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On Estimating the Seismic DisplacementCapacity of Timber Portal-FramesCristiano Loss a , Daniele Zonta a & Maurizio Piazza aa Department of Civil, Environmental and Mechanical Engineering ,University of Trento , Trento , ItalyAccepted author version posted online: 28 Feb 2013.Publishedonline: 18 Jun 2013.

To cite this article: Cristiano Loss , Daniele Zonta & Maurizio Piazza (2013) On Estimating the SeismicDisplacement Capacity of Timber Portal-Frames, Journal of Earthquake Engineering, 17:6, 879-901,DOI: 10.1080/13632469.2013.779333

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Journal of Earthquake Engineering, 17:879–901, 2013Copyright © A. S. ElnashaiISSN: 1363-2469 print / 1559-808X onlineDOI: 10.1080/13632469.2013.779333

On Estimating the Seismic DisplacementCapacity of Timber Portal-Frames

CRISTIANO LOSS, DANIELE ZONTA,and MAURIZIO PIAZZA

Department of Civil, Environmental and Mechanical Engineering,University of Trento, Trento, Italy

An analytical model is proposed to estimate the seismic displacement capacity, at serviceability andultimate limit states, of timber portal frame structures with dowelled joints. The predictions from thesimplified formula are compared with the results of numerical analyses carried out on a sample ofrepresentative cases. These cases result from a simulated design procedure, consistent with Eurocode8, and are generated via Monte Carlo sampling, with varying sizes, materials and load conditionsand also uncertainties in these parameters. Based on the outcome of the analysis we propose a setof practical formulas which allow prediction of the displacement with a given percentile of over-estimates. In addition, it is shown that these equations for displacement capacity can be used togenerate fragility curves for performance-based earthquake engineering applications. Prescriptionof an overstrength factor in code provisions is recommended to avoid brittle failure.

Keywords Displacement-Based Design; Wood Structures; Moment Resisting Joint; DesignDisplacement; Monte Carlo Method; Pushover Analysis

1. Introduction

This article aims to establish a simple but reliable way to estimate the seismic displacementcapacity of timber portal frames, in view of its implementation in a Displacement-BasedDesign (DBD) procedure. The current most used DBD approach was first presented byPriestley [1993] reacting to the limits of force-based methods commonly employed at thattime (and even today) in seismic design codes. This procedure is known as Direct DBD.The idea is to design structures in seismic zones using displacement as the input parame-ter, rather than force: displacement allows better control of the damage mechanism of thebuilding, prediction of its real performance and therefore, as suggested by Calvi [2003],the expected costs due to an earthquake. While we leave it to others to judge whether ornot DBD methods will replace traditional seismic design procedures, we must observe thatin the past years DBD has gained supporters [Priestley et al., 2007] and deserves someattention in view of its potential applicability. It is important to note here that timberstructures are particularly well suited to displacement-based design approaches becausedisplacements, rather than strength, are often a governing feature.

The fundamentals of Direct-DBD procedure can be seen in many publications—wesuggest Priestley [2003]—and include: identification of the inelastic deformed shape ofMDOF structures for a selected performance level (or limit state) and design displacement;the evaluation of the target displacement �d; and the definition of the matching hysteretic

Received 7 March 2012; accepted 13 February 2013.Address correspondence to Cristiano Loss, Department of Civil, Environmental and Mechanical

Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy. E-mail: christiano.loss@ing.unitn.it

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parameters such as the Equivalent Viscous Damping [Blandon and Priestley, 2005; Dwairiet al., 2007; Wijesundara, 2011; Pennucci et al., 2011]. Because target displacement is adesign input, a necessary condition for applying the DBD method is that it be possible toset a priori the target displacement of the building, regardless of those geometrical dimen-sions of members, which are unknown at the design phase. For an ultimate limit state,defining the target displacement may not be an obvious task, because the ultimate displace-ment capacity generally does depend on the member characteristics. The same holds trueeven for a serviceability limit state, when the limit state capacity is controlled by the struc-tural performance. An open area of research on DBD addresses the development of simpleand practical analytical models for estimating the design parameters (target displacementand equivalent viscous damping), for any structural typology and material [Priestley et al.,2007].

The Direct-DBD procedure has been developed and tested for some specific coveredconstruction technologies: reinforced concrete structures [Pettinga and Priestley, 2005;Sullivan et al. 2005, 2006], precast structures [Priestley, 2002; Pennucci et al., 2009] andsome forms of steel structures [Goggins and Sullivan, 2009; Garcia et al., 2010; Maleyet al., 2010].

In the case of timber structures, the state-of-the-art of these methods is discussedin Loss [2011]: the impression is that, while for some types of timber technologiesdisplacement-based methods are well established, for others we are still far from their fullapplicability. For example, a comprehensive analytical formulation of the displacement-based design method, applied to mid-rise timber frame panel systems (also known as woodframe) was developed by the research groups involved in the NEESWood project [Pang andRosowsky, 2009], coordinated by Van de Lindt. However, the state of development of theDirect-DBD method for timber structures which differ from the typical American multi-story wood frame is not clear. The authors of this article have recently investigated the caseof glue-laminated timber portal frames [Zonta et al., 2011] which are extensively used inEurope for the construction of commercial open-space buildings. The method is based onthe observation that, under certain conditions, the displacement capacity of the portal, bothat yield and at ultimate limit state, can be predicted with reasonable accuracy with a simpleequation which is to some extent independent from the dimensions of the members. Thesebaseline conditions include:

● brittle failure mechanisms are not allowed;● gravity loads are either negligible or do not affect the failure mechanism; and● Inelastic deformation is concentrated at the rafter-to-column joints.

The method has been tested in a number of case studies, and a first tentative applicationof this approach in the form of a code is found in Calvi and Sullivan [2009]. However,implementation of this method in engineering practice should be accompanied by a set ofadditional design rules which must be specified. In this article, we aim to define the limit ofapplicability of this method and to specify the design provisions which should accompanyits code implementation. Using a statistical-numerical Monte Carlo procedure here, weintend to verify the ability of the analytical expression to estimate the design displacement,accounting rationally for the uncertainty of the model at the design level and warranting anappropriate level of safety (Eurocode 0 [CEN, 2005]).

The prototype structure selected as a case study, a glulam single-story portal framebuilding, is specified in terms of construction and geometrical parameters in Sec. 2.Section 3 briefly presents and discusses the analytical formulation of the design displace-ment published by Zonta et al. [2011]. Section 4 reports the implementation of the FEModel used in nonlinear static analyses. Section 5 describes the Monte Carlo procedure

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Estimating the Displacement Capacity of Timber Portal-Frames 881

and interprets the data obtained with the simulation; specifically, it is explained how theestimation error may change based on the design overstrength factor and the gravity load.Next, in Sec. 6 we use the outcomes of the numerical analysis to calibrate practical for-mulas; these formulas, with the accompanying code provisions, can be used to estimatethe displacement capacity of the portal frame at the design stage. Lastly, some concludingremarks are reported at the end of this article.

