Finite semi-rigid study of light timber nailed structures.

Thierry Descamps - Assistant Professor

Sélim Datoussaïd - Professor

Laurent Van Parys - Assistant Professor

FPMs – Faculty of engineering, Dpt of Civil Engineering and Structural mechanics,

Rue du Joncquois, 53 - 7000 Mons, Belgium Tel +32(0)65.37.45.32 - Fax +32(0)65.37.45.28

Thierry.Descamps@fpms.ac.be

Summary

Widely used for light frame structures (e.g. with nails or screws), or for heavy laminated wood structures (e.g. with dowels), dowel-type fasteners are the most common form of connectors used in timber construction. The purpose of this work is to develop a tool for the semi-rigid analysis and design of such joints. The method consists in the introduction of a particular finite element, the so-called "Finite Semi-Rigid Element" between the ends of the connected members. The joint element consists of two nodes, each with three degrees of freedom. These nodes will be tied with the nodes end of assembled beams during the FE analysis. The stiffness of the FSRE is computed from the geometry of the joints and embedding strengths along and perpendicular to the grain. The gusset panels are supposed to be stiff, assuming that any displacement between the timber and the gusset plate is only due to the slip between the wood and the fasteners. To validate the finite element modelling, some experimental results on nailed connections are presented and compared with numerical analysis results.

1. Introduction

Informal languages commonly define a structure as a constructed assembly of joints separated by members. In order to simplify the structural analysis, joints are usually designed by assuming an ideally pinned or rigid behaviour, neglecting this way the influence of the joint stiffness on the deflections, or on the plastic load redistribution at the ultimate state. Due to the fact that the connection stiffness and strengths are relatively low in comparison to that of other structural components, connections play a major role in the load-response characteristics of light-frames (e.g. they add ductility and increase damping). The growing interest in taking into account the semi-rigid behaviour of joints in timber structures will lead to more safety in timber structures, a reduction of the amount of material, and so, contribute to the development of timber structures.

2. Timber frames analysis

To increase safety, connections which exhibit some ductility are generally preferred. Moreover, the introduction of the semi-rigidity in the computations allows a better prediction of the serviceability limit states and leads to meaningful reduction of cost by means of reduced cross-sections. It points in the direction of the need to develop suitable semi-rigid modelling techniques. In general, the finite element models for timber trusses are based on plane frame models, where the timber beams are modelled by means of beam or truss elements. The main differences between the models are found in the joint modelling:

• Ideal truss modelling: It is assumed that all timber members are connected by hinges.

• Semi-truss modelling: With the aim of introducing some stiffness in the model, web members are connected by a hinge, chords are continuous and stiff. In this case the truss is not an ideal truss but a semi-truss, however called truss in practice [1], [2].

• Semi-truss modelling with edge eccentricity: The hinge is located in the joint line, i.e. in the chord edge introducing an eccentricity in the joint [3].

With all these modelling approaches, timber frames are analysed under the assumption that joints are pinned.

2.1. Development of semi-rigid modelling methods

Besides the focus of building strong and ductile joints, it is important that the stiffness and the load capacities of the joints can be easily predicted by e.g. a finite element program or by analytical calculations. Translational stiffnesses are often assumed as infinite. This assumption usually does not affect significantly the stress distribution. The definition of the rotational stiffness may be a sensitive and arduous work. The principle approaches for the modelling of the semi-rigidity of timber frames are summarized here.

• Modified MOE: to easily take the translational stiffnesses of the joints into account, a reduced MOE is used, e.g. for secondary elements as webs. This fictitious MOE depends on the joint slips [4].

• Fictitious elements: in some cases, the simple assumption of a hinge is too penalizing for the design (e.g. heel joint or ridge joint). To add some kind of rotational stiffness in the analysis, a fictitious beam element is added. The properties and the location of the fictitious elements have to be experimentally estimated [2] [5] [6].

