2021 A new screw connection model and FEA of CFS shear wall panels

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A new screw connection model and FEA of CFS shear wall panels

M. Nithyadharan

^a,

⁎ , V. Kalyanaraman

^b

aDepartment of Civil and Environmental Engineering, IIT Tirupati, Tirupati 517506, India

bDepartment of Civil Engineering, IIT Madras, Chennai 600036, India

a b s t r a c t a r t i c l e i n f o

Article history:

Received 7 April 2020

Received in revised form 23 October 2020 Accepted 25 October 2020

Available online 13 November 2020

This paper presents the details and application of a newly developed variably oriented spring pair element (VOSPE) model and the Bouc-wen-Baber-Noori (BWBN) constitutive model, to represent the behaviour of the screw connections between the CFS framing and the sheathing in the non-linearﬁnite element analysis of the cold formed steel shear wall panels (CFSSWP) under the monotonic and cyclic loadings. The parameters of the BWBN model were obtained from the experimental load-displacement behaviour of the screw connection data, with the screw experiencing shear parallel to the free edge of the sheathing. The proposed VOSPE screw model was implemented in ABAQUS to analyse the behaviour of cold formed steel wall panels. Theﬁnite element analysis results compare well with the experimental results of the overall wall panels under both the monotonic and cyclic loadings, wherein the behaviour and the strength were essentially dictated by the non-linear behaviour and strength of the screw connections. Finally, a non-linear dynamic analysis of a wall panel under an earthquake loading is illustrated, using the proposed model of the wall panel.

Keywords:

Cold formed steel Shear wall panels Finite element analysis Screw connection

Monotonic and cyclic loadings Bouc-wen Baber-Noori model

1. Introduction

Wall panel system, based on sheathing boards made of different material attached to cold-formed steel (CFS) framing members, is an at- tractive alternative structural system for single and multi-storeyed building walls andﬂoors. They have been widely used in construction in USA [34], Canada [25], Europe [9,21] and Australia [16,38]. The high strength to weight and the stiffness to weight ratios are the main advan- tages of this system to resist earthquake forces. The research in CFS building system made up of the wall panels, is moving towards the performance based seismic design (PBSD) as a more rational seismic resis- tant structural design method [11,25]. The PBSD approach [6] is based on the coupling of the multiple performance levels and the ground motion intensities, thus generating and evaluating the performance objec- tives to be satisﬁed. The critical underlying assumption behind the PBSD philosophy is that the performance of the structure, when subjected to different level of earthquake ground motion, can be predicted with con- sistency and accuracy. The fundamental requirement to achieve this is a good numerical model to simulate the behaviour of the CFS structural system and its components, mainly the cold-formed steel shear wall panel (CFSSWP) under in-plane shear due to earthquake loading.

The in-plane shear behaviour and the strength of the CFSSWP are dictated by the behaviour of the CFS framing members, the anchor of such members, the connection between the framing members and the

sheathings serving as the shear membrane. The modelling of the wall panel can be either at the micro level, where the nonlinear behaviour of each individual component and its interactions are modelled to predict the overall response of the wall panel [5,7,8,15,17,18,40,41]; or at the macro level, where the nonlinear behaviour of both the structural and non-structural wall panels is modelled with the equivalent springs [12,35]. Modelling of the components such as the CFS framing members, the connections between framing members, the local deformation of framing members at the anchor points, and the sheathing material have been studied satisfactorily [5,38]. Useﬁet al. [38] has reviewed the different numerical models used in CFS systems and subsystems.

They concluded that the screw connection between the board and the CFS framing members is the major contributing element for the nonlinear behaviour and the energy dissipation characteristics of the CFSSWP under the cyclic loading. Although many studies have dealt with the connection modelling [7,14,18,29], a few limitations are observed in these constitutive models, as reviewed in detail later. The constitutive models currently used for simulating the screw connection behaviour in the FE model of the wall panel are mostly polygonal hysteretic models [38], which do not account for the effect of loading history on the subsequent load deformation behaviour [5,7,29]. Hence, this study focusses on modelling the non-linear behaviour of the screw connections between the CFS framing member and the sheathing in the FEA model of the wall panel. This paperﬁrst presents details of a newly developed variably oriented spring pair element (VOSPE) and the Bouc-Wen-Baber-Noori (BWBN) constitutive model, for realistically representing the screw connection behaviour in theﬁnite element analysis of the CFSSWP, under the monotonic and cyclic in-plane shear

⁎ Corresponding author.

E-mail addresses:[email protected](M. Nithyadharan), [email protected](V. Kalyanaraman).

Contents lists available atScienceDirect

Journal of Constructional Steel Research

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loadings. Then the VOSPE element based FEA model of the wall panel is validated with the experimental data of the CFSSWP with the calcium silicate board as the sheathing under the monotonic and reversed cyclic loadings. Finally, a non-linear dynamic analysis of a wall panel under an earthquake loading is illustrated, using the proposed model of the wall panel.

2. Literature review onﬁnite element modelling of CFSSWP

In a cold formed steel wall panel, the sheathing is stiff in its plane, and resists the in-plane shear with the board acting as a web member, and the CFS framing behaving as a chord member of a deep composite girder. The state of the art on the numerical modelling and the nonlinear analysis of the steel framing members using the linear, shell as well as hybrid of linear and shell elements has been well studied [4,19,36]. The non-linear analysis (including effects of residual stresses and imperfections) capturing the local, distortional and overall buckling behaviours (and their interactions), and the modelling of the local dis- tortions at the anchorages and connections of the CFS members in the CFSSWP under axial and lateral loading is well established [5,37–39].

The modelling of the sheathings using the shell elements [29,42] and the membrane elements is also well demonstrated [18]. This paper focusses on the modelling the behaviour of the discrete screw connection between the CFS framing and the sheathing alone, since it is known to be the major contributing factor to the non-linear behaviour and strength of well-designed walls panels [16,27,33]. Once the screw connection model is established, it is straight forward to integrate the effect of the other components on the overall behaviour, using the methods discussed in literature reviewed earlier.