2. Description of the Prototype Structure Investigated

The structural concept selected as a case study in this article is that shown in Fig. 1a. Thisis a typical one-story warehouse system with structural members built entirely of glulamtimber of type GL24h [CEN, 2000]: the structure scheme and details are taken from thetextbook by Piazza et al. [2005].

Being single story, and as the plan configuration is compact and symmetrical inboth principal directions, the building is classified as regular in elevation and plan as perEurocode 8 [CEN, 2004c]. In elevation, the building can be seen as a single-story structurewith mass located at the roof level. The principal structure consists of a number of equallyspaced portal frames and in this work we will focus on the in-plane behavior of such aframe. The building construction details are presented in Loss [2011], while additionaldetails on the secondary structure can be found in Piazza et al. [2005]. The building canaccommodate the static design actions and is in accordance with Eurocode 5 [CEN, 2004b].

In Sec. 4, we will perform a simulated design analysis to generate a sample of build-ings, constructed using this technology, representative of the variability expected in reallife. Some of the basic rules for determining this variability are explained here. First, thespan and height of the portal frame are limited by technical and economic restraints. Table 1summarizes the typical dimensional range found in typical European timber construction,based on the state-of-the-art analysis carried out by Loss [2011]. The same author observedthat when these geometrical limits are exceeded, different types of construction system areused. The roof slope αr, necessary for drainage, in general varies very little for indus-trial buildings, and has little or no effect on the structural behavior; we take the slopeas 10 degrees in this work. The spacing i separating individual frames typically rangesfrom 5–8 m; however, in practice a change in spacing simply results in a change in thegravitational and seismic mass transmitted to the main portal frame, thus the effect of

(b)(a)

FIGURE 1 Description of the case study; 3D isometric view of the main resistant struc-ture (a) and geometrical description of the rafter-to-column joint (b) (modified from Piazzaet al., 2005).

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TABLE 1 Geometrical limits of the portals commonly adopted in Europe

Parameter Min Max

Length L [m] 10.00 25.00Column height Hc [m] 3.80 9.00Typical commercial cross-section h [mm] 501 2209Internal rafter radius R [m] 6.60 143.00Roof slope αr [◦] 5 20Space between portals i [m] 5.00 8.00

spacing variability is equivalent to the effect of load variability. For this reason in the anal-ysis reported in Sec. 4 the spacing i is taken as constant and equal to 6.5 m, while all thevariability is assigned to the load.

Each portal frame consists of two columns hinged at the base to the foundation andconnected with a moment resisting joint to the cranked rafter at the top (Fig. 1b). Therafter-to-column connection is a moment resisting joint fastened with dowels located intwo concentric circles, acting as double shear plane, timber-to-timber connections. Thejoint also includes other types of steel fasteners, with tightening function only, that do notaffect the failure mechanism and are not therefore considered in the following discussion.The dowel material is commonly mild steel of class S235, S275, or S355, compliant with[CEN, 2004a]. It is interesting to observe that in normal design practice, there is no prefer-ence for a specific steel class: the designer simply selects the steel grade and dowel numberthat provide the necessary moment resisting capacity of the joint. In reality, as we willshow later in Sec. 5, the higher the steel grade, the higher the likelihood of a brittle ulti-mate mechanism. Therefore, low-grade steel should be preferred to improve the seismicperformance of the joint. For this reason, in the analysis below we will assume that thesteel chosen by the designer is class S355, which is the most critical in terms of seismicperformance.

3. Analytical Equations for Design Displacement

In this section, we recall the analytical formula for design displacement �d, as introducedand discussed in Zonta et al. [2011] for their own case study. At the design phase, neglect-ing gravity loads and overcoming the unknown of geometry of the elements and materialproperties, we can formulate the expression by Zonta et al. [2011]. In this simple formu-lation of �d we can assume that all the inelastic displacement is due to the joint rotation,while the timber member behavior is only considered elastic.

Hence, we can estimate the total design displacement �d as the sum of the displace-ment �j due to rigid rotation of columns, as a consequence of joint yield, and the elasticdeformation �s of the portal frame, calculated assuming the joints to be rigid, as illustratedin Fig. 2:

�d = �j + �s. (1)

It is demonstrated that the elastic component of the design displacement can beexpressed as:

�s∼= c2Hγ (λ + 1) , (2)

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Estimating the Displacement Capacity of Timber Portal-Frames 883

H = +

L

Δj

Δs,1Δs,2

Δs

u

Δd

FIGURE 2 Conceptual model for estimating the design displacement �d.

where H is the height of the portal frame, λ = H/L is the portal aspect ratio, γ = L/his the ratio between the portal length and its depth, and c2 is a numerical parameter that,under some reasonably conservative assumptions, also reported in Zonta et al. [2011], isestimated at 1/2000.

Conversely, the �j component depends on the performance level considered. An esti-mate is generally given by:

�j = δλγβ

1 + 1/λγβ, (3)

where β = h/rext is the ratio between the depth of the rafter and the external radius of thejoint and δ is the critical slip of the dowel which depends on the performance level. Morespecifically, at the ultimate limit state (ULS), the critical slip coincides with the ultimate slipvalue δu, while we can define the serviceability limit state (SLS) as the condition wherebythe critical slip is equal to its yield value δy.

The interesting aspect of these formulas is that all the parameters are either known orcan be estimated at the design stage: H and λ are obviously known to the designer; thedimensionless parameter β (=h/rext) can be estimated at the design stage; while γ (=L/h),the span-to-depth ratio of the rafter, is expected to be between 10 and 15 under normaldesign assumptions.

In Zonta et al. [2011] the reader can find details of the initial assumptions and thederivation of Eqs. (2) and (3). Here, it is worth noting that Eq. (3) is determined withthe “lower bound theorem”, assuming the stress in the rafter-to-column joint subject onlyto uniform shear VJ and moment MJ and negligible gravitational loads. Equation (2) isbased on the assumption that the joint connection is appropriately designed to allow ductilebehavior of the portal. In the design process, this is normally obtained by limiting thebending resistance of the joint to the resistant bending moment of the rafter, scaled with anoverstrength factor α.

In Eurocode 8 [CEN, 2004c], the overstrength factor is prescribed as a function ofsystem construction for steel and concrete [Park and Paulay, 1975], while curiously there isno design prescription in the case of timber structures. This is a point that needs attentionbecause, under seismic action, the behavior of a wood structure is very sensitive to theconnection performance: timber elements tend to show brittle failure, and the ductility andthe ability to dissipate energy can be provided by mechanical fasteners between the variousmembers. This issue will be more quantitatively discussed in Sec. 6 after examination ofthe results of the analysis.

We point out here that in Eq. (2) the shear component of deformation of timber ele-ments has been neglected for the sake of simplicity. We will see from the results of theanalysis reported in Sec. 5 that the shear elastic deformation could at least slightly affectthe elastic deflection at the serviceability limit state. However, it is virtually irrelevant when

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we want to estimate the ultimate deformation capacity. Indeed, the ultimate displacementcapacity is dominated by the rigid rotation of the columns due to joint deformations (�j),thus any refinement in the elastic deformation of members is an unnecessary sophisticationof the problem.