• Spring models: the stiffness of the joints is taken into account by the introduction of two translation springs acting for axial and shear forces and one rotation spring acting for the bending moment. The springs behave as mutual independent (no coupling effects) and connect nodes with identical location. Eccentricity introduced by the joint may be taken into account by some fictitious elements [7] [8] [9].

• Joint elements: This modelling is based on the definition of all components which contribute to the stiffness of the joint and the definition of special finite elements used to take the deformations of the joints into account [10] [11] [12]. This last approach was followed in this work.

2.2. Nailed joints for light frames

Nailed gusset plates joints have almost disappeared in timber construction with the advent of nail plates. However, these joints present many technical advantages. The use of gusset plates made of wood-based panels allows increasing both the stiffness and the efficiency of the joints. Large dimensions and thicknesses of wood panels allow doing any complex connection geometries. In an attempt of doing highly capable connections, plywood panels are preferable than OSB panels because of their strength, stiffness and resistance to thickness swell when exposed to moisture.

3. Development of the Finite Semi-Rigid Element

Fig. 1 Example of a heel joint modelling.

Because nailed joints with gusset plates only allow a connection between co-planar beams, 2D beam elements with three degrees of freedom were used. The elements are located on the system lines but some eccentricities are introduced if needed. To introduce the semi-rigidity a particular finite element, the so-called Finite Semi-Rigid Element (FSRE) is introduced between the ends of two connected members [10] [11] [12] [13] [14]. The FSRE is a mathematical entity that could connect two superposed nodes, and so, has no physical length. However, its length defined with the positions of an entry and exit nodes, influences the spans, and so influence the global behaviour of the structure. The modelling of a joint composed of n members needs the definition of n FSRE. In the joint, each FSRE is associated to a member and connects two nodes.

Fig. 2 Geometrical definition the FSRE.

The entry node is located at the centroid of the nail group and connects the gusset to the member. The exit node is located at the gusset centroid. For multi-members connections, all FSRE are connected together in this point. Most of the time, for simple symmetric connection, the exit node is located at mid-length of the joint line. In case of non-symmetric nailing respective to one member, the nails centroid may not coincide with the system line. In this case, a beam element is used as auxiliary elements, to transfer forces from the FSRE entry node to the system lines, as presented on Fig 1 and 2. These auxiliary elements have the same stiffness than members they connect to.

3.1. Contact between members

In the connection, contact between members may occur and modify the load transfer and so the ultimate load level and the. To introduce this contact stiffness in the computation, a contact element is generally introduced in the model [9][12][15]. Contact only have a slight influence on the ultimate load. In the semi-rigid definition of the connection, the contact stiffness will act in parallel to the gusset stiffness. Furthermore, according to the cutting of the members and the angle between them, contact may introduce some stresses acting perpendicular to the grain. These may lead to premature splitting of wood in the joint area. So, contact was avoid in experimentations and no contact elements were considered in the computations.

3.2. Finite semi-rigid study of continuity nailed joint

The purpose is here only to give an overview of the method on a simple example of co-axial members. The studied structure is a simple straight beam with a nailed splice joint introduced at mid-span. We propose to define the FSRE stiffness matrix in this simple case. The stiffness of the

Fig. 3 Nailed splice joint.

Fig. 4 Nailed joint with gusset plate.

FSRE is computed from the geometry of the joints and the embedding strengths along and perpendicular to the grain. The gusset panel is supposed to be stiff, assuming that any displacement between the timber and the gusset plate is only due to the slip between the wood and the fasteners. Because of the large deflections of the nails at the ULS, this assumption is realistic at least as long as no splitting has occurred in the wood.

