1.a Predictive Analytical Model for the Elasto-plastic Behaviour of a Light Timber-frame Shear-wall

Construction and Building Materials xxx (2015) xxx–xxx

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Construction and Building Materials

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A predictive analytical model for the elasto-plastic behaviour of a lighttimber-frame shear-wall q

http://dx.doi.org/10.1016/j.conbuildmat.2015.06.0250950-0618/� 2015 Elsevier Ltd. All rights reserved.

q This document is a collaborative effort.⇑ Corresponding author.

E-mail addresses: [email protected] (D. Casagrande), [email protected] (S. Rossi).

Please cite this article in press as: D. Casagrande et al., A predictive analytical model for the elasto-plastic behaviour of a light timber-frame sheaConstr. Build. Mater. (2015), http://dx.doi.org/10.1016/j.conbuildmat.2015.06.025

Daniele Casagrande, Simone Rossi ⇑, Roberto Tomasi, Gianluca MischiDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, Italy

h i g h l i g h t s

� The behaviour of timber shear walls subjected to horizontal forces in the elasto-plastic field is investigated.� The ductility of the light timber-frame walls is determined.� A numerical modelling of timber shear-wall is developed.� Some laboratory tests on timber shear walls are presented.

a r t i c l e i n f o

Article history:Received 12 February 2015Received in revised form 3 June 2015Accepted 10 June 2015Available online xxxx

Keywords:Light timber-frame wallElasto-plastic behaviourAnalytical modelDuctilitySeismic capacityExperimental tests

a b s t r a c t

This paper presents a predictive analytical model for the elasto-plastic behaviour of a light timber-framewall under horizontal loading. The possibility to represent the total force carried by all fasteners (allow-ing for their sequential yielding) in one nonlinear spring is shown to be a key benefit. The development ofthis spring was investigated via a parametric study in which the variables were the sheathing panelaspect ratio and the fastener spacing. By developing equivalent springs for the other components, a rhe-ological model for elasto-plastic behaviour of a sheathed timber-frame as function of the mechanicalproperties of connections was also defined.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

A fundamental step in the investigation of the seismic capacityof light timber-frame shear-walls structure is the study of the nonlinear behaviour of a single shear wall. The most common strate-gies in seismic analysis and design consider in fact that the struc-ture global capacity strongly depends on the local ductility of thestructural elements. Structures are hence designed so that the seis-mic energy dissipation is located in some structural componentswhich should be designed to yield during a seismic event (ductileelements). On the contrary, the other components must bedesigned to remain in the elastic range (brittle components)according to the capacity design approach [1–3].

Several seismic analysis methods are suggested by Standards,but linear elastic analyses are mostly performed in practice. The

nonlinear response of the structure to the seismic event is thenconsidered by modification of the elastic seismic forces via thebehaviour factor of the structure. This depends on the globalover-strength of the structure and on the global ductility, which,in turn, is related to the ductility of the local components. For thisreason, in order to define accurately the q value of a timber struc-ture (see [4,5]), the relationship between the ductility of the com-ponents (e.g. fasteners, hold-downs, angle-brackets, etc.) and theglobal ductility should be pre-determined. In several internationalStandards for seismic design of light timber-frame buildings, val-ues of q are suggested. However, a specific relationship betweenthe local and the global ductility of the structure, differently fromother structural types, e.g. concrete or steel, is not provided. Localcomponents, where the yielding is expected, should in fact bedesigned by reference to the ductility demand of the entire struc-ture: the greater the q value, the higher the ductility demand of thelocal components. An analytical relationship is hence necessary todetermine the local demand.

For this purpose a predictive analytical model for theelasto-plastic behaviour of a light timber-frame shear wall under

r-wall,

http://dx.doi.org/10.1016/j.conbuildmat.2015.06.025

mailto:[email protected]

mailto:simone.rossi1@ unitn.it

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2 D. Casagrande et al. / Construction and Building Materials xxx (2015) xxx–xxx

horizontal loading is presented in this paper. In particular, themain goal of this model is to link the local properties, i.e. ductility,of each component to the global properties of a single wall. Thisdoes not fully define the required relationship, but it representsthe first fundamental step. One of the key parts of the work pre-sented in the next sections is in fact the representation of themechanical behaviour of all sheathing-to-frame fasteners in onenonlinear horizontal spring. This approach reduces considerablythe complexity of the wall model and for this reason it may be verysuitable to analyse the non-linear behaviour of an entire building.The number of degrees-of-freedom and hence the run-time of themodel are in fact significantly lower than those of a model whereall fasteners are represented as non linear elements. The employ-ment of the non linear static analysis method on severaltimber-frame structures represents in fact one of the future possi-ble developments of the work presented in this paper, correlatingthe ductility of each wall to the global structure ductility.

2. Analytical models for the behaviour of light timber-frameshear-walls under horizontal loading

The elastic behaviour of a light timber-frame shear-wall sub-jected to a horizontal load can be obtained by means of severalanalytical expressions proposed in literature or in Standards, con-sidering different contributions to deflection from structural com-ponents. In [6] four contributions are taken into account, due tosheathing-to-framing connection, the shear deformation of panel,the wood-frame and the rigid-body rotation of the wall causedby the compression perpendicular to the grain of the compressedstud respectively. The same deformation contributions arereported in the [7] considering in the rigid-body rotation of thewall also the deformation of anchor devices subjected to a tensileforce. On the contrary, in [8] the rigid-body rotation contributionis calculated only from the total vertical elongation of the wallanchorage system. A similar approach was proposed in [9,10]defining an equivalent single degree of freedom model. In themodel proposed in [11] four contribution are considered too,neglecting the wood-frame contribution and the compression per-pendicular to the grain, but adding the rigid-body translation of thewall due to anchorage system and taking into account the stabiliz-ing effect of the vertical load acting on the wall.