4. Numerical Model

The efficiency of the analytical formulation of Eq. (1) is verified via numerical simulationusing a parametric finite element model of the portal frame, which reproduces the nonlinearpost-yield behavior of members and joints. The actual displacement capacity of the struc-ture is investigated by nonlinear static analysis (pushover). In literature there are severalmodels that describe a moment-rotation relationship for dowelled joints and a state-of-the-art review can be found in Foliente [1997]; some models are specifically developed forjoints with connectors with circle configuration, for example Ceccotti and Vignoli [1989,1990]. To overcome the simplifications inherent to this type of model, here we prefer tomodel the joint at the dowel lever, accenting the exact geometry of timber members andmetal connectors (Figs. 3a and 3b).

The portal frame is anchored to the ground with two hinges and analyses are limited toin-plane behavior. In detail, the timber elements of the portal are constructed by combininga number of linear finite elements (26–42 depending on the rafter or column dimension)of the beam type. The inelastic behavior of each individual connector is reproduced witha set of nonlinear spring elements, of type N-link, distributed on a realistic double-circle

0 x

z

(a) FE Model implementation of the portal frame

MODEL DEFINITION (In-plane behavior only)

•Frame elastic elements: 26 ÷ 42 •Non-linear in-elastic elements: 148 ÷ 328

•Restraints: 2

(b) Joint FEM implementation

Load

F

Slip δ

δy δu

0

Fy

Fy/2 Fy/2

Joint geometrical model Mode of dowelcollapse

Load-Slip relationship

uμδ

Fy

kULS1

FIGURE 3 Finite Element Model (FEM) implementation for structural model (a) andrafter-to-column joints with an elastic-perfectly-plastic load-slip relationship of thedowels (b).

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Estimating the Displacement Capacity of Timber Portal-Frames 885

configuration. An N-link element has a shear mechanism in the two main directions, radialand tangential, and its load-slip relationship, thought to reproduce the real local behavior ofthe metal dowel, is assumed elastic-perfectly plastic. The parameters of the load-slip curveare: the load-carrying capacity, Fy, the stiffness, kULS, and the ultimate slip ductility, μδ ,which is the ratio between the ultimate slip δu and the yield slip δy. The validity of the idealelastic-plastic relationship, assumed for the dowel slip model, is discussed in Loss [2011].

In the simulated design, Fy is calculated using the same resistance model for connec-tions of Eurocode 5 [CEN, 2004b], sometimes referred to as the European Yield Model,which is based on the Johansen model [1949]. This depends on dowel diameter d, thick-ness of timber members, timber density ρ and steel tensile strength fu. The stiffness kULS

is assumed equal to the ultimate stiffness as defined in Eurocode 5, which in turn alsodepends on dowel diameter d and timber density ρ. Note that, in general, the ultimate stiff-ness may vary significantly in doweled timber connections. However, when the expectedfailure mechanism is ductile, the connector follows a monotonic load-slip law where onecan easily recognize the yield point, thus Eurocode 5 prediction of kULS is accurate. Thereader will find in Ehlbeck and Larsen [1993] an exhaustive discussion on the experimentalorigin of the expressions for modulus kULS and its calibration.

Once Fy and kULS is determined, the yield slip of the connector δy is taken as the ratioof the two:

δy = Fy

kULS, (4)

while the ultimate slip of the dowel δu is assumed μδ times δy.The SLS displacement of the portal is simply calculated for the limit state condition

δ = δy at the most critical dowel. Conversely, to calculate the ultimate displacement ofthe portal, we consider two possible failure mechanisms: ductile embedding of the dow-els (ULS1) or brittle failure of a timber member (ULS2). Ductile failure (ULS1) occurswhen the maximum slip demand δ at the most stressed dowel exceeds the maximum slipcapacity δu (Fig. 4b); formally the limit state equation reads δ = δu. According to the brittlemechanism (ULS2), the maximum portal displacement is limited by the maximum bendingmoment in the joint MS,j developed by the rafter or the column (Fig. 4a); in this case thelimit state is given by the equation MS,j = MR,b, where MR,b is its resisting moment. The twomechanisms can be considered mutually independent, so that the first to fail determines thefailure of the portal.

We need not remind readers that a ductile failure mode is the most desirable in seismicdesign, because it allows higher displacement capacity. Not surprisingly, most seismic codeprovisions, including those of Eurocode 8 [CEN, 2004c], are thought to prevent brittle fail-ure. Yet we must observe that, in Eurocode 8, in principle there is no explicit requirementto verify rafter and column members with an overstrength factor respect to the joint resis-tance. Thus, a brittle failure cannot be excluded a priori, even if the structure has beencorrectly designed, but can occur under some unfortunate, but not unlikely, combination ofmaterial properties. This concept will be better clarified in the next section.

5. Numerical Validation of the Design Displacement

In this section we discuss the ability of Eq. (1) to predict the displacement capacity of a realstructure. To this aim we will first define a sample of structures representing the possiblevariability in dimensions and material properties; we predict the displacement capacity,both the serviceability and ultimate limit states, using Eq. (1) and we compare this capacity

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Displacement Δ

Force P

Δ1 Δ2 Δ3 Δi Δi+1

Δd(2) Δn

P1

P2

P3

Pi

Pi+1

Pn

Force-displacement relationship (P-Δ)

(a) Brittle collapse mode, Δd,1

Moment-displacement relationship (M-Δ)

Displacement Δ

Δ1 Δ2Δ3 Δi Δi + 1

Δd(2) ΔnM1

M2

M3

Mi

Mn-1Mi+1

Mn

MR,b(2)

Moment M

Joint, Sx

Joint, Dx

Δd(1)

MR,b(1)

MR,b(1)

MR,b(2)

Δd(1)

(b) Ductile collapse mode, Δd,2

0 x

z

0 x

z

F

δ

δy δu

0

F

δ

δy δu

0

FIGURE 4 Brittle collapse mode (a) and ductile collapse mode (b).

with that obtained from the nonlinear model described in Sec. 4. The distribution of thescatter between the two values is an indicator of the reliability of the simplified model ofEq. (1).

5.1. Variability of Design Parameters and Material Properties

In order to generate a sample of portals, representative of the possible configurationsexpected in real life, we need to consider both the variability of the design dimensionsexpected in a real design process and also the random variability of material properties.

The dimension variability can be tackled following a simulated design process fortimber elements and rafter-to-column connections. In detail the design geometrical valuesassumed are: the length L and high H of the portal; the rafter cross-section dimensions hand bc; the column cross-section width bb (while the column depth is assumed identicalto that of the rafter). The normal portal spans observed in typical European constructionrange from 10–25 m, as reported in Table 1, and the span distribution can be assumedflat in this range. In the same table the height of the portal is shown as generally variablefrom 3.8 to 9 m; however there is a correlation between height H and span L; to reproducethe typical design demand, we assumed that variable H has triangular distribution between3.8 and 9 m with maximum value at 9 m or L/2, whichever is the lowest. The designchoice for the rafter cross-section deserves explanation: in principle one may think thatthe designer chooses cross-section dimensions based on both span length L and load q.In reality, by empirical examination of real single-story portal buildings, we observe that[Loss, 2011]: the cross-section depth h depends essentially on L; the cross-sectional widthis independent of span; the design load does not affect significantly the designer’s choiceof the rafter. We reproduce these observation assuming: for h, triangular and symmetricdistribution with extremes L/14 and L/11; for bc, triangular distribution between 120 mmand 200 mm, with mode at 120 mm (Fig. 5d).