The proposed method defines two FSRE to model the splice joint presented in Fig. 3. For a general purpose, a connection between n members will be modelled with help of n FSRE. Each FSRE is composed of two nodes as presented in Fig 4. The first one is called node 1 or entry node, and the second one is called node 2 or exit node. Node 1 is located at the centre of the nail groups respectively to member 1. Node 2 is at the geometric centre of the gusset. The local coordinates systems associated to these two nodes are parallel to each other and to the

orthotropic directions of timber. During the finite element analysis, the entry node of the FSRE will be connected with a beam element while the exit node will be connected to one or more other FSRE. With the aim of establishing the stiffness matrix of the FSRE connecting nodes 1 and 2, each nail i is located in the local coordinate systems 1 and 2:

If the nodal displacements and force vectors at nodes 1 and 2 are respectively denoted U and F, the element equilibrium relation can be written as:

The first step consists of the definition of the displacements of any fasteners i with respect to the entry coordinate system. So, assuming that the fastener i is linked to the entry local coordinate system, one can write:

Similar relation can be written to express the displacement of any fasteners i in respect with the exit coordinate system. Displacements Ui,2/1 and Ui,2/2 are obviously linked together by means of a matrix T1,2 which simply refers to the offset between the coordinate systems 1 and 2. The resulting displacement of the fastener at node 1, expressed in local coordinate system 1:

This residual displacement produces a force at node 1. Once displacement and local stiffness of the fastener i are defined, the summation of the forces associated to each fastener gives the nodal resulting force and the bending moment created by the local forces Fi,x and Fi,y at the reduction nodes 1 and 2.

where n is the number of fasteners and:

(1)

with (2)

with

(3)

(4)

(5)

k11 and k22 are the translational stiffnesses of the fasteners parallel and perpendicular to the grain, the so-called embedment stiffnesses. As no moment is defined at one nail, kθ,θ=0. Analogous equation can be written according to the exit node 2.

In the expression of the matrix 1

iK , we made the assumption that timber is an orthotropic material, characterized by only two stiffness parameters. These translational stiffness of the fastener, k11 and k22, are measured experimentally from the fastener load-slip curves, respectively for a loading parallel and perpendicular to the grain. No stiffness interaction terms k12 or k21 are considered, which means a total decoupling of the orthotropic directions of timber. For a good prediction of the joint behaviour at ULS, the full non-linear slip-curves have to be considered. This means that the orthotropic stiffness k11 and k22 are a function of the displacement leading to a non-linear problem. The resolution of this non-linear system is achieved with a Newton Raphson algorithm. Timoshenko's beam elements are used to model the timber members. To fine down the post-processing work, an automatic diagnostic according to Eurocode 5 is available.

4. Application

For an easily check of numerical results, simple structures subjected to an axial force, shear force or pure bending moment, have been studied. Only most pertinent results are presented here.

4.1. Moment resisting connection

Splice joints allow making taller members with small timber elements. In case of important spans, trusses are often produced with splices in the upper chord and in the lower chord.

• Geometry: The studied structure is a straight 2,1m long spruce beam with a nailed gusset joint at mid-span as presented in Fig. 5. The gusset plates are 17 plies plywood panel, 18mm

Fig. 5 Geometry of the nailed splice joint.

thick. Thirty-two nails of 3mm diameter and 80mm long are used. End spacing and distance between nails are respected according to EC5. A gap of 2cm is introduced between members. To ensure the symmetry of the connection, nails are placed on both sides of the connection. For the FE analysis, the beam is simply supported and divided in six beam elements and two FSRE. Two symmetric loads F are applied on both sides of the connections to create a pure bending moment.

• Experimental campaign: Ten beams with nailed gusset connections were tested. Before cutting the beams and introducing the connections, the MOE of each of them was measured by a four points bending test according to EN408. All tested beams present the same curvilinear behaviour with a large plastic deflection. Elastic stiffness is mainly influences by the timber MOE. Small variations of the plastic stiffnesses are observed with regard to the variations of the MOE. The observed failure modes correspond to a brittle crack along the grain, after important plastic deflections of the nails.

• Results and discussion: The connection as an obvious influence on the vertical displacement at mid-span. We can also observe a important loss of bending stiffness due to the introduction of the connection. Indeed, relative rotations between beam element and FSRE present an important fluctuation. Figure presents the evolution of the total displacement computed at each nail of the FSRE until the ULS is reached. The results presented here are computed with the measured timber MOE of 12178 N/mm². The ultimate load reached in simulation is Fmax=11,2kN (no convergence).