Regarding the non-linear behaviour, many studies were con-ducted to investigate the capacity of timber shear-walls under

(a)Fig. 1. Light timber-frame wall: (a) simplifi

Please cite this article in press as: D. Casagrande et al., A predictive analyticalConstr. Build. Mater. (2015), http://dx.doi.org/10.1016/j.conbuildmat.2015.06.0

horizontal loads simulating seismic actions, see [12–16].However, most of them were focused on experimental tests oradvanced numerical modelling rather than on the proposal of ananalytical expression to predict the non-linear behaviour.

In [17] a plastic model is proposed for the analysis of fullyanchored light frame-timber shear-walls applying the upper andlower bound methods and evaluating the plastic strength of a fullyanchored wall. The expressions proposed in [10], characterised bylinear relationship to the sheathing-to-fastener deformation en andto the anchorage connection deformation da, might be used also inthe non-linear range since the displacement of the wall due tothese two contributions was obtained by geometrical considera-tions on the deformation of the wall.

In the paper an alternative approach is proposed to predict ana-lytically the elasto-plastic behaviour of a timber shear-wall bymeans of the definition of a rheological model. The deformationcontributions taken into account refer to the elastic model sug-gested in [11] which includes the mathematical model proposedin [18] for the sheathing-to-framing fastener deformation.However, as reported in next sections, more deformation contribu-tions can be added to the rheological model if these cannot beneglected.

3. Rheological model for the elasto-plastic behaviour of a lighttimber-frame shear wall

According to [11] the elastic behaviour of a light timber-frameshear wall under a horizontal force F and a uniform distributed ver-tical load q, can be represented by a simple pinned frame, bracedby a horizontal spring of stiffness equal to KSH representing thesheathing-to-framing connection see Fig. 1. The contribution givenby the devices, which prevents the horizontal translation of thewall, is represented by horizontal spring of stiffness KA connectedto the ground, whereas the rigid-body rotation, arising from thehold-down device, is taken into account by means of a verticalspring of stiffness equal to kh. The replacement of thesheathing-to-framing connection fasteners with a single horizontalspring (KSH) allows in fact a considerable reduction of the degreesof freedom of the model.

In [11] the frame internal equivalent spring is characterised by astiffness equal to KSP , which accounts for the sheathing-to-framingconnection stiffness KSH and the sheathing shear deformation KP .Because KP is usually much greater than KSH we get:

(b)ed numerical model, (b) configuration.

model for the elasto-plastic behaviour of a light timber-frame shear-wall,25


D. Casagrande et al. / Construction and Building Materials xxx (2015) xxx–xxx 3

1KSP¼ 1

KPþ 1

KSHffi 1

KSH. Is important to note that to consider even the

stiffness contribution of the sheathing panel KP;KSH has to bereplaced in the following sections by KSP .

The implementation of the wall model in the non-linear range isquite simple and straightforward: each spring is described in gen-eral by a non-linear curve. In order to obtain a simple analyticalexpression relating the behaviour of each individual connectionand of the wall, the non-linear behaviour of each spring is assumedto be described by an elasto-perfectly plastic idealised force vs dis-placement curve, characterised by stiffness, strength and ductility.

Assuming an elasto perfectly plastic behaviour of each modelspring, the non-linear mechanical behaviour of the wall is there-fore described by a bi-linear or tree-linear curve, as showed here-inafter, depending on the magnitude of the vertical distributed q.

In order to obtain a simple analytical relationship between theparameters (stiffness, strength and ductility) of each non-linearspring and the mechanical parameters of the wall, the modelhas been substituted by a rheological model characterisedby means of two in-series non-linear horizontal springs(sheathing-to-framing KSH and rigid translation KA) and a third ele-ment, placed in series with the described horizontal springs, madeup by a non-linear horizontal spring KH (representing therigid-body rotation) placed in parallel to a friction block (Fq)standing for the vertical load contribution, see Fig. 2.

The main goal of the paper is to propose a general approach torelate the local mechanical properties to the wall ones in a generalway. However, the rheological model can be updated adding otherelasto-plastic springs if other deformation contributions were to beconsidered. The same approach reported in next sections (devel-oped considering only the contributions due to hold-down, anglebrackets and fasteners) can be adopted. The analytical expressionscan be in fact easily modified.

3.1. Rigid-body rotation connection

The parameters, describing the mechanical behaviour of therigid-body rotation connection, can be obtained by geometricaland mechanical observations of the simplified numerical modelof the wall [11], depending on the vertical load q, on the geometry

Fig. 2. Rheological model.

Fig. 3. Overturning and


of the wall (height h and length l) and on the mechanical parame-ters that characterise the hold-down (stiffness kh, strength RH andductility lh), Fig. 5.

The hold-down device, used to prevent the wall from the rigidrotation, is loaded by a tensile force only in the event that the over-turning moment Movt , caused by the horizontal force F, is greaterthan the stabilizing moment Mstb, resulting from vertical load q,see Fig. 3. This condition occurs when:

Movt ¼ F � h P Mstb ¼q � l2

2ð1Þ

Hence the value Fq of the horizontal force which characterisesthe friction block (Fig. 4) is given by:

Fq ¼q � l2

2 � h ð2Þ

If the horizontal force F is lower than Fq, the hold-down deviceis not in tension and the wall undergoes no rotation. On the con-trary, if F is greater than Fq , the hold-down device is in tensionand the wall deformation is also characterised by the rigid-bodyrotation contribution.

The mechanical behaviour of the hold-down horizontal spring,see Fig. 4, can be obtained from the hold-down elasto-perfectlyplastic curve, see Fig. 5, by-means of some simple analyticalexpressions. The hold-down horizontal elasto-perfectly plasticcurve is characterised by the stiffness KH , the strength RH and theductility lH , whereas the hold-down connection devices curve isdescribed by the stiffness kh, the resistance RH and ductility lh.