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Estimating the Displacement Capacity of Timber Portal-Frames 887

(n) Tensile strength of thedowel (S355)(Normal distribution)

(i) Modulus of rupture(GL24h)(Lognormal distribution)

(h) Modulus of elasticity perpendicular to grain (GL24h)(Lognormal distribution)

(l) Timber density (GL24h)(Normal distribution)

(m) Shear modulus parallel tograin (GL24h)(Lognormal distribution)

(g) Modulus of elasticityparallel to grain (GL24h)(Lognormal distribution)

CV = 13%

f (L)

L [m]

bc [mm]

HC [m] h [mm]

f (H) f (h)

120

f (bc)

200

(a) Length of the portal(Constant distribution)

(d) Base of the column(Triangular distribution)

(b) Height of the portal(Bi-Triangular distribution)

(c) Height of the beam(Bi-Triangular distribution)

f (E0)

E0 [Mpa]

f (E90)

E90 [Mpa]

f (fm)

fm [Mpa]

L/12.5L/2

10 25 3.8 9 L/14 L/11

E0k = 9400 E90mean = 390 fmk = 24

bb [mm]140

f (bb)

200

(e) Base of the beam(Triangular distribution)

q [KN/m2]

(f) Snow load(Constant distribution)

F (q)

2 5

f (ρ )

ρ [Kg/m3]

f (G)

G [Mpa]Gmean = 720ρk = 380

μ [-]

(o) Static slip ductility (GL24h and S355)(Lognormal distribution)

f (μ)

μk = 6

f (fu)

fu [Mpa]fuk = 510

CV = 13% CV = 25%

CV = 4%CV = 13%CV = 10%

CV = 7%

WhenL>18m

WhenL 18m

FIGURE 5 Trends of input variables selected in the MC simulation; Probability distribu-tion f (x) for the function x; Probabilistic models for properties of glued laminated timberin accordance with JCSS [2007].

The total distributed design load q on the rafter includes the dead load, which dependson the portal spacing, and the live load; in the design simulation, the total is assumeduniformly variable between 13.0 kN/m and 32.5 kN/m.

The joint ductility depends on the combination of dowel diameter and materials (steeland wood). Unlike the timber member dimensions, some of the design variables are

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888 C. Loss, D. Zonta, and M. Piazza

assumed to reproduce the most unfavourable situation for connector ductility. Specifically,the fixed parameters used are: dowel diameter d = 12 mm, wood strength class GL24h[CEN, 2000] and steel S355 [CEN, 2004a]. The random variability of the mechanicalproperties of the materials is illustrated in Figs. 5g–n, based on the nominal values (charac-teristic or mean) reported in CEN [2000] and CEN [2004a], and the distribution suggestedby the JCSS (Joint Committee on Structural Safety) in the “Probabilistic Model Code”[JCSS, 2007].

The distribution of dowel slip ductility μδ deserves specific discussion. In dissipativezones, where the inelastic capacity of the structure is concentrated, Eurocode 8 [CEN,2004c] prescribes a minimum value for static slip ductility of 6, in order to allow theassumption of high ductility class (DCH) for the structural system. The same Eurocode8 states some general requirements for high ductility class. These conditions are: (a) thediameter of the dowels is 12 mm or less, and (b) the thickness of the members is at least10 times the dowel diameter d. What happens in practice is that these are the conditionsnormally followed by the designer to achieve ductile behavior when designing a dowelledstructure in a seismic zone. For this reason, we will assume the dowel diameter as constantand equal to 12 mm. As for the slip ductility, there is an implicit assumption that when con-ditions (a) and (b) are simultaneously satisfied, then the ductility slip should be at least 6 ormore. This value is justified by the experimental background of Eurocode 8, as reportedin Ceccotti and Larsen [1988] and Ceccotti [1989]. To put this observation into statisticalterms, we can state that the distribution of μδ has, for example, a 5% fractile value of 6. Thedistribution is assumed lognormal and a coefficient of variation of 7% is judged appropriatein this case.The ultimate slip of the connection, δu, is conventionally defined by:

δu = δyμδ , (5)

where δy is calculated from the mechanical properties of materials and the geometry of theconnection, using Eq. (4).

We must remark at this point that the sole Eurocode 8 prescription does not preventbrittle failure mode of the portal. In fact, since there is no formal verification of the strengthhierarchy between member and connection, the bending moment at the joints can be greaterthan the resisting moment of the rafter and/or column.

5.2. Monte Carlo Simulation

The analytical model for the evaluation of design displacement �d, Eq. (1) has been numer-ically validated through Monte Carlo simulation. This method evaluates the effect of designvariability and uncertainties in the random parameters defined in Sec. 5.1, which control themodel implemented for the calculation of �d. In the Monte Carlo method all the mechan-ical and geometrical variables and other uncertainties associated with the design processare described with probabilistic distributions to take into account the effect of uncertain-ties. We refer readers, unfamiliar with the background of the Monte Carlo method, to thetechnical literature such as Robert and Casella [2004] and Elishakoff [2003].

Using a Monte Carlo simulation, wherein the geometry of the members and theirmechanical properties are randomly selected, we can validate the predictions of Eq. (1) bycomparing its values with the numerical pushover (nonlinear static) results. Because anyparametric variation requires generation of an individual finite element model, to handlea large number of cases, the process of model generation needs to be controlled auto-matically. In detail, we have created a software program using VBA (Visual Basic for

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Estimating the Displacement Capacity of Timber Portal-Frames 889

Applications) called “Portal Frame Control Software” (PFCS), which controls the struc-tural analysis program SAP 2000 [2006] and Microsoft Excel [2012]. The PFCS softwareis organized in blocks, to simplify its extension to other geometric configurations and typesof wooden structures. PFCS generates all the parameters of the FE Model, runs the anal-ysis and extracts the displacement value for each model of the data set created. PFCSthen compares the displacement, estimated via the analytical formula (Eq. (1)), with thedisplacement provided by nonlinear static analysis. Finally, PFCS creates a statistical dis-tribution of the final data value, including the main statistical indices, and creates a databasefile.

The sampling, operated by the PFCS software, is done by an internal algorithm thatuses pseudo-random numbers (distribution in the range 0–1) and the inverse distributionfunction method. From the probability density function, the associated distribution func-tion F(x) is defined and its analytically inverse function F−1(u) is obtained. The generic ith

random number, xi, is evaluated from the pseudo-random value, ui, directly from the curveof F−1(ui). The problem of correlation between input variables is treated according to the“subjective method” proposed by Hertz [Robert and Casella, 2004] (conditional sampling).Thus, in carrying out the Monte Carlo simulation, for each iteration a value for the inde-pendent variable is first generated; then the specific distribution function is selected and thedependent variable value extracted.