All nails do not have the same displacement, showing the unequal distribution of the load between the nails in the connection. Actually, nails 3,5,11 and 13 are the less loaded and nails 2,8,10 and 16 are the most loaded. The distribution of the load between the nails underlines that the instantaneous centre of rotation of the FSRE is its entry node. Numerical simulation slightly overestimates (12%) the ultimate load observed during test. Figure 7 presents the comparison between experimental and numerical displacement at mid-span. The computed displacement at the ULS is twice than experimentally observed. However, FSRE model gives an accurate prediction of the global connection behaviour until 22% of the ultimate experimental load, and a conservative and good prediction until 89%.

Elastic properties of timber members (MOE) have a poor influence on the global behaviour of the studied structure. Results are very sensitive to embedment stiffness used for the definition of the FSRE. Ratio of embedment stiffness parallel to the grain to embedment stiffness perpendicular to the grain, plays a major role in the ultimate load reached. This underlines the importance of the orthotropic definition of stiffness properties for nailed connections.

4.2. Unbraced structure

• Geometry: The studied structure is a square structure assembled with four nailed gusset joints as presented in Fig. . A gap is introduced between the members to avoid any contact. The structure is simply supported in C4 and loaded in C1. The structure is divided in eighteen beam elements and eight FSRE.

• Experimental campaign: Four structures have been tested. The MOE of different beams used in the structure were measured by a four points bending test. The shortening and lengthening of diagonals C1C4 and C2C3 were measured by means of dial gauges. For all the tested structures, the observed failure modes correspond to a brittle crack appeared in the connection after important nails yielding. Local defects may explain the failure location. As expected, nails far form the centre of rotation present more important deflections.

Fig. 6 Beam with FSRE in bending:

displacement at nails until ULS. Fig. 7 Comparison of experimental

and numerical result.

Fig. 8 Square unbraced structure with

four nailed connections.

Fig. 9 Experimental and numerical

shortening of the diagonal C1C4

• Results and discussion: Fig. 9 presents the comparison between experimental and numerical shortening of the diagonal C1C4. The numerical modelling seems conservative as the ultimate load is underestimated. At 25% of the experimental ultimate load, the maximum error on the computed displacement is about forty-six percents, which represents an underestimation of the initial stiffness of the structure of the same value. The results presented here are computed with the measured MOE of constituent beams.

Good adequacy was found between experimental and numerical results, for both shortening and lengthening of diagonals C1C4 and C2C3. Finite semi-rigid models give satisfactory conservatives results. Connections exposed to combined loads, e.g. bending moment, axial and shear forces, can be modelled with FSRE. Brittle failures encountered after the fastener yielding is not predicted. An ending criterion based on the maximum relative displacement for each nail must be introduced in the model. Under this assumption, post-elastic behaviour of any connections can be modelled with FSRE.

5. Conclusion

Connections that exhibit some ductility are suitable for many aspects. However, timber has a relatively low stiffness to strength ratio resulting in "flexible" structural systems. Insuring that joint strength criteria are fully achieved by non-compliance with stiffness criteria, is likely to be the more common reason for problem arising during the life of the structure.

An opportunity to provide some ductility, consists in introducing in the connection slender fasteners, forming plastic hinges at failure. In this regard, double shear nailed connections with lateral plywood gussets present many advantages among others, their strength, stiffness and geometrical flexibility.

The proposed connection model can be easily introduced in any structural analysis. Because of its general definition with help of sub-elements, no limitations in terms of connection geometry, number of connected members, size or type of fasteners exist. Some example are presented in Fig 10. The model is fitted to all dowel-type connections without any major modifications, making it

Fig. 10 Connection examples.

easy to introduce eventually new component types. It should be emphasize that in the FSRE definition, the connected members may have different cross-sections and that each member may hold a different number of fasteners, without any problems. This enlarges the application field of the model, e.g. for static and dynamic studies, and allows the design of truss or unbraced structures. The semi-rigid model consists in the definition of a particular finite element, whose particular stiffness matrix is computed according to the location of the fasteners, and their embedment strength. A basic advantage of the FSRE definition is that it needs for computation only the orthotropic embedding property, which is a well known property of any connection. Moreover, as the full non-linear load-slip curves contain all information about strength and stiffness, no additional information are needed for ULS verifications. Because ductile connections present a short elastic range, the FSRE is modelled with non-linear properties. This definition also assumes that the gusset panel is rigid, which seems realistic as for slender fasteners, nail and timber yielding predominate.