The strength RH can be directly calculated from the hold-downstrength rh as:

RH ¼ nh �rh � s � l

hð3Þ

- RH is the hold-down strength;- l is the length of the wall;- nh is the number of hold-downs for each corner of the wall;

stabilizing moment.

Fig. 4. Rigid-rotation (horizontal) spring mechanical behaviour.



Fig. 5. Hold-down mechanical behaviour.


- s represents the internal level arm ratio, usually between0.95–1.

The yield displacement DY ;H can be obtained by the hold-downyield displacement dy;h, by means of a simple geometricaltransformation:

DY;H ¼dy;h

s � l � h ð4Þ

The stiffness KH is therefore given by:

KH ¼RH

DY ;H¼ nh � kh �

s � lh

� �2

ð5Þ

Similarly, the ultimate displacement DU;H can be obtained as:

DU;H ¼du;h

s � l � h ð6Þ

therefore the ductility lH is equal to the hold-down ductility lh,according to the following expression:

lH ¼DU;H

DY ;H¼ du;h

dy;h¼ lh: ð7Þ

Fig. 7. Rigid translation spring mechanical behaviour.

3.2. Rigid-body translation connection

The rigid-body translation of the wall is usually prevented bymeans of metallic angle brackets (nailed or screwed to the wall)or inclined screws. If the devices are placed along the wall lengthwith constant spacing ia the number of devices na can be obtainedby:

na ¼lia

ð8Þ

The idealised elasto-perfectly plastic linear curve of each devicecan be obtained by a numerical model or by the bi-linearisation ofthe experimental curve, defining its strength ra, its stiffness ka andits ductility la (Fig. 6). The parameters, which characterise themechanical behaviour of the horizontal non-linear spring KA ofthe rheological model (Fig. 7), can be obtained by isolating the con-tribution of the wall rigid translation:

DY;A ¼ dy;a ð9Þ

DU;A ¼ du;a ð10Þ

Fig. 6. Angle brackets (or screws) mechanical behaviour.


RA ¼ra � l

ia¼ ra � na ð11Þ

KA ¼ka � l

ia¼ ka � na ð12Þ

lA ¼du;a

dy;a¼ la ¼

DU;A

DY ;Að13Þ

3.3. Sheathing-to-framing connection

The sheathing-to-framing connection, represented by the hori-zontal non-linear spring indicated with KSH , see Fig. 9, takes intoaccount the deformation contribution given by the fasteners (nailsor staples) which connect the wood frame to the sheathing panel.However, the mechanical behaviour of the connection (and thusthe strength RSH , the stiffness KSH and the ductility lSH) does notdepend only on the mechanical behaviour of the fasteners (thestrength f c , the stiffness kc and the ductility lc , see Fig. 8), but itis also strongly influenced by their disposition. Since fastenersare generally placed with constant spacing along the edge of thepanel, only the spacing s and the ratio between the height andthe length of the panel h=b can be considered. Hence, in general,the mechanical behaviour of each fastener is not equal to thesheathing-to-framing connection one.

The fastener elasto-perfectly plastic curve can be obtained, forexample, by experimental tests (monotonic or cyclic test, in thesame way of angle brackets or hold-downs); performing experi-mental tests on full-scale walls, considering all the possible cases(varying the type of fastener, the fastener spacing s and the ratioh=b of the panel) results anyway burdensome and expensive.Consequently an analytical expression, relating the mechanicalbehaviour of the fasteners to the sheathing-to-framing connectionone, is required.

In European Standard for timber structures [19] a relationshipbetween the strength of fasteners f c and the sheathing to panelconnection strength RSH is reported. This equation was obtainedby means of the limit analysis static theorem assuming an equaldistribution of the shear stresses on the edge of the panel andhence a constant shear action on each fastener. For a lighttimber-frame wall braced by several panels with a width equalto bi the expression is given by:

Fig. 8. Fastener mechanical behaviour.



Fig. 9. Sheathing-to-framing connection spring mechanical behaviour.


RSH ¼ nbs � rc �P

bi � ci

sð14Þ

where:

- nbs is the number of the wall braced sides (1 or 2);

- ci ¼1 if a < 2a2 if 2 < a < 40 if a > 4

8<:

- a ¼ hb is the panel shape parameter;

- bi is the panel width.

Moreover, the sheathing-to-framing connection strength RSH canbe increased by a factor equal to 1.2 according to the [19].

When the wood frame can be assumed rigid, namely the flexu-ral deformation of studs and plats is negligible (this hypothesis canbe assumed realistic in most cases, see [20]), the stiffness KSH canbe obtained directly by the model proposed in [18], known thestiffness kc of fasteners, the spacing s, the panel parameter a andthe wall length:

KSH ¼nbs � kc

s �P kiðaiÞ

bi

ð15Þ

where kjðajÞ ¼ 0:810þ 1:855 � aj, see [11].According to Eqs. (14) and (15), the yield displacement DY ;SH is

therefore calculated as:

DY ;SH ¼RSH

KSHð16Þ

whereas the ultimate displacement DU;SH is given by:

Fig. 10. Trilinear mechanical


DU;SH ¼ lSH � DY ;SH ð17Þ

Concerning with the sheathing-to-framing connection ductilitylSH , Standards do not specifically suggest an expression for its cal-culation, known the fastener ductility lc.

The equivalent single degree of freedom model [10], relates lin-early the sheathing-to-framing connection deformation DSH ¼ Dnail

to the nail deformation dc according to linear geometricalassumptions:

Dnail ¼ kswn � dc ð18Þ

where kswn depends on geometrical properties of the panel. Hence itis not difficult to show that the sheathing-to-framing connectionductility lSH results equal to fastener ductility lc . However thismodel, differently from the model proposed in [18], is based onthe assumption that the nails along the perimeter of the panel areequally stressed.