In this work we simulated a total of 1,000 cases; a large enough sample to make theset statistically representative while keeping the computation time within acceptable limits.For each case, we estimated both the serviceability limit state displacement and the ulti-mate displacement achieved by pushover analysis, �push, and their corresponding drift θ =�push/H. To give an idea of the extent of the population generated, in Fig. 6 we report thecumulative distribution of the drift limit for a SLS and an ULS. In particular, we observethat, for a SLS, the average drift limit is 1.2%, with a coefficient of variation of 16.2%,while for the ULS the average drift of the population is 3.2%, with a coefficient of variationof 20.9%.

For each case, we can also calculate the prior limit displacement estimate, �d usingEq. (1). A comparison of the two displacement sets of values, �d and �push, allows us toassess the reliability of Eq. (1), as explained in detail in the next ection.

0.000.100.200.300.400.500.600.700.800.901.00

0.00

%

1.00

%

2.00

%

3.00

%

4.00

%

5.00

%

6.00

%

7.00

%

8.00

%

Cum

ulat

ive

freq

uenc

y

drift θ [%]

Pushover(SLS)

1000cases

Pushover(ULS)

θ =1.2% θ = 3.2%

1000cases αd =1.1 (707 cases)

αd =1.2 (532 cases)

αd =1.3 (359 cases)

No αd requirements

FIGURE 6 Cumulative frequency distributions of the SLS and ULS drift capacities fora 1000 sample of portal frame; ULS fragility curves are plotted in the case of no designoverstrength αd and for αd equal to 1.1, 1.2, and 1.3.

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5.3. Analysis Results

To quantitatively compare the two quantities, �push and �d, we simply use the ratio:

r = �d/�push. (6)

In practice, this ratio indicates the accuracy of the design displacement �d predicted a pri-ori with Eq. (1) based on the assumption that the displacement �push, extracted by thenonlinear static analysis, represents the “real-life” behavior of the structure. The resultingerror distribution is plotted in Fig. 7a for a serviceability limit state and in Fig. 7b for anultimate limit state.

Figure 7a shows that Eq. (1) in average estimates correctly the SLS capacity, but with acoefficient of variation of 24.0%. We also note that the error distribution is rather symmet-rical and could be approximated with a Gaussian distribution. The cumulative distributionof the rate r for a SLS is also reported in Figs. 8a and 8b.

Contrary to the SLS displacement, for an ultimate displacement (Fig. 7b) we imme-diately observe that the distribution is clearly asymmetrical, with a tail towards positivevalues, i.e., there are many cases where Eq. (1) overestimates the displacement capacity.Because of this tail, the mean value of the rate is slightly greater than 1, although the medianis 0.89. The reason for this distribution shape is clearly understood when we recognize the

(b) ULS(a) SLS

0

25

50

75

100

125

150

175

200

225

0.52

0.57

0.62

0.66

0.71

0.76

0.81

0.86

0.90

0.95

1.00

1.14

1.27

1.41

1.55

1.68

1.82

1.96

2.10

2.23

2.37

Fre

quen

cy

Δd/Δpush

μ = 1.02σ = 0.25

(1000 cases)

0

25

50

75

100

125

150

0.62

0.66

0.69

0.73

0.77

0.81

0.85

0.89

0.92

0.96

1.00

1.25

1.50

1.75

2.01

2.26

2.51

2.76

3.01

3.26

3.52

Fre

quen

cy

Δd/Δpush

Ductility mechanism Brittle mechanism

μ = 1.01σ = 0.41

(142 cases)(858 cases)

FIGURE 7 Histograms of predicted to real displacement ratio r = �d/�push (μ, averagevalue; σ , standard deviation).

(a) Without gravity coefficient cG (b) With gravity coefficient cG

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

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Fre

quen

cy

R2 = 0.48

0.00

0.25

0.50

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

1.60

Δ j /Δ d

Δd/Δpush

μ = 1.02σ = 0.25

μ + σμ−σ μ

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

ive

Fre

quen

cy

R2 = 0.09

0.00

0.25

0.50

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

1.60

Δ j/Δd

Δd/Δpush

μ + σμ−σ μ

μ = 0.90σ = 0.18

FIGURE 8 Cumulative frequency distributions of ratio �d/�push and �j (joint displace-ment) to �d (total deformation) ratio for a SLS: (a) without and (b) with gravity coefficientcG (μ, average value; σ , standard deviation; R2, coefficient of determination; the dottedcurve is the best Gaussian approximation).

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Estimating the Displacement Capacity of Timber Portal-Frames 891

failure mode associated with each case. Equation (1) assumes ductile structural behavior,while, as mentioned in Sec. 4, the real structure can undergo two possible failure modes:ductile or brittle. In our simulation, 142 cases of 1,000 are dominated by a brittle failuremechanism. Interestingly, we observe in Fig. 7b that the cases where Eq. (1) overestimatesare all “brittle.” Hence, the design seismic demand is greater than the column or raftersection capacity.

We have already observed that brittle behavior occurs when the resisting moment ofthe joint exceeds the member resistance. We also remarked that current seismic code pro-cedures for timber structures, e.g., Eurocode 8 [CEN, 2004c], or national seismic codes(CS. LL. PP., 2008; IBC, 2006), provide criteria and rules for the design of reinforcedconcrete and steel elements according to the Design Capacity philosophy, but not for thedesign of timber portal frame structures. The outcome of the simulated design processdemonstrates that the absence of specific design prescriptions to control geometrical andnominal mechanical properties of members and connections, together with the expectedvariability of material properties, can sometimes result in brittle behavior of the portaland unexpectedly low displacement capacity. This is a problem because the displacementcapacity, as targeted and expected at the design stage, in practice is not always achieved inreal life.

5.4. Effect of Design Overstrength Factor

An easy solution to this problem is to design the member capacity based on the real jointcapacity with an appropriate overstrength factor. The overstrength design factor, αd, isdefined as:

αd = MR,b/MR,j, (7)

where MR,b is the resisting moment of the most critical member, typically the rafter, andMR,j is the bending capacity of the rafter-to-column joint. For members with rectangularcross-section, as considered in this study, MR,b is estimated with:

MR,b = (bbh2)/6fm,k, (8)

where bb and h are, respectively, the width and depth of members, and fm,k is the character-istic bending strength. For dowelled connections arranged in circle configuration, when theeffect of the vertical load is negligible, the resisting moment of the connection MR,j can beestimated with some approximation with:

MR,j = nextrextFy + nintrintFy, (9)

where next and nint are, respectively, the number of external and internal dowels for the joint,rext and rint are, respectively, the external and the internal radius of the dowel circles and Fy

is the characteristic load-carrying capacity of the dowel. These parameters are all known atthe design phase.