With special care to avoid any premature brittle failure, the model is accurate in failure as well as in deflection design. However, for the definition of embedment properties needed for the FSRE, the

characterization of fastener stiffness should be graded according to the timber stress grade. This will reduce the dispersion on input data and benefit given that the numerical results are highly sensitive to embedding properties. Despite of the fact that no distinction is generally made for embedding stiffness along or perpendicular to the grain, for slender connections, experimentations show a slight difference. As the numerical results are highly sensitive to the ratio of parallel to perpendicular stiffness, the orthotropic behaviour of the connection must be taken into account. The definition of the geometrical model of the FSRE is of major importance. It modifies free spans of connected members but also the stiffness of the connection as the direction of the soliciting loads relative to the FSRE is affected. The definition of the FSRE between the centroid of gusset panel and the centroid of the nail group relative to a member gives satisfactory results. To guarantee a security margin, an ending criterion based on the maximum relative displacement computed at nails in the FSRE, should be defined. This criterion seems reasonable as it has non sense to admit relative displacement at nails larger than 5mm, as well as the embedment strength is relative to the maximum load at 5mm of embedment.

6. References

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[2] Riberholt H., Guidelines for Design of Timber Trussed Rafters., Proceedings of CIB W-22, 22-14-1, Berlin, 1989.

[3] Brynildsen O.A., Structural models for trussed rafters., Norsk Treteknisk Institute, Oslo, Norway. Report 59. 22 pp., 1979.

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[5] Riberholt H., Guidelines for Static Models of Trussed Rafters., Proceedings of CIB W-18, 15-14-1, Karlsruhe, 1982.

[6] Suddarth S.K., Wolfe R.W., Purdue Plane Structures Analyser II – A – Computerized Wood Engineering System., Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. General Technical Report FPL-40, 1983.

[7] Maraghechi K., Itani R.Y., Influence of Truss Plate Connectors on the Analysis of Light-Frame Structures., Wood and Fibre Science., 1984, Vol.16 (3).

[8] Olsson A., Rosenqvist F., Comparative computation of deflections in wooden roof trusses connected with nail plates (in Swedish)., Lund University, Division of Structural Mechanics, Report TVSM-5066, 1996.

[9] Kevarinmaki A., Semi-rigid Behaviour of Nail Plate Joints., PhD Thesis, 2000, Helsinki University of Technology, TKK-TRT-109.

[10] Foschi R.O., Analysis of Wood Diaphragms and Trusses. Part II: Truss Plate Connections., Canadian Journal of Civil Engineering., Vol.4, 1977.

[11] Nielsen J., Stiffness Analysis of Nail-Plate Joints Subjected to Short-Term Loads, PhD Thesis, 1996, Dept. of Building Technology and Structural Engineering, University of Aalborg, Denmark.

[12] Ellegaard P., Analysis of Timber Joints With Punched Metal Plate Fasteners., PhD Thesis, 2001, Department of Building Technology and Structural Engineering, Aalborg University, Denmark.

[13] Foschi R.O., Truss Plate Modelling in the Analysis of Trusses., Proceedings of Metal-Plate Wood-Truss Conference, 88-97, 1979.

[14] Poutanen T., Analysis of Timber Nail Plate Components., PhD Thesis, 1995, Publication 165 - Tampere University of Technology, Tampere, Finland.

[15] Ellegaard P., Effect of chord splice joints on force distribution and deformations in trusses with punched metal plate fasteners., Holz als Roh und Werkstoff, Vol. 65, (6), 469-475, 2007.