In Section 4, an alternative analytical expression based on themodel developed in [18] is proposed, evaluating the evolving stressin each nail due to the load increasing on the wall and demonstrat-ing that in several cases the sheathing-to-framing connection duc-tility lSH results lower than the fastener ductility lc. A linearrelationship is however confirmed.

3.4. Definition of the idealised elasto-perfectly plastic curve of a wood-framed wall

After defining the idealised elasto-perfectly plastic curve ofeach element of the rheological model (the three horizontal springsand the friction block), the elasto-plastic curve of the entire model,and hence of the wall, can be obtained. Known the mechanicalbehaviour of each component of the model, the mechanical prop-erties, which define the model curve, can be calculated by meansof simple mathematical expressions. As reported in Fig. 10, theparameters of the curve are the friction block yield force Fq, thewall strength RW , the wall stiffness Ktot;nt , when the rotation contri-bution is not considered (block friction not yielded), the wall stiff-ness Ktot , when the rotation contribution is considered, the wallsecant stiffness KW , the wall displacement Dq;W when the frictionblock yields, the wall yield displacement DY ;W , the wall ultimatedisplacement DU;W .

curve wood-framed wall.




The friction block yield force Fq can be calculated according toEq. (2) depending on the wall geometry and the vertical load q.

The wall stiffness Ktot;nt depends only on the sheathing-to-framing and the rigid-body translation contribution. It can beobtained by:

1Ktot;nt

¼ 1KSHþ 1

KAð19Þ

The displacement Dq;W of the system for which the friction blockyields results:

Dq;W ¼Fq

Ktot;nt¼ q � l2

2 � h �1

KSHþ 1

KA

� �ð20Þ

The wall strength RW is defined as the minimum value from thestrength of each connection (sheathing-to-framing, translation androtation) according to the following expression:

RW ¼ minðRH þ Fq; RA; RSHÞ ð21Þ

The weakest connection, which firstly yields since characterisedby the minimum strength, can be identified by the index i, definedas:

if RW ¼ RH þ Fq ! i ¼ H

if RW ¼ RA ! i ¼ A

if RW ¼ RSH ! i ¼ SH

8><>: ð22Þ

If the wall strength RW is greater than the friction block yieldforce Fq, the curve of the mechanical behaviour of the wall is char-acterised by an additional linear elastic segment and for this reasonit is described by a three-linear curve (Fig. 10).

The wall stiffness Ktot can be calculated taking into account thestiffness of each model component (sheathing-to-framing, transla-tion and rotation) according to the following expression:

1Ktot¼ 1

KSHþ 1

KAþ 1

KHð23Þ

Therefore the wall yield displacement DY;W can be obtained by:

DY;W ¼Fq

Ktot;ntþ RW � Fq

Ktot¼ RW

Ktot� Fq

KHð24Þ

Hence the wall secant stiffness KW , defined as the ratio betweenthe wall strength RW and the yield displacement DY ;W , is given by:

Fig. 11. Bilinear mechanical c


KW ¼RW

DY;W¼ RW

RWKtot� Fq

KH

¼ 1Ktot� Fq

KH � RW

� ��1

ð25Þ

Substituting:

n ¼ Fq

RWð26Þ

for n < 1 we get:

1KW¼ 1

Ktot� n

KHð27Þ

When the wall strength RW is lower than the block friction acti-vation force Fq (n > 1) the friction block does not yield. This condi-tion usually occurs in case of weak fasteners or a high vertical load.The mechanical curve of the wall is therefore bi-linear (Fig. 11),and the secant stiffness KW results equal to Ktot;nt .

1KW¼ 1

Ktot;nt¼ 1

KSHþ 1

KAð28Þ

The wall yield displacement DY ;W can be obtained by:

DY;W ¼RW

Ktot;nt¼ RW

1KSHþ 1

KA

ð29Þ

Therefore, considering the two different cases, the wall secantstiffness KW and the wall yield displacement can be defined bythe following expressions:

n P 1! 1KW¼ 1

Ktot;nt

n < 1! 1KW¼ 1

Ktot� Ktot

KH

(ð30Þ

The wall yielding displacement DY;W can be calculated in bothcases as:

DY;W ¼RW

KWð31Þ

For the evaluation of the wall ductility lW , the plastic displace-ment of the rheological model is equal to the plastic displacementof the weakest (and hence yielded) connection. For this reason, anincrease of the system displacement is caused only by the stretchof the spring representing the weakest yielded connection. In fact,an increase of the stretch of the other elastic spring would requirean increase of the external force F. Therefore we get:

urve wood-framed wall.



(a) (b)

Fig. 12. (a) Fasteners bi-directional spring and yielding surface; (b) elastic-perfectly


Dpl;W ¼ Dpl;i ð32Þ

The wall ductility lW is defined as the ratio between the wallultimate displacement DU;W and the wall yield displacementDY;W . Because the ultimate displacement DU;W is given by thesum of the yield displacement DY;W and the plastic displacementDpl;i we obtain:

lW ¼DU;W

DY ;W¼ DY ;W þ Dpl;W

DY;W¼ 1þ Dpl;W

DY ;W¼ 1þ Dpl;i

DY ;Wð33Þ

The plastic displacement of the weakest connection Dpl;i can becorrelated directly to the yield displacement of the same connec-tion DY ;i and to its ductility li according to the followingexpression:

Dpl;i ¼ DU;i � DY;i ¼Ri

Ki� li � 1� �

ð34Þ

Substituting Eqs. (31) and (34) in the Eq. (33), we obtain:

lW ¼ 1þRiKi� ðli � 1Þ

RWKW

ð35Þ

If the weakest element is represented by thesheathing-to-framing connection or the rigid translation connec-tion, the wall strength RW is given by:

RW ¼ Fi ð36Þ

with:

i ¼ SH or A

Therefore, the wall ductility can be obtained by the followingsimplified equation:

lW ¼ 1þ KW

Ki� ðli � 1Þ ¼ 1þ j � ðli � 1Þ ð37Þ

As the j parameter is lower than 1, the ductility of the weakestconnection li is always greater than the wall ductility lW .