The choice of the overstrength factor affects the chance of undesired brittle behav-ior, and evidently the higher the overstrength assumed at the design level, the lower theprobability of a brittle portal. Introducing a design overstrength factor is incompatible withthe design solution generated using the simulated design process above. So, to simulatethe impact of the overstrength factor in the simulated design, we can simply filter outof the sample of portal frames generated, those design solutions which are incompatible

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892 C. Loss, D. Zonta, and M. Piazza

(a) Without gravity coefficient cG (b) With gravity coefficient cG

R2 = 0.000.00

0.50

1.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

Δ j/Δd

Δd/Δpush

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

ive

Fre

quen

cy

0.00

0.20

0.40

0.60

0.80

1.00C

umul

ativ

eF

requ

ency

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

ive

Fre

quen

cy

μ + σμ−σ μ

μ = 0.96σ = 0.39

μ = 1.01σ = 0.41

μ + σμ−σ μ

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

ive

Fre

quen

cy

R2 = 0.140.00

0.50

1.00

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

Δ j/Δd

Δd/Δpush

μ = 0.93

σ = 0.09

αd = 1.1

μ + σμ−σ μ

3 4

1 2

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

ive

Fre

quen

cy

R2 = 0.130.00

0.50

1.00

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

Δ j/Δd

Δd/Δpush

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

ive

Fre

quen

cy

R2 = 0.140.00

0.50

1.00

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

Δ j/Δd

Δd/Δpush

μ = 0.92

σ = 0.08

αd = 1.2

μ + σμ−σ μ

5 6

7 8

μ = 0.90

σ = 0.07

αd = 1.3

μ + σμ−σ μ

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

ive

Fre

quen

cy

R2 = 0.090.00

0.50

1.00

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

Δ j/Δd

Δd/Δpush

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

ive

Fre

quen

cy

R2 = 0.020.00

0.50

1.00

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

Δ j/Δd

Δd/Δpush

R2 = 0.010.00

0.50

1.00

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

Δ j/Δd

Δd/Δpush

αd = 1.1

αd = 1.2

αd = 1.3

μ = 0.89

σ = 0.07

μ = 0.87

σ = 0.07

μ = 0.86

σ = 0.06

μ + σμ−σ μ

μ + σμ−σ μ

μ + σμ−σ μ

R2 = 0.000.00

0.50

1.00

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

Δ j/Δd

ΔdΔpush

FIGURE 9 Cumulative frequency distributions of ratio �d/�push and �j (joint dis-placement) to �d (total deformation) ration for an ULS: (a) without and (b) with gravitycoefficient cG (μ, average value; σ , standard deviation; R2, coefficient of determination; thedotted curve is the best Gaussian approximation).

with the specific overstrength. We repeated this exercise with a limiting overstrength factor(αd) varying from 1–1.5 with incremental steps of 0.05. The new distributions obtainedare of the type reported in Fig. 9a. A comparison of these graphs shows that prescribingeven a small overstrength design factor prevents the brittle failure and that the probability

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Estimating the Displacement Capacity of Timber Portal-Frames 893

of overestimation rapidly reduces with an increasing overstrength factor. For example forαd = 1.2 we observe that an overestimation occurs in less than 2.0% of the cases and theoverestimate error is never greater than 7.0%.

Although there is no overstrength specification in Eurocode 8, we note here that theEuropean Commission Working Group TC250/SC 5 (see the attached article by Follesaet al., 2011) has proposed overstrength factors varying from 1.3–1.6 for various timberconstruction systems (not including that investigated in this article). These values are gen-erally conservative and suggested for pre-design of structures, and are based on Fragiacomoet al. [2011] and Jorissen and Fragiacomo [2011].

In our specific case, we observe that the number of brittle cases drops drasticallywhen even a minimal overstrength design factor is considered, and an overstrength αd =1.1–1.2 is more than enough to ensure ductile behavior. This suggests to us that, at leastfor timber portal frame systems, the conservative recommendations of EC-WG-TC250/SC5 could be relaxed to αd = 1.2. Finally, we remind the reader here that the choice of theoverstrength design factor, αd, is critical to control the ultimate displacement capacity, butit is obviously irrelevant to the SLS performances of the portal.

In the same graphs of Figs. 8a and 9a we plotted the rate r against the joint displace-ment component �j normalized to the displacement capacity �d. It is interesting to observehow the two quantities are correlated, meaning that the higher the joint displacement capac-ity, the lower the percentage estimate error which affects Eq. (1). We will discuss thisobservation in more detail in Sec. 6.

5.5. Effect of Gravity Load

Another potential source of error, both for the SLS and the ULS, is that Eq. (1) does not takeinto account the gravitational load on the portal. Qualitatively, we understand that the effectof the gravitational load is to engage the dowel joint, reducing their embedding capacityunder seismic action, and thus reducing the overall rotation capacity of the joint and dis-placement capacity of the portal. We want to see here if we can find an easy correction ofEq. (1) to take account of this fact. The higher the gravitational load, the lower the dis-placement component �j allowed by joint rotation. Formally, we can correct the quantity�j, introducing a gravity-load coefficient cG that reduces the displacement capacity basedon q. In equation:

�j,q (q) = CG(q) �j, (10)

where the quantity �j,q means the joint displacement capacity in the presence of a gravityload q. To provide an estimate for coefficient cG we can suggest the following: first, thereis an ultimate gravity load qlim which engages the full moment capacity of the joint, whichis to say that when q = qlim there is no further elastic rotation capacity at the joint; second,once known load qlim we can assume that the elastic rotation capacity of the joint, and thusthe corresponding elastic displacement capacity of the frame �j,el, varies linearly with thegravity load q. Keeping in mind that �j,el = �j,0/μδ , then the q-dependent displacementcapacity reads:

�j,q = �j,0 − �j,elq

qlim= �j,0

(1 − q

μδ qlim

), (11)

where μδ here represents the dowel ductility demand relevant to the limit state considered.Consistently with the assumptions above, μδ = 1 for a serviceability limit state and μδ =

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894 C. Loss, D. Zonta, and M. Piazza

6 for an ultimate limit state. By comparing Eq. (10) with Eq. (11), coefficient cG can beformally expressed as:

cG = 1 − q

μδ qlim, μδ =

{1 for DLS6 for ULS

. (12)

To estimate a priori (i.e., at the pre-design stage) the value of qlim we can write theequilibrium of the dowelled joint at the ultimate state:

Fy = FM + FV . (13)

As shown in Fig. 10, we can conservatively assume that at the ultimate limit state, themaximum bending moment at the joint is MJ = qlimL2/12, and the corresponding shearVJ = qlimL/2.