The screw connections between the framing member and the sheathing resist the relative deformation between them and transfer the shear between the CFS framing member and the sheathing. The failure of the screws under in-plane shear loads, initiates with the tilting, bearing andﬁnally mostly ends with the complete pull through of the screw, causing local damage to the sheathing around the screw/fastener location [26,27,33]. The discrete screw connection behaviour is complex and involves multiple failure modes. Hence, the detailed local level micro mechanical modelling of the screw, the board, the CFS framing and the interaction between them using the FEA is neither computationally economical nor adequately accurate.

Therefore, it is desirable to have an appropriate simple constitutive model of the screw behaviour for use in theﬁnite element modelling of the wall panel. Such a screw connection model has to properly simulate the experimental results on the screw connections, and experience forces similar to that in the wall panels sub-assemblage under the monotonic and cyclic loadings. Further, the screw connection model has to deal with the fact that (1) the screw force in the connection always acts along the direction of the latest relative deformation between the points connecting the board and the CFS framing member, as well as (2) the constitutive model used to evaluate the magnitude of the shear force depends upon the prior loading history, and exhibit the stiffness and strength degradations as well as pinching.

2.1. Screw connection element

Early literature on the FE model of the wall panel, the sheathing to (CFS/wood) framing (screw/nail) connections were represented as a non-oriented spring pair element (NOSPE) [8,15–17,40]. In the NOSPE model, the screw connection is represented using a pair of orthogonal springs along the geometric axis (in the vertical and horizontal directions) of the panel, with the constitutive properties based on the screw/nail connection sub-assemblage tests. In the non-linear FEA of such a model using the NOSPE, the resultant of the orthogonal components of the shear force in the screw is not along the direction of the resultant relative displacement between the nodes on the sheathing and

framing members, connected by the screw. Hence, the wall panel strength and stiffness are overestimated in this model.

To overcome this limitation in the NOSPE model, Judd and Fonseca [18] proposed an oriented spring pair element (OSPE), to model the nail connection in the FEA of wooden wall panels. In the OSPE model also the sheathing-to-framing connection is represented using two independent orthogonal nonlinear springs. However, the main spring in the pair is oriented along the initial relative displacement trajectory of the nodes in the wood framing and the sheathing connected by the screw/nail, which is obtained initially by a linear analysis. However, the spring orientation is kept constant in the non-linear analysis. But the experimental studies on the wall panels indicate that the direction of this relative deformation changes during the test [27], due to the large displacement and non-linear behaviour. Hence, the shear force component is also obtained orthogonal to the direction of current relative displacement, under large deformation before failure, although to a lesser extent than in NOSPE model. Judd and Fonseca [18] implemented the OSPE model in CASHEW (Cyclic Analysis of Shear Wall) program developed by Folz and Filatrault [14], to study the wall panels under the quasi-static cyclic loading. Xu and Dolan [41] extended the OSPE model of Judd and Fonseca [18] to include the hysteretic behaviour in the nail connections to model the wood wall panel behaviour under the quasi-static loading. Buonopone et al. [7] proposed a single radially symmetric non-linear spring element from the element library in the Open Sees [22]. Buonpone et al. [7] and Padilla- Llano [29] modelled the nonlinear behaviour of the screw connection with the rule based hysteretic model of the“Pinching 4”element in open Sees [22].

Substantial variation was observed between the numerical and experimental cyclic load-deformation behaviours as well as the cumulative energy dissipated, particularly at the immediate post peak deformation ranges [7].

Hence, there is a need for improving the screw connection model used in the FEA of the wall panel under in-plane cyclic shear loading and comparing not only the hysteretic load deformation behaviour but also the energy dissipation in each cycle.

2.2. Constitutive modelling of the screw connection

The constitutive model of the screw connection between the board and the CFS framing is based on the experimental load-displacement behaviour of the screw connections under the reversed cyclic loading.

In the CFSSWP, the screws between the sheathing and the CFS framing members are installed at an edge distance from the free edge of the sheathing. Fulop and Dubina [15] generated the screw connection load-displacement curves for the lap connections between the steel to steel sheathing and the steel to OSB sheathing. Serrete et al. [32], Fiorino et al. [10], Miller and pekoz [23] and Peterman et al. [30] presented the results of the behaviour of the screw connection between the CFS framing and the sheathing, with the screws experiencing the shear perpendicular to the nearest free edge of the sheathing, as shown inFig. 1a.

In the wall panels, under the in-plane shear loading, most of the screws connecting the CFS framing and the sheathing (excepting a few screws closer to the panel corners) experience shear nearly parallel to the free edge of the sheathing [5,7,27]. However, most of the experimental data reported in literature, are on the screw connections with the screw shear force acting perpendicular to the free edge of the sheathing (Fig. 1a). Nithyadharan and Kalyanaraman [26] presented a screw sub assemblage experimental data, wherein the screws experienced the shear force acting parallel to the nearest free edge of the sheathing, as shown inFig. 1b. This is closer to the direction of forces experienced by most of the screws in the connection between the CFS framing and the sheathing in the wall panel tests. Recently, Useﬁ et al. [38]

summarised the constitutive models used to model the screw behaviour in the FEA model of the CFSSWP. Most of the material models are either piecewise linear or non-linear rule-based hysteretic models [5,7,14,29,30]. Recently, Xu and Dolan [41] used a modiﬁed form of

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the differential BWBN model, proposed by Foliente [13], to represent the nail connection behaviour in the wall panelﬁnite element analysis.

The unknown parameters of the constitutive model were determined using the experimental load-displacement behaviour of the screw/

nail connection under the monotonic loading or any particular reversed cyclic loading protocol [28,30].

The screw connections at the different locations of the wall panel experience different loading protocol, under a general earthquake loading.

The constitutive model of the screw connection element therefore has to be independent of loading protocol. The differential BWBN model of the screw connection, based on the experimental data using a speciﬁc loading protocol, has shown good agreement with experimental load- displacement behaviour of the screw connections driven by other loading protocols also [2,28]. But the other phenomenological (hysteretic rule based and piecewise linear) screw connection models can trace the behaviour adequately only under the speciﬁc loading protocol, used for identifying the hysteretic behaviour model parameters [30].