If the weakest connection is represented by the sheathing-toframing or the translation connection (i ¼ SH or A), in case ofn P 1 we get:

j ¼ KW

Ki< 1! 1

Ki<

1KW! 1

Ki<

1KSHþ 1

KAð38Þ

whereas in case of 0 < n < 1 we obtain:

j ¼ KW

Ki< 1! 1

Ki<

1KW! 1

Ki<

1KSHþ 1

KAþ 1� Ktot

KHð39Þ

If the weakest connection is represented by the rigid-body rota-tion one (i ¼ H), we obtain:

RW ¼ RH þ Fq > RH ð40Þ

In this case the wall ductility lW can be calculated by means ofthe following expression:

lW ¼ 1þ RH

RH þ Fq� KW

KH� ðli � 1Þ ¼ 1þ i � j � ðli � 1Þ ð41Þ

Since j < 1 and i < 1, as shown previously, the weakest con-nection ductility is greater than the wall ductility. For this reason,in order to maximise the wall ductility, the stiffness of the strongconnections which has to remain in the elastic range, should beas great as possible so that the parameter j tends to 1.

Known the wall ductility, the wall ultimate displacement DY ;W

can be obtained by:

DU;W ¼ lW � DY;W : ð42Þ


4. Non-linear mechanical behaviour of a fully anchored lighttimber-frame wall

A light timber-frame wall is defined fully-anchored if the stiff-ness constraints, which prevent the rigid-body motion of the wall(KA;KH), can be considered infinitely rigid. The wall deformation,according to the models described in the previous section, is repre-sented only by the sheathing-to-framing connection contribution(KSH).

In this section a fully anchored wall model is used to obtain ananalytical relationship between the non-linear behaviour of thesheathing-to-framing connection ductility lSH and the fastenerductility lc . This relationship is in fact necessary, as reported inSection 3.3, for the rheological model making.

The analytical expression was carried out by means of anelasto-plastic analysis, increasing step by step the external hori-zontal force F and assuming a redistribution of the forces of thefasteners.

The analysis was performed also defining a kinematic mecha-nisms of the wall as well, up to the wall failure condition relatedto the achievement of the acceptable ultimate displacement du;c

of one fastener at least.The mathematical model proposed by [21] was used to perform

the analysis at each step, assuming the wood-frame as rigid.According to that, the wall frame was represented by pinnedbeams (the frame is hence not restrained for horizontal loads)whereas the sheathing panel was assumed like a rigid-body. Thefasteners were modelled by bi-directional elastic spring: the inter-nal force of the fasteners was hence linear to their displacement,see Fig. 12. The fasteners position was described considering areferring system placed in the centre of gravity of the fasteners.Generally, the fastener disposition is symmetric (the fastenersare placed along the edge of the panel with an equal spacing),therefore the origin of the referring system was placed at the cen-tre of the panel. The external force F was applied at the top cornerof the frame.

At each step the structure was analysed assuming its elasticbehaviour (the solution was obtained by means of the method ofminimum of the potential energy) after updating the stiffnessmatrix of the model in order to consider that some fasteners hadalready yielded at the previous steps.

When the stiffness matrix of the model becomes singular, fur-ther elastic steps are no more achievable: the structure is to besolved by means of the kinematic theory.

According to [19], the yielding surface on the mathematicalmodel of fasteners was assumed circular since fasteners have asmall diameter and hence their mechanical properties are notinfluenced by the grain orientation. The yielding condition occurswhen global displacement, obtained summing the relative dis-placement Dj, is equal to dy;c. After the fastener has yielded, aninternal constant force equal to rc is assumed, independently onthe direction of the fastener relative displacements in the plasticphase Djþ1 . . . Dn, see Fig. 12 (the ultimate displacement surface is

plastic behaviour of fasteners.



Fig. 13. Fully anchored wall kinematic mechanisms.

Table 1Fully-anchored wall ductility a ¼ 1; n case which the collapse displacement occursbefore the kinematic.

Ductility lSH with a ¼ 1

lc s/b

1/2 1/4 1/6 1/8 1/12 1/25

1.00 1.00 1.00 1.00 1.00 1.00 1.001.50 1.38 1.29 1.27 1.27 1.26 1.262.00 1.73 1.63 1.60 1.60 1.60 1.592.50 2.07 1.96 1.92 1.92 1.91 1.913.00 2.40 2.28 2.23 2.23 2.22 2.223.50 2.73 2.59 2.54 2.54 2.53 2.524.00 3.05 2.89 2.84 2.84 2.83 2.824.50 3.37 3.20 3.14 3.14 3.13 3.125.00 3.69 3.50 3.44 3.43 3.42 3.425.50 4.00 3.80 3.74 3.73 3.72 3.716.00 4.32 4.10 4.03 4.03 4.01 4.016.50 4.63 4.40 4.33 4.32 4.31 4.307.00 4.94 4.70 4.63 4.62 4.60 4.597.50 5.26 5.00 4.92 4.91 4.89 4.898.00 5.57 5.29 5.22 5.21 5.19 5.18


assumed circular as well). As reported in [22], the assumption ofcircular surfaces to represent the internal state of the fastenerscannot be represented by two perpendicular springs, each of oneis characterised by an elasto-plastic curve. In this case in fact theyielding surface is squared and the yielding condition is dependenton the direction. Assuming for example a loading direction of fas-tener equal to 45�, the yielding condition is reached when in bothsprings a displacement equal to dy;c is achieved. This corresponds to

a global displacement equal toffiffiffi2p

dy;c , greater then dy;c .Analyzing the common cases in practice, three different types of

kinematic-model for a light timber-frame wall can be defined. Ineach kinematic-model, the lateral stability of the wall is not longerensured because of the progressive yielding of the fasteners. It isalso important to remark that the stiffness of a yielded fasteneris assumed equal to zero, according to the constitutive law usedfor it (see Fig. 12).

