and axially unrestrained cases is due to the compressive arching effect, which is induced by the presence of the axial restrains and tends to increase the sustained load levels at initial loading stages (Izzuddin 2005). It is recalled that the compressive arching effect was not considered in the current analytical model, which explains the good agreement between the analytical predictions and the axially unrestrained curves. When the applied load keeps increasing, the joints start to experience plastic rotations, and the axially restrained system enters the catenary stage. In this stage, the analytical predictions gradually deviate from the axially unrestrained curve and then join the axially restrained curves. The axially unrestrained 20

ACCEPTED MANUSCRIPT FE models give a lower load carrying capacity because no catenary action is developed in

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those models.

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Fig. 7 Verification of beam deflection profile

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Fig. 8 Verification of load-joint rotation response

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Generally speaking, the analytical model can well capture the load-joint rotation responses of axially restrained double-span beam systems, especially at the catenary stage.

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Considering possible different levels of axial restraints, the analytical model may offer an upper bound of the load carrying capacity. Nevertheless, the condition of low axial restraints is unlikely the case for common steel structures where sufficient horizontal ‘ties’ (and sometimes vertical ‘ties’ as well) are mandatory under the current building regulations for robustness design (SCI, 2011). Therefore, the proposed analytical model is considered to be applicable to most typical steel structures where the catenary action can be developed due to the presence of axial restraints offered by the neighbouring sub-structural system. Even for the case where the axial restraints are not sufficient to develop the catenary action, the proposed analytical model may still be used if a reduction factor of the load carrying capacity is considered. According to the current FE study results, the load carrying capacities of the 22

ACCEPTED MANUSCRIPT axially unrestrained models are around 20% to 40% lower than those of the axially restrained models; hence a conservative load reduction factor of 0.6 may be applied on the analytical predictions for the case of no axial restraints. This may provide a lower bound for the

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prediction of the load carrying capacity. However, the derivation of reliable reduction factors

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needs further investigations in future studies, which is not within the scope of this paper. 3.5 Discussion of analytical results

Following the verification study, more analytical load-joint rotation predictions (both

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static and dynamic) of double-span beam systems with varying parameters are presented in this section, where model B1 is considered as the reference case. Fig. 9 through Fig. 11

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illustrate the load-joint rotation curves of the models with varying initial joint stiffness k1 (but with the same joint plastic rotation θp), post-elastic joint stiffness k2, and beam lengths L. It is

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observed from Fig. 9 that the load carrying capacity prior to θp is almost proportional to the initial joint stiffness k1, while the post-elastic response is not affected by k1. The load-joint

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rotation responses of the model with varying post-elastic joint stiffness k2 given in Fig. 10 clearly show the benefits of the strain hardening effect of the joints. With increasing joint

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rotation beyond θp, the load carrying capacity is significantly increased due to the consideration of the post-elastic joint stiffness k2. At a joint rotation of 0.06 radians for the dynamic case, the load carrying capacity of the reference system B1 can achieve as much as twice of that with no post-elastic joint stiffness. Furthermore, it is found that when the post-elastic joint stiffness k2 is ignored, a case which is recommended in Eurocode 3 (2005) for elastic-plastic global analysis of steel frames with semi-rigid joints, the sustained load is still gradually increased beyond θp. This reflects the beneficial effects of the catenary action on the loading carrying capacity, even in the absence of the contribution directly from the joint moment resistance. Fig. 11 shows that the load carrying capacity is sensitive to the length of the double-span beam system, where an increase of the span leads to decreasing sustained load at the same joint rotation. 23

ACCEPTED MANUSCRIPT For all the considered cases, the load carrying capacity of the system subjected to sudden column loss (dynamic case) is evidently less than the static load carrying capacity at the same joint rotation. When the same load level is considered, the dynamic ductility

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demand is significantly increased compared with the static one. Taking the reference case (B1) for instance, at the load level of 100 kN, where the joints for both static and dynamic cases

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are within the elastic range, the static joint rotation is around 0.004 radians, and the dynamic joint rotation is approximately twice of the static rotation. This relationship between the static

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and dynamic responses is typical for a linear system. At an increased load level of 200 kN, the static joint rotation reaches 0.0085 radians, which is still within the elastic range.

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However, the dynamic joint rotation attains almost 0.045 radians (more than four times larger than the static rotation) at the same load level. This indicates that compared with the

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condition of gradual (static) column loss, a sudden column loss scenario leads to a

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considerably higher joint ductility demand, especially when the joints undergo plastic

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rotations.