Following the same path as in Zonta et al. [2011], we assume that the ultimate distri-bution of forces in dowels is proportional to that in the elastic state. Hence, moment (FM)and shear component (FV) of dowel reaction are related to moment and shear through thefollowing equations:

FM = MJ

nrext, FV = Vj

n, (14a, b)

where n is the total number of dowels in the connection. The dowel number n is, inprinciple, not known before designing the connection, but we can estimate it as follows.Assuming the dowels arranged on two circles of radius rint and rext, and mutually spacedidowel, the number of dowels n is:

n = 2πrext

idowel+ 2πrint

idowel. (15)

These parameters cannot be taken arbitrarily but should satisfy the specification inEurocode 5 [CEN, 2004b]; in particular, Eurocode requires that the two circles be spacedat least a2 times d, where a2 depends on the angle between force and the direction of grain.Assuming exactly rint = rext −a2 d, Eq. (15) changes to:

Where:

lh

Jh

Jlk =

12

qlimL2

M2 = M3 = −

Portal Frame with Rigid Joints

Design scheme

If k 0:Always k < 0.5

4(2k + 3)−

qlimL2

M2 = M3 =

Bending moment diagram

FIGURE 10 Design scheme assumed to estimate the qlim parameter.

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Estimating the Displacement Capacity of Timber Portal-Frames 895

n = 2πrext

idowel+ 2π (rext − a2d)

idowel= π

(4rext − 2a2d)

idowel≈ 4πrext

idowel(16)

where the right-hand term is obtained observing that, under normal design conditions,quantity 2 a2 d is negligible respect to 4rext (for instance, in the thousand cases simulated,the amount 2 a2 d is always less than 5.5% of 4rext.). Going back to the limit state equation,on the assumption that component FM is dominant on FV, Eq. (13) reduces to:

Fy∼= FM = Mj

nrext(17)

and with further manipulation, taking account of for Eq. (15) and keeping in mind thatrext = L/γ β, we eventually obtain:

Fy = qlimL2/124πrextidowel

rext

= qlimL2/124πL2

idowel(γβ)2

= qlimidowel (γβ)2

48π. (18)

From which the ultimate gravity load qlim that the joint can carry is:

qlim = 48π

idowel

Fy

(γβ)2 . (19)

Equation (19) shows that, in the end, the maximum load capacity of the portal is con-trolled by dowel capacity and by simple geometrical relationships that are known or canbe assumed at the design stage. Particularly, we have already seen in Sec. 3 that acceptableestimates of γ and β are 10–15 and 2–3 respectively, while idowel can be normally assumed5 times d to comply with Eurocode 5 specifications.

In conclusion, to account for the gravitational load effect, we can reduce displace-ment �j by a factor cG, which in turn depends on the ultimate load capacity of the portalestimated with Eq. (19). Figures 8b and 9b show how the design displacement error dis-tribution changes, when we take into account the coefficient cG in Eq. (1): the number ofoverestimate cases is further reduced, and the standard deviation of the error is generallylower, especially in the SLS case.

6. Suggested Code Implementation of Displacement CapacityPredictive Formula

Based on the outcomes of the previous section, we now state a practical formulation forthe target displacement that can be implemented in a seismic design code. Although notdeterministic, this formulation estimates the target displacement with a known degree ofconfidence and a given percentile of overestimate cases, for the cases considered in thisstudy. Our aim is also to prevent any large overestimate which would fail the designexpectations.

We have seen in Sec. 5.4 that the higher the displacement, the lower the error of theanalytical expression. Figures 7b and 8b confirm that this holds true, both for serviceabilityand ultimate capacity, even taking into account the gravity coefficient cG. An explanationof this is that the estimate of �j is generally more accurate than �s, thus higher percent-age estimate errors are expected for smaller displacements when the elastic component �s

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896 C. Loss, D. Zonta, and M. Piazza

dominates. Based on this we propose here a more refined expression of the portal displace-ment capacity in Eq. (1) which, without losing simplicity, estimates more accurately theplastic/elastic deformation and does consider the gravity load. As in the original formu-lation, this model can be used at the design phase when the dimensions of the structuralmembers are not yet defined. In general the formula can be written as:

�d = c1�jδy

δ(μδ − cG) + c2�s

= c1δy (μδ − cG)λγβ

1 + 1/λγβ+ c2Hγ (λ + 1) ,

μδ ={

1 for SLS6 for ULS,

(20)

where c1 and c2 are two parameters, cG is the gravity reduction factor, and �j and �s are aspreviously defined in Sec. 3. The two parameters c1 and c2 should be chosen in view of thedistribution of the design displacement error we can tolerate. For example, we may want tocalibrate the formula with the aim of minimizing the scatter between “real” capacity andits analytical prediction. Or, if we are using this equation as a pre-design tool, we may needa conservative estimate, to control the possibility of overestimation (or negative errors).

To start, let us first calibrate the equation with the following criteria: (a) the mean valueof the displacement error is zero, (which is to say that the mean value of rate r is equal to1) and (b) the standard deviation of rate r is minimal. In practice this is a standard leastsquare optimization problem. Using the data generated by the Monte Carlo simulation andassuming an overstrength factor of αd = 1.2, we obtain: c1 = 1.07 and c2 = 1/1540 for theULS; c1 = 1.02 and c2 = 1/1780 for a SLS. The resulting distributions are those shown inFig. 11. It is demonstrated that these distributions are approximately normal, with statisticalproperties as shown in Table 2.

The previous parameters are calibrated to minimize the prediction error, or in otherwords, so that the formula overestimates and underestimates with equal probability. Thisis commonly acceptable when we design against a serviceability limit state (for example,see Eurocode 0; CEN, 2005). However, against an ultimate limit state, we normally preferto use a conservative model for the displacement capacity, where the probability of anoverestimate is minimal. If, for example, we tolerate overestimation in up to 5% of cases,

(b) ULS(a) SLS

0.00

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1.00

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R2 = 0.08

0.00

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1.35

1.40

1.45

1.50

1.55

1.60

Δ j/Δd

Δd/Δpush

μ + σμ−σ μ

μ = 1.00σ = 0.20

0.00

0.20

0.40

0.60

0.80

1.00

Cum

ulat

ive

Fre

quen

cy

R2 = 0.02

0.00

0.50

1.00

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

Δ j/Δd

Δd/Δpush

μ = 1.00σ = 0.08

αd = 1.2

μ + σμ−σ μ

FIGURE 11 Cumulative frequency distributions of ratio r = �d/�push for the calibratedformula to estimate design displacement �d at SLS (a) and at ULS (b).

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Estimating the Displacement Capacity of Timber Portal-Frames 897

TABLE 2 Summary of the main statistical quantities for the predicted to real displacementratio r = �d /�push distributions

SLS ULS

Median med(�d/�push) 0.97 0.99Average m(�d/�push) 1.00 1.00Standard deviation ds(�d/�push) 0.20 0.08Percentile 5% P5%(�d/�push) 0.73 0.87Percentile 2% P2%(�d/�push) 0.68 0.84No. of cases NTOT 1000 532No. of cases <1 N(�d/�push)<1 559 264

TABLE 3 Calibration parameters and sampling statistical parameters of ratio r = �d/

�push for various target percentiles

SLS ULS

Percentile c1 c2 μ σ c1 c2 μ σ

50% 1.02 1/1780 1.00 0.20 1.07 1/1540 1.00 0.085% 0.95 1/2565 0.73 0.15 0.97 1/1780 0.89 0.072% 0.94 1/2790 0.68 0.14 1.00 1/1985 0.87 0.071% 0.94 1/2985 0.64 0.13 0.98 1/2000 0.86 0.07

we should recalibrate the parameters to obtain an error distribution with 5% percentileequal to zero. We can repeat the calibration process with different target percentiles (50%,5%, 2%, and 1%) and the resulting values are reported in Table 3.