The limitations of the FEA studies on the wall panels reported in literature are summarised below

• The screw connection under the cyclic loading exhibits non-linearity, strength and stiffness degradations and severe pinching characteristics, which are dependent on the prior loading history.

The phenomenological material models (Bian at al [5], Buonopane et al. [7], Padilla- Llano [29]) have not been proved to be independent of the loading protocol. The cyclic load deformation behaviour obtained from such numerical models deviate considerably from the experimental results, especially at the advanced stages of the loading.

Further, most of the studies do not compare the energy dissipation, which is important to model the behaviour under the earthquake loading.

• The constitutive property of the screw connection is obtained from the experimental sub-assemblage tests, wherein the screws experience the shear perpendicular to the nearest free edge of the sheathing (Fig. 1a). This is not a realistic representation of the actual behaviour of most screws in the panels under in-plane shear [7,27], which experience shear more parallel to the nearest free edge of the board.

• The screw connection shear force and stiffness are actually mobilised only in the direction of the current relative displacement between the points in the board and the framing member connected by the screw [26,27]. Whereas, the NOSPE models assume the screw forces to be along the two orthogonal geometric axis, the OSPE models assume

the screw forces to be in the direction of initial relative displacement between these points. Hence, they fail to represent the correct direction of screw forces, thus over estimating the strength and stiffness.

• Landolfo [21] also, in his review on the seismic challenges in lightweight steel framed systems, has identiﬁed that a fully satisfactory analytical modelling is still lacking. He observed that a more sophisticated numerical models are required to adequately predict the system behaviour and for developing the performance-based design methods.

These issues are addressed in this paper by developing and demon- strating a new variably oriented spring pair element (VOSPE), based on the differential BWBN constitutive model. The BWBN model parameters were derived from the experimental database of the screw connection that are tested under the monotonic and the reversed cyclic loadings by Nithyadharan and Kalyanaraman [26], where the screws were subjected to shear force parallel to the nearest free edge. Also the use of VOSPE model for the screw connection behaviour is demonstrated, which properly models the direction of the connecting screw resultant force.

This paper presents comparison of the results of theﬁnite element analysis of the CFSSWP using the proposed VOSPE model with experimental response of the wall panel having the size of 1.2 m × 2.4 m (an aspect ratio of 1:2), under the monotonic and cyclic loadings. Further, an analysis of the behaviour under a simulated earthquake loading, using the wall panel constitutive model is demonstrated. Both the experimental results used and the analytical study presented focus on the behaviour of the panel as dictated by the screw connections between the board and the framing members. As stated earlier, the other failure modes have been adequately covered in literature and hence are not dealt with in this study.

3. Variably oriented spring pair element (VOSPE) model

The VOSPE is a two noded, six degrees of freedom (DOF) element with three translational DOF at each node. The proposed variably oriented spring pair element (VOSPE) is an extension of the OSPE model, originally proposed by Judd and Fonseca [18]. The difference between the VOSPE and OSPE models is that, in the VOSPE model the orientation of the spring pair is modiﬁed at every load step, with the main spring being oriented along the direction of the latest relative displacement Fig. 1.Screw test setup and direction of screw forces.

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between the nodes on the board and the CFS member connected by the screws.

In the VOSPE model, the sheathing-to-framing screw connection is represented by using two orthogonal nonlinear springs, connecting the CFS frame at node B′and the sheathing at node A', as shown in Fig. 2. This is similar to the OSPE model. The stiffness of the spring in theZ-direction (orthogonal to the board surface) is taken to be 1.0×

10¹⁰kN/mm, which is signiﬁcantly large than the in-plane stiffness, and is not shown inFig. 2, and in the VOSPE element formulation. The connection tangent stiffness,KuandKv, and the connection forces,Fu

andFvare a function of the latestuandvdirection relative displacements, respectively. The element stiffness matrix in the local coordinate system of the spring pair corresponding to the screw connection is given by,

k⁰ h i¼

Ku 0 −Ku 0 Kv 0 −Kv

Ku 0

Sym: Kv

2 66 4

3 77

5 ð1Þ

whereK_uandK_vare the current stiffness of the screw connection in the direction of the relative displacement and theﬁctitious spring in the orthogonal direction, respectively.

The transformation matrix of the VOSPE element to the global coordinate direction is given by,

½ ¼T

cosφ sinφ 0 0

−sinφ cosφ 0 0

0 0 cosφ sinφ

0 0 −sinφ cosφ

2 66 4

3 77

5 ð2Þ

whereϕis the angle between the global‘x' axis and the latest relative displacement direction of the nodes (A' and B'), which is connected by Kusprings after each load increment. The angleϕbetweenuand the geometric axisx,is written as,

φ¼ tan⁻¹ Y⁰_B−Y⁰_A X⁰_B−X⁰_A

!

ð3Þ

where (XA′,YA′) and (XB′,YB′) are the current coordinates of the screw connection node at the sheathing material and the CFS member, respectively.

The global stiffness matrix corresponding to the spring pair is, [K] = [T'][k'][T]. In the VOSPE model,ϕis calculated after every load step in the non-linear analysis. The main spring with stiffness,Kuis always oriented along the latest direction of the relative displacement between the points A' and B′. This eliminates the contribution of the orthogonal

spring stiffness,Kvin the screw model, since the relative displacement in the v direction is zero. The value of the force in theﬁctitious orthogonal spring,F_vis always nearly equal to zero, and is present only to ensure numerical stability of the simulations under the monotonic and reverse cyclic loadings.

3.1. VOSPE constitutive model

Useﬁet al. [38] reviewed eight hysteretic models used in the study of the CFS framed structures under lateral cyclic loading and concluded that although the models take into consideration many of the features such as strength and stiffness degradations, pinching etc., they do not adequately account for the history of the past loading on the subsequent behaviour.