Fig. 9 Influence of joint initial stiffness k1

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Fig. 10 Influence of joint initial stiffness k2

Fig. 11 Influence of beam length L

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4. Illustrative design example

A simplified design example is given in this section to illustrate the robustness assessment process of steel building frames using the proposed analytical model. The considered four-story building is shown in Fig. 12, and the details of the structural elements are identical to those considered in model B1. In this simplified example, the coupling effect between 2D slabs and beams is neglected, such that to provide a quick estimation for the progressive collapse potential of the structures in practical design. The roof is considered to be identical to the other floors. The factored UDL applied along the beams is 30kN/m, and an internal column at the ground floor is suddenly removed due to blast loading. The directly affected bay can be isolated and assumed as an axially restrained sub-assembly, as shown in

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ACCEPTED MANUSCRIPT Fig. 12. The sudden removal of the column is in effect close to suddenly applying the gravity load on the same structure in the absence of the column at the beginning. This sudden application of gravity loading is associated with dynamic effects, where the dynamic ductility

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demands must be accommodated to avoid progressive collapse. As the affected floors have identical structural geometry and gravity loading, a further reduced model consisting of an

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individual floor system can be considered due to a synergetic load resisting mechanism among the four floors with an identical deflection (Izzuddin et al., 2008, Fang et al. 2013b).

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In addition, the suddenly applied UDL can be converted to a concentrated load applied at the mid-span above the removed column with a non-dimensional weighting factor of a = 0.5.

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Therefore, the suddenly applied concentrated load is P = 30 kN/m × 10 m × 0.5 = 150 kN. The next step is to derive the key joint parameters k1, k2, and θp, which in practice can be

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obtained from either detailed FE studies or codified methods (e.g. the component method).

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Employing the former method, the values of k1, k2, and θp are obtained which have been

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given in Fig. 6 (k1 = 30000 kNm/rad, k2 = 5500 kNm/rad, and θp = 0.009 radians). Incorporating the joint parameters and those for the steel beams (i.e. L = 5m, E = 205000 MPa, I = 0.000213 m4, A = 0.007104 m2) into Eq. (20), the dynamic load-joint rotation

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responses of the double-span beam system can be derived, which are reproduced in Fig. 12. It can be easily obtained from the curve that the ductility demand is 0.0265 radians under P = 150 kN.

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Fig. 12 Example structure for illustration of design process The obtained ductility demand needs to be compared with the ductility supply stipulated in relevant design guidelines, where the one published by DoD (2009) is used in this example. The double-span beam system which provides the capacity of the structure to resist collapse due to the removal of a vertical load-bearing element should be considered as ‘primary’ elements. According to the Nonlinear Acceptance Criteria, the allowable plastic rotation angle of extended end-plate connections should fall into the category of Partially Restrained Moment Connections (Relatively Stiff). Considering the fact that the behaviour of the connection is governed by flexure of plate, the allowable plastic rotation angle is obtained as 0.035 radians according to ‘Table 5-2’ in DoD (2009). As this ductility supply is greater than the ductility demand, the structures is considered to be safe. Finally, the capacity-demand 27

ACCEPTED MANUSCRIPT ratio (CDR) can be calculated based on the dynamic load-joint rotation curve. CDRs are normally used to indicate the potential of the structure for progressive collapse, which are expressed in terms of the structural resistance at the point of failure compared to the applied

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loading, where a ratio exceeding unity indicates a safe structure. Considering the overall ductility supply of 0.035 + 0.009 = 0.044 radians (plastic + elastic), the corresponding load

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carrying capacity is 198 kN according to Fig. 12. Therefore, the CDR for the considered double-span system is CDR = 198 kN / 150 kN = 1.32, which indicates a safe structure.

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5. Summary and conclusions

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A ductility-centred analytical model was proposed in this paper for robustness assessment of axially restrained double-span steel beam systems subjected to sudden column

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loss. Through taking the response of semi-rigid beam-to-column joints into consideration and

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employing the energy balance concept, a convenient yet reliable procedure was proposed for obtaining the relationship between the load carrying capacity and the corresponding ductility

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demand (i.e. maximum joint rotation). The analytical model was validated through comparisons against detailed 3D FE models, and good agreements were generally observed.

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The FE results also showed that the effect of catenary action is critical in the load-resisting mechanism, where evident increases of the load carrying capacity were observed for the axially restrained models compared with their axially unrestrained counterparts. The analytical results were found to agree very well the results obtained from the axially restrained models, especially at the catenary action stage. When a reduced level of axial restraints was considered, the analytical model could offer an upper bound of the load carrying capacity estimation. Following the verification study, the analytical predictions of the double-span beam systems with varying parameters were further presented. The influences of joint and steel beam properties on both static and dynamic analytical results were discussed. A detailed illustrative design example was finally given to demonstrate the use of the analytical model for robustness assessment of structures. 28

ACCEPTED MANUSCRIPT As a general remark, the proposed analytical model is applied within the multi-level ductility-centred robustness assessment framework. While sophisticated robustness assessment approaches (e.g. detailed nonlinear dynamic analysis) can now be employed in

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design practice, these approaches are computational expensive and require experienced engineers with extensive advanced structural engineering knowledge. Therefore, the current

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model can be considered as an efficient supplementary approach, which helps practicing engineers quickly judge the progressive collapse potential of structures, at least for the

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preliminary design stages.

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Acknowledgements

The authors gratefully acknowledge financial support from the International Science &

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Technology Cooperation Program of China (No. 2015DFE72830), the National Nature

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Science Foundation of China (Nos. 51579221, 51279180 and 51079127), Science

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Technology Department of Zhejiang Province (Grant No. 2013C33043) and the Hong Kong

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Polytechnic University Central Research Grant (G-YL86).

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pp. 278-294.

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