The first kinematic model is defined vertical rod, see Mode 1 ofFig. 13, because the fasteners are placed only along the intermedi-ate vertical stud of the wall. For this reason the sheathing panelacts like a vertical rod, whose rotation is equal to the wood frameone.

The second kinematic model is defined horizontal rod, see Mode2 of Fig. 13. In this case the sheathing panel is connected to theframe by means of only two fasteners placed in the middle pointof both perimeter studs. The sheathing panel acts like a horizontalrod characterised by a rigid-body horizontal displacement equal tohalf of the displacement of the top horizontal displacement of thewood frame.

The third kinematic model is defined not restrained panel, seeMode 3 of Fig. 13, since no fastener connects the panel to theframe. The sheathing panel is in fact completely released fromthe wood framed.

The kinematic analysis is carried out by increasing the kine-matic degree of freedom (usually represented by the top horizontaldisplacement of the frame) up to the failure condition, defined bythe achievement of the ultimate displacement of one fastener atleast.

However, the failure condition might be reached before thekinematic mechanism of model occurs since one fastener mightachieve its ultimate displacement when some other fasteners arenot yielded yet. This condition usually occurs when the fastenerspacing is little.

The force–displacement curve obtained by the elasto-plasticanalysis is represented by a piecewise-linear curve. Each line seg-ment is characterised by a gradually decreasing slope. The kine-matic mechanism is represented by the last line segmented,characterised by an horizontal slope.


The analyses are based on the hypothesis that wood-frame isassumed rigid. As reported in [23], this assumption can be consid-ered true for most of typical European walls where massive studsand plates are used. On the contrary, in case of small size ofwood-frame elements (i.e. 2’’x4’’) might be necessary to takeaccount of the real stiffness of studs and plates in the analysis.The mathematical model proposed by [18], in case of fully flexibleframe, could be adopted. In this case a lower ductility of the wall isexpected because of the lower stiffness than a rigid-frame wall.

4.1. Elastic - plastic analysis of a fully anchored wall with EATW

The Matlab program EATW (Elasto-plastic Analysis ofTimber-frame Walls), specifically developed by authors, allowedto analyse several light timber-frame walls with several dimen-sionless fastener spacings (s=b), panel geometrical parameters (a)and fastener ductilities (lc).

The output data are represented by all parameters characteris-ing the mechanical behaviour of the wall at each step of the anal-ysis (wall displacement, external force, panel rotation, framerotation, fastener internal forces, fastener nail-slip) and by the wallforce–displacement piecewise-linear curve.

Every force–displacement piecewise-linear curve of the wall,us-ing the elasto-plastic energy strain approach, was then



0.5

1

Fasteners position


bi-linearised to obtain the value of ductility of each fully-anchoredwall.

In Tables 1–3, the relationship between the fully-anchored wallductility (lSH) and the fastener one (lc) is reported (depending onthe fastener spacing and the panel geometrical parameter).

As an example, the elasto-plastic analysis, performed by theEATW Program, of a fully anchored wall characterised bya ¼ 2; s=b ¼ 1=2 and lc ¼ 5 is reported. For each step of the analy-sis, the wall force–displacement piecewise-linear curve isobtained, see Figs. 14–17. Circular and triangular dots representunyielded and yielded fasteners respectively. In Fig. 18 thebi-linear curve of the analysed wall is shown and the fastener posi-tion scheme is substituted by the wall kinematic model.

The obtained results were plotted in order to define the rela-tionship between the sheathing-to-framing connection ductilitylSH and the fastener ductility lc . As shown in Fig. 19, a linear rela-tionship can be assumed. The sheathing-to-framing connection

Table 2Fully-anchored wall ductility a ¼ 2; n case which the collapse displacement occursbefore the kinematic.


lc s/b

1/2 1/4 1/6 1/8 1/12 1/25

1.00 1.00 1.00 1.00 1.00 1.00 1.001.50 1.39 1.35 1.34 1.34 1.34 1.342.00 1.83 1.77 1.76 1.76 1.76 1.762.50 2.26 2.18 2.17 2.17 2.17 2.173.00 2.69 2.59 2.57 2.57 2.57 2.573.50 3.11 2.99 2.97 2.97 2.97 2.964.00 3.53 3.39 3.37 3.36 3.36 3.364.50 3.94 3.79 3.76 3.76 3.76 3.755.00 4.36 4.19 4.16 4.15 4.15 4.155.50 4.77 4.58 4.55 4.54 4.54 4.546.00 5.18 4.98 4.95 4.94 4.93 4.936.50 5.59 5.38 5.34 5.33 5.32 5.327.00 6.01 5.77 5.73 5.72 5.71 5.717.50 6.42 6.17 6.12 6.11 6.10 6.108.00 6.83 6.56 6.52 6.50 6.49 6.49

Table 3Fully-anchored wall ductility a ¼ 3; n case which the collapse displacement occursbefore the kinematic


lc s/b

1/2 1/4 1/6 1/8 1/12 1/25

1.00 1.00 1.00 1.00 1.00 1.00 1.001.50 1.43 1.40 1.40 1.40 1.40 1.402.00 1.90 1.85 1.85 1.85 1.85 1.852.50 2.36 2.30 2.29 2.29 2.29 2.293.00 2.82 2.74 2.73 2.73 2.73 2.733.50 3.27 3.18 3.17 3.17 3.16 3.164.00 3.73 3.62 3.61 3.60 3.60 3.604.50 4.18 4.06 4.04 4.04 4.03 4.035.00 4.64 4.49 4.48 4.47 4.47 4.475.50 5.09 4.93 4.91 4.90 4.90 4.906.00 5.54 5.37 5.35 5.34 5.33 5.336.50 5.99 5.80 5.78 5.77 5.76 5.767.00 6.44 6.24 6.21 6.20 6.20 6.207.50 6.89 6.68 6.65 6.63 6.63 6.638.00 7.35 7.11 7.08 7.07 7.06 7.06


ductility lSH is not significantly influenced by the fastener spacings=b whereas, it increases with the panel geometrical parameter a.