If 2% or fewer overestimates can be tolerated, the equation simplifies, with c1 =1.0 and c2 = 1/1985, and this simplicity suggests that this target percentile should beadopted in code provisions. Using an overstrength αd = 1.2 with the 2% target percentile,the overestimation error is never greater than 10%. In the authors’ opinion this is fullyacceptable at the pre-design stage.

In conclusion, when we design a timber portal frame under seismic action, using adisplacement based method, a suggested estimate of the design displacement capacity isgiven by:

�d = δy

(1 − q

qlim

)λγβ

1 + 1/λγβ

+ 1

1780Hγ (λ + 1) for SLS

�d = δy

(μδ − q

qlim

)λγβ

1 + 1/λγβ

+ 1

2000Hγ (λ + 1) for ULS.

(21)

The use of this analytical model should be accompanied by a formal verification of thecapacity design rule. Namely, the design value of the rafter and column end moment Mb,d

shall be determined in accordance with the capacity design rule, based on ultimate statejoint equilibrium, corresponding to the maximum embedment of the most critical dowel.In particular:

Mb,d = αd MR,j , (22)

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898 C. Loss, D. Zonta, and M. Piazza

(b) ULS(a) SLS

0.000.100.200.300.400.500.600.700.800.901.00

0.00

%

0.50

%

1.00

%

1.50

%

2.00

%

Cum

ulat

ive

freq

uenc

y

drift θ [%]

Analytical P50%(μ = 1.18%;σ = 0.12%)

Pushover(μ = 1.21%;σ = 0.20%)

Analytical P1%(μ = 0.75%;σ = 0.08%)

Analytical P2%(μ = 0.80%;σ = 0.09%)

Analytical P5%(μ = 0.86%;σ = 0.10%)

0.000.100.200.300.400.500.600.700.800.901.00

0.00

%

2.00

%

4.00

%

6.00

%

8.00

%

Cum

ulat

ive

freq

uenc

y

drift θ [%]

Pushover(μ = 3.57%;σ = 0.78%)AnalyticalP50%(μ = 3.57%;σ = 0.83%)

Analytical P1%(μ = 3.09%;σ = 0.75%)Analytical P2%(μ = 3.13%;σ = 0.76%)Analytical P5%(μ = 3.19%;σ = 0.75%)

FIGURE 12 Comparison between predictive and FEM generated fragility curves of thedrift θ : SLS (a) and ULS (b).

where αd is the overstrength factor for the joint, taken equal to 1.2; MR,j is the resistingmoment of the joint, which can be estimated with Eq. (9); and Mb,d is the design momentfor the timber column and rafter members.

To qualitatively appreciate the quality of the estimation, in the graphs of Fig. 12 wecompare the cumulative distributions of the limit drift for the entire sample with the corre-sponding distributions from the predictive formula, at the SLS (Fig. 12a) and at the ULS(Fig. 12b). From Fig. 12b we observe that the distribution from the analytical formulawith αd = 1.2 and 50% percentile virtually coincides with the real capacity distribution;thus the predictive formula can also be used as a good tool to generate fragility curvesfor performance-based earthquake engineering applications. However, remember that thefact that the distribution coincides doesn’t entail that each pair of corresponding samplescoincide. On the contrary, the SLS displacement prediction has a coefficient of variation of10.5%, in contrast with the 16.2% of the “real” distribution.

7. Conclusions

The validity of a formulation to estimate the serviceability and ultimate displacements oftimber portal frame structures has been investigated. The theoretical prediction from thesimplified formula has been compared with the results of numerical analyses on a sam-ple of cases generated through a simulated design procedure consistent with Eurocode 8.This procedure takes account of design variability and dimensional uncertainty, of mate-rial properties and load conditions. Based on the outcome of the analysis we can draft thefollowing conclusions.

1. The degree of approximation of the theoretical formulation is generally acceptable,provided that the dominant failure mechanism is actually ductile: joint dowel yieldshould occur before bending failure of the timber members. Conversely, the sim-plified prediction is highly overestimated when the failure mechanism is brittle,particularly in the ULS case.

2. Although ductile behavior should be targeted by any consistent seismic design pro-cedure, the simulated design shows that the design provisions of the current versionof Eurocode 8 are not sufficient to exclude brittle failure mechanisms, as there isno formal control over the overstrength factor of timber members with respect tojoints. More specifically, the simulation highlights that the failure is not ductile in14% of cases (Fig. 7b). The Eurocode 8 detailing provisions, for dowel diameters

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Estimating the Displacement Capacity of Timber Portal-Frames 899

and member thickness, ensure that the local dowel failure mechanism is ductile, butdo not exclude a brittle member failure before dowel yield.

3. This issue can be resolved by introducing provisions for direct verification of anappropriate overstrength factor, similar to the Eurocode 8 prescription for steeland reinforced concrete structures. When the gravity load is neglected, but theoverstrength factor is set to 1.2, the simulations show that the predictive formulaunderestimates the pushover displacement capacity in 98% of cases with a meanratio of �d/�push, at ultimate limit state, of 0.92 (Fig. 9b-5).

4. To account for the gravitational load we propose alternative expressions which,without losing simplicity, estimate more accurately the plastic/elastic deformationand take account of the gravity load. Different expressions are provided for giventarget overestimate percentiles. For a SLS, the 50% target percentile seems the mostappropriate choice in design. For an ULS, the 2% target percentile equation is par-ticularly simple, and it is suitable for incorporation in design provisions. With anoverstrength factor of 1.2 it is demonstrated that the overestimate error by the 2%-equation is never greater than 10%; in the authors’ opinion this is fully acceptableat the pre-design stage.

5. We note here that the simplified formulation is a tool to estimate, with an accept-able degree of confidence, the expected displacement capacity of a timber structure.More specifically, the SLS formula prediction has a 95% confidence of divergingless than the 38.7% from the FEM value; while the ULS formula prediction, witha design overstrength factor of 1.2, has a 95% confidence of diverging less than15.4%. This estimation can be done at a design stage when the member dimen-sions and the structural details have not been yet determined. Prior estimation ofdisplacement capacity is typically needed in Direct DBD procedures. It is equallyunderstood that the simplified expression is not an alternative to more refined calcu-lation methods that should be used to verify the real ductile capacity of a structuralsystem, once design dimensions and detailing are fully established.

Acknowledgments

This work was carried out with the financial contribution of the Italian EarthquakeEngineering Laboratory Network (RELUIS), Project Reluis-DPC 2005-2008, Line IV,“Development of a Model Code for Direct Displacement-Based Seismic Design.” Theauthors wish to thank all the researchers who contributed to development of the contentreported in this article, and in particular Dr. Timothy Sullivan.

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