The differential smoothly varying Bouc-Wen Baber-Noori (BWBN) model, originally proposed by Foliente [13], is used as the constitutive model in this study, to represent the stiffness of the screw connections between the CFS framing members and the sheathing, under static and cyclic loadings. The choice of the BWBN model in the present study is due to its ability to model the experimental response under any general loading, other than the loading protocol used for the parameter identiﬁcation [28]. In the screw connection model, the BWBN model is represented as a single degree of freedom system, with a linear viscous damper and a nonlinear hysteretic element, which are attached parallel with the mass, as shown inFig. 3. The hysteretic force consists of a linear restoring force,Fk(=αk u) and a nonlinear hysteretic restoring forceFh(=(1−α)k z), whereuis the displacement,kis the stiffness co- efﬁcient andαis the rigidity ratio (a weighing parameter in the range 0

≤α≤1). The hysteretic displacementz,is a function of the time derivative ofu. The time derivative ofzis aﬁrst-order nonlinear differential equation, given by Eq.4.

_

z¼h zð Þ Au_−ν β j ju_ j jzⁿ⁻¹zþγu z_j jⁿ η

8<

:

9=

; ð4Þ

In Eq. 4, parameterAregulates the ultimate hysteretic strength and the tangent stiffness;β,γ,nare the hysteresis shape parameters. The parametersνandηrepresent the strength and stiffness degradation functions andh(z)represents the pinching function. In the BWBN model, the degradation and pinching are represented in terms of hysteretic energy dissipated. The hysteretic energy dissipated,ε(t), is the continuous integral of the hysteretic force,Fhover the actual scalar dis- placementu, as given by Eq. 5.

Fig. 2.Variably oriented spring pair element (VOSPE) at angleϕ. Fig. 3.Hysteretic SDOF system for the BWBN Model [13].

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εð Þ ¼t ^{u T}Z^{ð Þ}

uð Þ0

Fhdu¼ð1−αÞω^o²^{u T}Z^{ð Þ}

uð Þ0

zdudt

dt¼ð1−αÞω^o²Z^T

0

z uð ;tÞu dt_ ð5Þ

The strength and stiffness degradation functions are respectively de- ﬁned as,

ν εð Þ ¼1þδνεð Þt

η εð Þ ¼1þδηεð Þt ð6Þ

whereδνandδηare the strength and stiffness degradation rates.

The pinching function,h(z)is given by, h zð Þ ¼1−ζ1e−ðzsgnð Þ−u_ qZ_uÞ²=ζ2

h 2i

ð7Þ

In Eq. 7, sgn u^: is a signum function ofu^:, andZuis the ultimate hysteretic displacement, given by.

Zu¼ A vðβþγÞ 1=n

ð8Þ

The two pinching functionsζ1(ε)andζ2(ε) control the progress of pinching, and can be written as.

ζ1ð Þ ¼ε ζs½1−e^−pε ð9Þ

ζ2ð Þ ¼ε ψ0þδψε λþζ1

ð Þ ð10Þ

In Eqs. 9 and. 10,ζs,p,q,ψ0andλare the pinching parameters.

The 14 parameters of the BWBN model are the hysteretic shape parameters (n, A,β,γ,α), the strength and stiffness degradation parameters (δν,δη), the pinching parameters (ζs,p,q,ψ0,λ) and the stiffness coefﬁcientk. These are to be determined using the load-displacement data from the screw connection sub-assemblage tests.

The BWBN model represented in a state vector form (Eq. 11) results in a stiff set of linear ordinary differential equations (ODE) (Eq. 12).

y¼ y₁ y2

y₃ 8<

: 9=

;¼ u z ε 8<

: 9=

; ð11Þ

_

y₁¼V ð12:aÞ

y⁰₂¼h zð Þ AV−ν β j jVj jy₂ⁿ⁻¹y₂þγ:V yj j₂ⁿ η

8<

:

9=

; ð12:bÞ _

y₃¼ð1−αÞk y₂V ð12:cÞ

whereV¼u^: is the time derivative of the input displacement in the loading protocol and the restoring force is given by.

F¼α:k:uþð1−αÞ:k:z ð13Þ

The Semi-implicit Rosen Brock method [31] was used to integrate the set of stiff ODE's. The solution vector of the set of ODE's was used to obtain the restoring force (Eq.13) in the constitutive model.

3.2. Parameter estimation for BWBN model

The unknown parameters of the BWBN model were identiﬁed using the experimental load-displacement data of the screw connections tested by Nithyadharan and Kalyanaraman [25] by adopting the system identiﬁcation technique as described in [28]. The objective is to deter- mine the parameters of the BWBN hysteretic model by minimizing the error between the computed force based on the BWBN model and

the experimentally measured force in the screw tests, over the full range of the displacement function values. The objective function to be minimized is deﬁned as the square of the error between the experimental and the BWBN model values of the force summed over the total discrete displacements from the screw connection experimental data [25]. The objective function is given by.

E¼ 1 M

X^M

i¼1

FExpi−FBWBNin;A;α;β;γ;δη;δν;ζs;p;q;ψ0;δψ;λ;k;ui

2

ð14Þ

whereFExpiis the force at thei^thmeasurement point from the experiment,FBWBNithe computed force from BWBN model at the displacement corresponding to thei^thdata. The computed force is a function of the model parameters andMis the number of data points. The evaluation of the unknown parameters of the model equations are carried out, so that theEis minimized. The force in the BWBN model is a function ofu, zas in Eq.13, which can be obtained only by integrating the ODE at each evaluation point. In this study, the Nelder and Mead's simplex algorithm [24] is used to identify the unknown parameters of the BWBN model.

The values of the BWBN parameters are obtained from the test data on the screw connections under both the static and cyclic loadings [25].

In these tests the screws were loaded with the shear force parallel to the nearest free edge of the sheathing, as encountered in most of the screws in the wall panels. The load deformation behaviour of the screw connection with 8 mm, 10 mm and 12 mm thick calcium silicate boards and the CFS member, with 25 mm edge distance of the screws from the nearest free edge of sheathing, as obtained from the experiments and the BWBN model is shown inFigs. 4 and 5for the monotonic and cyclic loadings, respectively. The parameter values of the BWBN model are also presented inFigs. 4 and 5.