Therefore, the analytical relationship between thesheathing-to-framing connection ductility lSH and the fastenerductility lc can be obtained by means of the following linearequation:

lSH ¼ qðaÞ � lc þ mðaÞ ð43Þ

where a is equal to h=b.

−0.5 0 0.5

−1

−0.5

0

x [b]

y[b

]

Fig. 14. Fasteners position on the panel.

−0.5 0 0.5−1

−0.5

0

0.5

1

x [b]

y[b

]

Fasteners position

0 5 10 15 200

0.5

1

1.5

2

2.5

Displacement Δ [δv]

For

ceF

[fv]

F - Δ

Fig. 15. First step of analysis: fastener lingered in the elastic phase and graphicload–displacement.

−0.5 0 0.5−1

−0.5

0

0.5

1

x [b]

y[b

]

Fasteners position

0 5 10 15 200

0.5

1

1.5

2

2.5


For

ceF

[fv]

F - Δ

Fig. 16. Second step of analysis: fastener lingered in the elastic phase and graphicload–displacement.




The parameters q and m, which depend on the panel geometricparameter a, can be obtained by means of an interpolation of thecurves wall ductility vs fastener ductility, getting the followingexpressions:

−0.5 0 0.5−1

−0.5

0

0.5

1

x [b]

y[b

]

Fasteners position

0 5 10 15 200

0.5

1

1.5

2

2.5


For

ceF

[fv]

F - Δ

Fig. 17. Last step of analysis: fastener lingered in the elastic phase and graphicload–displacement.

0 5 10 15 200

0.5

1

1.5

2

2.5


For

ceF

[fv]

F - Δ

Fig. 18. Kinematic mechanism.

2 4 61

2

3

4

5

6

μc

μS

H

sb = 1

2sb = 1

8sb = 1

25

(a)

21

2

3

4

5

6

μ

μS

H

sb =sb =sb =

(b

Fig. 19. Fully-anchored wall ductility vs fasten


q ¼ �0:054 � a2 þ 0:350 � aþ 0:305m ¼ 0:068 � a2 � 0:415 � aþ 0:753

(ð44Þ

Known the ductility of the sheathing-to-framing connectionlSH , the ultimate displacement DU;SH can be calculated as:

DU;SH ¼ lSH � DY ;SH ¼ lSH �RSH

KSHð45Þ

The mechanical behaviour of the sheathing-to-framing connec-tion is hence completely defined.

5. Experimental investigation

5.1. Test program and test specimens

Four light timber-frame walls were tested to asses their yieldingpoint, ultimate displacement, maximum force as well as their duc-tility. The specimens (see Table 4), which had the same geometricaldimensions, were made of the same materials: OSB/3 for thesheathing panels and wood C24 for the frame (see [24,25] respec-tively). On the contrary, the fasteners spacing was different foreach specimen; the fasteners used were annular-ringed nails2.8 � 80.

The specimens were tested under the same boundary condi-tions. The walls, in fact, were connected to the ground trough aheavy hold-down and a rectangular steel tube, see Fig. 20. Thehold-down, a prototype specifically developed for the test cam-paign, was made of steel S355 according to [26], it had flangesof thickness equal to 15 mm and it was designed for a strengthof 250 kN. Whereas, the rectangular tube had a cross-section of120 � 80 � 5 mm and it was made of steel S255 according to[26].

In order to reproduce the fully-anchoring condition,according to Section 4, both the connection devices were designedto be stronger than the sheathing-to-framing connectionand to avoid both the rigid-body rotation and the rigid-bodytranslation.

The specimens were tested in The Materials and StructuralTesting Laboratory of the University of Trento, applying themonotonic-test method under displacement control, with adisplacement velocity of 0.1 mm/s (see [27]). The tests wereperformed using a MTS-244 hydraulic-actuator, see Fig. 21.

4 6c

1218125

)

2 4 61

2

3

4

5

6

μc

μS

H

sb = 1

2sb = 1

8sb = 1

25

(c)

er ductility: (a) a ¼ 1; (b) a ¼ 2; (c) a ¼ 3.



Table 4Specimens mechanical and geometrical properties

M-D-B M-D-B2 M-D-B4 M-D-B8

Test number 1 2 3 4Wall base b [mm] 1250 1250 1250 1250Wall height h [mm] 2500 2500 2500 2500Sheathing panel type OSB/3 OSB/3 OSB/3 OSB/3s.p. thickness [mm] 15 15 15 15Braced sides 1 1 1 1Nail type 2.8x80 2.8x80 2.8x80 2.8x80Fastener spacing [mm] 1150 575 288 144

Fig. 20. Connecting-devices: hold-down and locking rectangular tube.

Fig. 21. Global view of the set-up used for the tests.


5.2. Test set-up and instrumentation

The test set-up (see Fig. 21) was composed by a heavy timberreaction frame on which a system of four lever-arms was con-nected (see [28]) and by a steel sole-plate anchoring the walls tothe ground.