It can be seen that the BWBN model of the screw connection has captured the experimental behaviour of the screws reasonably well. The strength and stiffness degradations and the pinching characteristics are also modelled properly by the BWBN model. The lack of symmetry about the horizontal axis in the experimental response curve is due to the damages accumulated in the prior positive loading cycle. Whereas, the BWBN model assumes the behaviour to be symmetric, causing some discrepancy in the amplitude of the force, particularly in the positive direction in the cycles. This BWBN constitutive model of the spring is used in the VOSPE model of the screws in the FEA of the wall panel.

3.3. Implementation of the VOSPE in analysis software

The ABAQUS software [1] (ABAQUS 2017) was used for the computational modelling and the analysis of the CFSSWP. The inbuilt SPRINGA element and the constitutive model in the ABAQUS material library can- not model the strength and stiffness degradations and the pinching characteristics, that are experienced by the screws under the reversed cyclic loading, and hence could not be used. The VOSPE with the BWBN based hysteretic model was implemented as a user element,

*UEL subroutine and was integrated with ABAQUS/Standard. The Newton-Raphson (NR) method was used in the non-linear FEA. After convergence at every displacement,u^(r−1), the NR algorithm increments the displacement by a small magnitude,∂u^(r)in ther^thiteration, and the total displacement at the end of the iteration is given by.

u^{ð Þ}^r ¼u^ð^r⁻¹^Þþ∂u^{ð Þ}^r ð15Þ

The nonlinear analysis requires the evaluation of the tangent [KT] and the secant stiffness [KS] matrices of the screw element after con- verged(r-1)^thdisplacement iteration. The coefﬁcient of the tangent and secant stiffnesses of the BWBN model are obtained by integrating the ODE (Eq.12) with the semi-implicit Rosenbrock method from the

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(r-1)^thstep to ther^thstep. Then the solution vector obtained at ther^th step, (u,z,ε)ris updated from the previous step solution vector, (u,z,ε)

r−1, and stored in the SVARS array in the UEL subroutine.

From the current solution vector, the resulting force in the spring is obtained as.

F¼αKuþð1−αÞKz ð16Þ

From the restoring force in (Eq. 16), the tangent stiffness coefﬁ- cient,KTii, and the secant stiffness coefﬁcient,KSii, are calculated using Eq. (17),

KT_ii ¼dF

du¼αkþð1−αÞ dz

du ð17:aÞ

Fig. 4.Screw connection monotonic BWBN model and its parameters for differing sheathing thickness & constant 25 mm edge distance.

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where, _du^dz ¼h zð Þ ^{A−ν β:}ð ^sgnð Þ^u^: ^{j j}^zⁿ⁻¹^zþγ^{j j}^zⁿÞ

η

.

KSii¼F

u¼αkuþð1−αÞkz

u ð17:bÞ

4. Finite element modelling of CFSSWP in ABAQUS

The load-displacement data from the experimental studies carried out by Nithyadharan and Kalyanaraman [27] on the 1.2 m × 2.4 m CFSSWP with calcium silicate board sheathing, under monotonic and reversed cyclic loading was used to validate the FEA model. The Fig. 5.Screw connection reversed cyclic BWBN model and its parameters for differing sheathing thickness & constant 25 mm edge distance.

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experimental study on the CFSSWP included, the calcium silicate board of different thicknesses (8, 10 and 12 mm) and different edge distances (10 and 25 mm) of the screws connecting the CFS framing and the sheathing, from the nearest free edge of the sheathing. The intermediate stud is located at 600 mm at the mid-width and the blocking at the mid- height of the wall panel, as shown inFig. 6a. In the experimental study by Nithyadharan and Kalyanaraman [27], the CFSSWP was designed to ensure that the screw connection between sheathing-to-wall framing governs the overall wall panel behaviour and strength. All the other failure modes prior to that were designed out, since the modelling of these failure modes has been well studied. The details of the CFS sections and the other components used in the CFSSWP are given inTable 1and shown inFig. 6a. The end chord members are two lipped channels back to back to carry the chord forces, under the in-plane shear loading.

The calcium silicate board was attached to the CFS frame on both sides as sheathing, using the self-drilling tapping bulge-head screws of 3.9 mm diameter and 35 mm length. The screw spacing was 150 mm c/c at the periphery (starting at 75 mm from the ends) and 300 mm c/c at the interior members. The International Standard Organisation (ISO) loading protocol based on ASTM E-2126 [3] as shown inFig. 7was used to conduct the reversed cyclic tests on both the screw connections and the wall panels. InFig. 7, Dpeakwas considered as 32 mm for all the range of the wall panels tested with 8 mm, 10 mm and 12 mm board thicknesses, under the cyclic reversed loading.

The B31 beam element was used to model the studs, the top and bottom tracks as well as the blockings. The B31 is a 2-noded 3-D linear beam element modelled with the young's modulus, E = 200kN/mm² and the shear modulus,Gs= 77 kN/mm². The cross-sectional properties of the CFS framing members, consisting of the thin walled open cross section as given inTable 1and shown inFig. 6a, were represented using the ABAQUS [1] built-in option named“arbitrary section”. The cross section in this implementation of ABAQUS is deﬁned by a series

of points corresponding to the corners in the thin walled cross-section and these are connected by line elements to deﬁne the open section.

The three-dimensional 4-noded M3D4R membrane element was used to model the calcium silicate boards. The sheathing material was

Fig. 6.Details of CFSSWP used for FEA evaluation.

Table 1

Details of CFSSWP used for FEA.

CFSSWP components Speciﬁcations

Sheathing material Calcium silicate Board Brand name - HILUX

8, 10 and 12 mm Thickness boards Cold-formed steel members Steel Grade and Strength:

Hot dipped galvanized steel (Zinc coated)

Brand - TATA GALAVANO Nominal yield strength 310 N/mm2 Nominal tensile strength 415 N/mm2 Wall members

(i) Studs C-100 × 50× 20 × 1.2 (2400 mm long)

(ii) Chord members 2 No's C-100 × 50 ×20 × 1.2 (back to back)

(iii) Tracks U-103 × 50 × 1.2 (1200 mm long)

Steel to steel connections

(i) CFS framing member connections 4.8 × 16 mm self-drilling wafer-head screws.

(ii) Chord members to Hold-down connector

12 No's 8 mmΦGrade 8.8 HSFG bolts Frame to foundation connection

(i)Hold-down Purposely designed welded steel

hold-down

(ii) Hold-down anchors to foundation 2 No's 12 mmΦGrade 10.8 HSFG bolts (iii) Shear anchors 8 No's 12 mmΦGrade 10.8 HSFG bolts Steel to sheathing connections 3.9 × 35 mm self-drilling bulge-head

screws

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Fig. 7.ISO loading protocol as per ASTM E2126-19 [27].

Table 2

Elastic modulus and failure Stress for calcium silicate board used as sheathing.

Thickness E1 E2 v12 G12 FT1 FC1 FT2 FC2 S12

mm kN/mm² kN/mm² – kN/mm² N/mm² N/mm² N/mm² N/mm² N/mm²

8 3.52 4.61 0.14 1.00 4.04 14.81 2.78 13.74 7.46

10 4.34 6.64 0.16 1.00 3.99 15.68 2.08 12.50 7.46

12 4.93 4.93 0.19 1.00 4.63 13.45 2.78 10.10 7.46

FT1, FT2- Tensile failure stress in 1 and 2 direction.

FC1,FC2- Compressive failure stress in 1 and 2 direction.

S12= Shear stress in 1–2 plane.

Fig. 8.Deformed shape ofﬁnite element wall panel model.

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wmodelled using orthotropic plane stress element. The constitutive properties, the tensile, compressive and shear failure stresses along the longitudinal (1) and transverse (2) directions of the calcium silicate board are given inTable 2[20]. FromTable 2, it is observed that the sheathing material exhibits different strengths in tension and

compression in the 1 and 2 directions. Tsai-Hill failure criterion [1]

as given in Eq.18, was used. In Eq. 18,σ11,σ22are the normal stress in the 1 and 2 directions, respectively andσ12is the shear stress in 1–2 plane. The values ofIFgreater than or equal to 1.0 implies failure in the sheathing material.

Fig. 9.Comparison of CFSSWP monotonic F-d Response between FEA and Experiments.

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IF¼σ²11

X² −σ11σ22

X² þσ²22

Y² þσ²12

S² <1:0 where

X¼ X¼FT1 if σ11>0

¼FC1if σ11≤0

; Y¼ Y¼FT2 if σ11 >0

¼FC2if σ11≤0

& S¼S12

The VOSPE with the BWBN constitutive model, which was embed- ded in ABAQUS [1] through a user element *UEL subroutine, was used to represent the screw connection between the CFS framing members and the sheathing. Both the experiments and FEA results showed that the stresses in the CFS framing members and the calcium silicate board were well within the elastic limit, except at the screw connections (which is already accounted for by the screw connection model).

In the experimental studies, the bare CFS frame without the sheathing offered little resistance to the in-plane shear and deformed as a mechanism into a parallelogram shape. Hence, the connection between the stud and the track was modelled as a hinge. The out-plane DOF at all nodes in the wall panel model were restrained, since the panel was lat- erally restrained at the top and bottom in the tests. In the experimental study on the wall panel, the end studs were connected to the bottom track via hold-downs, and anchored to the reaction beam. The tracks were connected to the reaction beam with the shear anchors between the end and intermediate studs, as shown inFig. 6a. The hold-downs and shear connections were idealized as pinned and roller supports as shown inFig. 6b in the FEA model of the wall panel.

To avoid the numerical instability near and post-ultimate loads, and to track the post-peak behaviour, aﬁctitious horizontal spring was in- troduced in the FEA model along the horizontal in-plane shear loading direction at the top of the wall panel, using the SPRING2 element in ABAQUS. The displacement was applied at the top-left track stud joint, similar to that in the experiment. The deformed shape at an intermediate stage of loading is shown inFig. 8. The user element corresponding to the screw connection between the framing and sheathing elements is represented as‘X' inFig. 8. In the FE model, a coarser mesh with nodes at the screw locations in the wall panel was used for the boards, since the nonlinearity of the CFSSWP is governed essentially by the screw connections. At very early stages of loading in the experimental studies by Nithyadharan and Kalyanaraman [27], the boards experienced diagonal cracks along a line connecting the two end screws at the adjacent orthogonal edges of the corners, where the tensile stresses are experienced. The screws at these cracks were ineffective thereafter. The FEA of the model with these screws also indicated the diagonal failure of the sheathing at these end screw locations, at an early stage of loading.

Thereafter, these screws at the board corner cracking were made ineffective in the subsequent non-linear analysis.

4.1. Behaviour of CFSSWP under monotonic loading

The FE modelling and the analysis of the CFSSWP are carried out under monotonic loading with the VOSPE, NOSPE and OSPE model of the screw connections, and are compared with the experimental results of the CFSSWP with 8 mm, 10 mm and 12 mm board thickness and 25 mm edge distance from the free edge of the sheathing inFig. 9a, b and c, respectively.

Comparison of the experimental andﬁnite element results corresponding to the NOSPE, OSPE and VOSPE element models of the screw connections are presented inFig. 9, in terms of the in-plane shear forceF, versus the corresponding net in-plane sway displacement,dof the wall panel.The net in-plane sway displacement from the experiments is obtained by deducting the rigid body components of the panel deformation [27]. The values of the ultimate load,Fuand the corresponding net in-plane sway displacement,D_Fu, of the three panels tested under the monotonic loading [27], as obtained from the experiments and the FEA, are presented inTable 3.

Based onTable 3andFig. 9, the following observations are made:

• The initial stiffness of the wall panel as obtained from all the three FE modes results compare well with the experimental values. At this stage the behaviour of all components is essentially linear elastic.

The differences between the three FEA screw connection models

Table 3

Comparison of the ultimate load and the corresponding displacement.