The reaction frame leverage system can be used to apply verti-cal dead load on the specimens; anyway, in the campaign-testdescribed no vertical load was applied. The sole-plate was madeby two UPN240 S235 welded together.

Each test was monitored by-means of five instruments placedas shown in Fig. 22:

- the load-cell of the actuator to measure theapplied-horizontal-force (1);

- a LVDT transducer incorporated in the actuator to control theapplied-horizontal displacement (2);

- a wire potentiometer to measure the horizontal displacement(3);

- two LDVT to measure the uplifting (4) and the rigid-body trans-lation (5) respectively.

5.3. Results from experimental tests and main observations

Despite the use of the heavy hold-down and the use of thelock-plate, the deformation contributions due torigid-body-rotation and the rigid-body-translation have not beencompletely avoided. Anyway, these are considerably small com-pared to the total horizontal deformation and therefore they canbe considered negligible, see Fig. 23.

The Force-vs-Displacement Curves, see Fig. 24(a), are the mainresult of the tests. Analyzing these curves, it is possible noting


how the strength of the specimen, as well as their stiffness,increase with the decrease of the fasteners spacing. This confirmswhat stated in Section 3.3, in which both the strength and the stiff-ness of the wall, see Eq. (14) and Eq. (15) respectively, are consid-ered inversely-proportional to the fastener spacing. Using the Eqs.(14) and (15) to evaluate the yielding-point, this can be shown tobe not dependent on the fastener spacing:

DY;SH ¼RSH

KSH¼

nbs � rc �P

bi �ci

snbs �kc

s�PkiðaiÞ

bi

¼rc �P

bi � ci �P kiðaiÞ

bi

kcð46Þ

Consequently, both the ultimate displacement and the ductilitydoes not change, as demonstrated in Eqs. (45) and (43) respectively.

The results of the test campaign back this theory. The Fig. 24(a)shows that yielding point and ultimate displacement can be con-sidered constant for each test-specimen; the curves, therefore, con-firm the hypothesis presented in the paper that the behaviour of a



Fig. 22. Test instrumentation set-up.

Fig. 23. Test results: uplift, translation-vs-imposed displacement.


timber-frame wall due to the sheathing-to-framing connection innot influenced by the fastener spacing.

In particular, for the bi-linear curves obtained from the specifi-cations of [27] see Fig. 24(b), the yielding displacement is includedin a gap of �4mm with respect to the mean value, whereas the ulti-mate displacement is included in a gap of �6mm with respect tothe mean value. These gaps could be considered acceptable for tim-ber and allow to consider yielding point, ultimate displacementand consequently ductility as constant parameters.

6. Concluding remarks

In the paper an analytical method, based on a simplified rheo-logical model, was presented with the aim to describe theelasto-plastic behaviour of a light timber-frame wall under a hor-izontal force. The proposed expressions allow to define an analyt-ical relationship between the mechanical properties of thestructural elements (fasteners, angle brackets and hold-downs)and the mechanical properties of the entire wall. With particularreference to a seismic design, these expressions can be useful to


relate the local ductility (i.e. fastener) to the global wall ductilityand hence to define the ductility demand of the components(weak) where a energy dissipation is expected. The rheologicalmodel takes account of the deformation contributions of fasteners,hold-downs and angle-brackets but simply it can be modified add-ing other elastic or elasto-plastic springs positioned in seriesregarding other contributions (wood frame flexibility, sheathingpanel shear deformation, compression perpendicular to the grainof the bottom plate, etc.).

Another innovative matter regards the relationship between theductility of a fully anchored light timber-frame wall and the ductil-ity of fasteners. By means of the mathematical model proposed in[18], assuming the timber-frame as rigid, several elasto-plasticanalyses of different types of light timber-frame shear walls werecarried out with EATW in MatLab, changing the geometrical prop-erties of the wall and the fastener spacing. A simple linear equationwas proposed for the researched relationship, depending on thegeometrical properties of the sheathing panels and the fastenerductility. As expected, the wall ductility results lower than the fas-tener ductility. It is not significantly influenced by fastener spacingwhereas the geometrical shape of the panel is to be taken into



(a) (b)

Fig. 24. Test results: force-vs-displacement and Bi-linear curves.


account. This key aspect has been also investigated by means offour laboratory tests which have demonstrated that the yieldingdisplacement and the ultimate displacement can be consideredconstant with the fasteners spacing. The proposed formula forthe ductility, in combination with the expression for the strengthand stiffness, allows to define the elasto-plastic behaviour of anequivalent horizontal spring representing the mechanical beha-viour of all fasteners used to connect the wood-frame to thesheathing panel. The ability to represent the total force carriedby all fasteners (allowing for their sequential yielding) in onenon-linear spring is shown to be a key benefit especially when anon-linear analysis (e.g. pushover) of multi-storey multi-walls tim-ber buildings is required, reducing hugely the number of degrees offreedom of the model. At each level for each wall only oneelasto-plastic spring is sufficient to represent the elasto-plasticbehaviour of the all fasteners. Otherwise each fastener should berepresented by a suitable elasto-plastic spring, increasing a lotthe complexity of the model.

Acknowledgments

The presented research has been carried out in the frameworkof the ReLUIS-DPC 2015 project. Support from the ReLUIS-DPC net-work, the Italian University Network of Seismic EngineeringLaboratories and Italian Civil Protection Agency, is gratefully.Authors would also to thanks Prof. M. Piazza for his precious tech-nical suggestions, to Ph.D. Eng. Tiziano Sartori for his help duringthe laboratory test, to Eng. F. Vinante for his help during the paperwriting and to technicians of the Material and Structural TestingLaboratory of the University of Trento for their support.

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http://dx.doi.org/10.1016/j.engstruct.2013.08.039



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Documents

1.a Predictive Analytical Model for the Elasto-plastic Behaviour of a Light Timber-frame Shear-wall