Sheathing to frame connection element representation

Fu Dpeak

kN %

Difference

mm %

Difference 8 mm board & 25 mm edge distance

Experimental values 22.11 – 29.82 –

NOSPE 23.13 −4.61 30.67 −2.82

OSPE 22.92 −3.65 34.57 −15.90

VOSPE 21.43 3.07 32.57 −9.19

10 mm board & 25 mm edge distance

NOSPE 27.77 −15.19 33.72 −8.51

OSPE 27.31 −13.25 37.17 −19.58

VOSPE 25.75 −6.80 34.87 −12.18

12 mm board & 25 mm edge distance

NOSPE 35.01 −25.81 39.32 −25.92

OSPE 34.30 −23.24 41.72 −33.58

VOSPE 32.70 −17.49 40.06 −28.26 Fig. 10.Orientation of the screw forces acting from screws to the sheathing near ultimate load (in degrees with respect to nearest free edge).

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Fig. 11.Comparison of cyclic FEA results with wall panel experiments with 8 mm sheathing thickness.

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become more pronounced in the post-elastic range. The sub-ultimate deformation of the VOSPE model is more for the same load compared to the NOSPE and OSPE models, indicating the stiffening effect of the extra shear force in the screws orthogonal to the main spring in these other two models. The curve corresponding to the VSOPE screw model is generally closer to the experimental curve than the curves corresponding to the other two models. The ultimate strength corresponding to the VOSPE model is also closer to the experimental values.

• The displacement corresponding to the ultimate strength,DFu, from the FE model are between 3 and 33% higher in the FEA models compared to the experimental results (Table 3). At this stage almost all the screw connections along the perimeters have failed and the stiffness is mostly from the cold-formed steel framing members. Since the connections within the CFS framing members is assumed as hinged in the FEA model, there is an under estimation of the stiffness of the panel when the screw connections to the board have failed and hence the over estimation of the deﬂection in the VOSPE.

The wall panel results with the VOSPE model of the screw connections is generally superior to the OSPE and NOSPE screw connection models, due to its (a) better kinematic representation of the force between CFS framing and sheathing (b) non-linear modelling of the screw connection with the BWBN model, which is based on the entire prior loading history and (c) the screw constitutive model identiﬁed from the screw test results wherein the screws experience shear force parallel to the free edge of the sheathing [26]. The comparison of the load-displacement behaviour of the wall panel FE models with the experiments clearly shows that the performance of VOSPE model is better than the other screw models, for all the range of wall panel with different sheathing thicknesses, presented here.

The orientation of the VOSPE model screw forces nearer to the ultimate load, as obtained from the FEA model of the CFSSWP with 10 mm board thickness, is shown inFig. 10. It can be observed that majority of the screws in the vertical studs and the tracks (excluding the screws closest to the corners of the panel) experience forces within ±25° of the line parallel to the nearest free edge. Similar screw force orientation was also observed by Buonopane et al. [7] and Bian et al. [5] from their computational model of the wall panel. Hence, the use of a screw constitutive model based on tests wherein the shear in the screw is parallel to

the nearest board edge, is more appropriate than the one wherein the screw force is perpendicular to the nearest board edge.

4.2. CFSSWP under reversed cyclic loading

Figs. 11, 12, 13, show the comparison between the FEA results and the results of the experimental studies carried out on CFFSWP with 8 mm, 10 mm and 12 mm thick sheathing under net displacement loading protocol [28]. Theﬁgures show (a) the net in-plane displacement loading protocol, (b) the comparison between the in-plane shear force F,and the corresponding net in-plane sway displacementd, (c) the comparison of the load-displacement loop at the ultimate cycle alone and (d) the comparison of the cumulative energy dissipated over the cycle history.Table 4presents a typical comparison of the average of the ab- solute peak value of the maximum positive and negative forces and the corresponding average displacements for each complete cycle, as obtained from the FEA and the experimental values for the CFSSWP with 10 mm thickness and 25 mm edge distance from the free edge of sheathing.

Based onFigs. 11, 12 and 13andTable 4the following observations are made:

• The cyclic load deformation curves obtained from the FEA compare reasonably well with the experimental results. The differences are mainly due to the sharper peak values obtained in FEA compared to the strain hardening behaviour seen in experimental results in each cycle. The comparison is superior to other similar test comparisons reported in literature.

• The peak forces obtained from the FEA are generally slightly larger than that obtained from the experiments, again due to the sharper peaks in FEA results.

• The BWBN model assumes that the peak load under cyclic loading to be reached at the peak displacement in the cycle. Hence, the FEA results of the wall panel behaviour based on the BWBN model exhibit the peak load and the peak displacement to be occurring simultaneously. Whereas, the experimental results of both the screw connections and the wall panel indicate that a small strain softening occurs before the peak displacement, in the non-linear loading ranges.

• There is no known comparison of the energy dissipation during the cyclic loading in literature, as represented inFigs. 11d,12d and13d.

Both the energy dissipated per cycle and the cumulative energy Fig. 12(continued).

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dissipated, as obtained from the FEA are less compared to that obtained from the experiments. This is partly due to the lower energy dissipation (around 5–30%) seen in the BWBN models compared to that in the screw connection experiments in predicting the unknown deformation trajectory under cyclic loading [28], and partly due to the energy dissipated in connection within framing members (not included in the analysis).

Thefigures demonstrate a reasonably good validation of the FEA results of the wall panels in terms of both hysteretic trace and energy dissipation. The correlation coefficient between the FEA generated load displacement curves of the wall panel and the experimental data is 0.96. In addition, fromTables 4, it is seen that thefirst 4 to 5 cycles (usually in the linear elastic range), the error in peak load as obtained from the FEA compared to the experimental values is nearly 50%, indicating that the initial elastic load deformation is not adequately captured by the BWBN constitutive model based on cyclic loading. For the early cycles in the elastic range the model based on monotonic loading gives a better comparison. In the 5th to 10th cycle the error in peak force from FEA is around 20 to 35%. From the 10th cycle onwards and until the specimen reaches the ultimate load under the cyclic load, the error in the peak load from FEA is usually within 15%. It is clear that a model calibrated using the full cyclic loading of the test data of the screw connection is likely to be influenced by all the data and hence may show larger error in the ranges where the number of data points used for calibration are lesser. Similar comparison of the peak loads was observed for the other CFSSWP with 8 mm, 10 mm and 12 mm board thickness and 25 mm edge distance from free edge of sheathing.

In summary, the FEA results capture the trends in the load deformation and the energy dissipation very well, although error in actual numbers is to be noticed. However, the variation of the FEA results compared to the experiments is much higher in the other literature in this area.

Hence, it can be concluded that the proposed screw connection modelling strategy, based on the VOSPE with the BWBN constitutive model of the screw connection is capable of capturing the wall panel experimental behaviour under the static and cyclic in-plane shear loadings, with reasonable accuracy for the range of wall panel results examined.

5. Nonlinear time history analysis

An objective of developing a good FEA modelling capability of the wall panel based on experimental data of screw connections, is to study the performance of the wall panel building system under earthquake loading. This would help with the development of the performance-based design methods for earthquake loads. In this section, a nonlinear time history analysis of a CFSSWP under an earthquake loading is demonstrated, by adding the dynamic equation of motion in the state vector form to the set of ODE's in Eq.12 [13,28]. The wall panel macro model is essential for conducting nonlinear time history analysis. In the absence of the CFSSWP experimental load- displacement data, which is expensive to develop experimentally for each case, it is expeditious to develop such a load-displacement response through FE modelling of the wall panel with VOSPE model for the screw connections.

A nonlinear time history analysis of the wall panel with 10 mm thick board and 25 mm screw edge distance when subjected to the Kobe 1995 earthquake time history [42], is demonstrated as an example. A weight of 125 kg is used to model the mass at the top and a damping ratio of 5%

is assumed. To understand the adequacy of the micro mechanical VOSPE screw model in the nonlinear time history analysis, two results are compared; (1) uses the BWBN model calibrated from the experimental data of the full wall panel (referred to as Exp-BWBN) [27], and (2) uses the BWBN model of the load–displacement response of the wall panel obtained from the micro mechanical FEA model of the full wall panel with the VOSPE screw connection, (referred to as FEA-BWBN), (Fig. 12b). The nonlinear response comparison under the Kobe earthquake is presented inFig. 14. It is observed that the mass normalized force at the top of the panel is essentially similar for the two wall panel models (Exp-BWBN and FEA-BWBN) (Fig. 14a). The comparison of the mass normalized force and the deformation from the nonlinear time history response is shown inFig. 14b.

Due to the differences in the hysteretic energy dissipated in the two models of the wall panel in each cycle (Fig. 14b), the cumulative energy dissipated varies by 25–32% (Fig. 14c). This study shows that it is possible to use the sub-element models of the screw connections from screw connection experiments, and obtain a good estimate of the response of the wall panel building system under random earthquake loading, without having to resort to expensive wall panel tests.

6. Summary and conclusions

In this paper, a VOSPE element was proposed for modelling the screw connections between the CFS framing and the sheathing, to analyse the behaviour of cold formed steel wall panels under monotonic and cyclic loading, as affected by the screw connections. The differential BWBN hysteretic model was used to represent the constitutive relation- ship of the screw connections, which at any stage is based on the cumulative time history up to the current time. The VOSPE element was included as a user element *UEL in ABAQUS for the FEA of the wall panel. The responses obtained from the micro mechanical FEA of the wall panels were compared with the monotonic and cyclic loading experimental results on the wall panels under in-plane shear deformation.

The FEA results of the CFSSWP with the screw connections modelled using VOSPE compare better than the existing screw connection models such as NOSPE and OSPE. The cumulative energy dissipated by the wall panel under cyclic loading as obtained from the FEA is about 25–32%

less compared to experimental values. This is a good when compared to results presented in literature on this topic. The differences in the model values may be due to additional energy dissipated by the frame end connections in advanced loading range, not considered in the analysis. In order to model the screw connection even more accurately, it Table 4

Comparison of peak displacements (Upeak) and the corresponding peak forces (Fpeak) in each cycle from experiment and FEA in the wall panel.

Cycle No. Upeak FPeak

Experiment FEM Experiment FEM % Difference

mm mm kN kN

1 0.25 0.26 1.68 0.80 −53

2 0.51 0.51 2.97 1.58 −47

3 0.96 0.97 4.70 3.01 −36

4 1.45 1.46 5.77 4.53 −22

5 2.03 2.03 6.73 6.24 −7

6 4.36 4.38 9.94 12.75 28

7 4.44 4.47 9.52 12.68 33

8 4.48 4.50 9.30 12.22 31

9 8.78 8.87 15.66 21.75 39

10 8.91 8.99 14.90 18.66 25

11 9.02 9.08 14.51 16.37 13

12 13.25 13.43 19.75 22.87 16

13 13.69 13.79 18.02 20.65 15

14 13.87 14.01 17.04 18.67 10

15 18.23 18.95 20.49 23.52 15

16 19.33 19.50 17.72 20.84 18

17 19.62 19.81 16.48 18.17 10

18 24.31 25.15 19.04 21.28 12

19 25.44 25.87 16.13 18.51 15

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may be necessary to generate data on the screw behaviour, where the screw shear is at different angles to the free edge and the two orthotropic axis of the board and use them in the FEA model.

Finally, it was shown that it is possible to numerically generate and study the performance of wall panels and the building systems under the monotonic, cyclic and random earthquake loadings, using the

micro modelling and the BWBN model for the screw connections derived from tests. These results can be used in the development of performance-based design method for these structures. Further, vali- dating the proposed VOSPE model with independent studies on wall panels having different aspect ratio, sheathing materials with various thickness, is in the scope for future work in this area.

Fig. 14.SDOF Nonlinear response of CFSSWP model under Kobe earthquake.

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Declaration of Competing Interest

The authors declare that they have no known competingﬁnancial interests or personal relationships that could have appeared to inﬂu- ence the work reported in this paper.

Acknowledgements

This research did not receive any speciﬁc grant from funding agen- cies in the public, commercial, or not-for proﬁt sectors.

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