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Mathematical modelling of timber-framed wallsstrengthened with CFRP strips

Miroslav Premrov *, Peter Dobrila

University of Maribor, Faculty of Civil Engineering, Smetanova 17, 2000 Maribor, Slovenia

Received 1 November 2005; received in revised form 1 January 2007; accepted 16 February 2007Available online 24 February 2007

Abstract

This paper provides mathematical modelling for prefabricated timber-framed walls composed of a timber frame andfibre-plaster boards. Because the tensile strength of the fibre-plaster boards is approximately 10-times lower than the com-pressive one, it is convenient to strengthen the boards in their tensile diagonal direction with carbon fibre-reinforced poly-mer (CFRP) strips, which are glued to the boards. Based on analysis of experimental research results [M. Premrov, P.Dobrila, B.S. Bedenik, Analysis of timber-framed walls coated with CFRP strips strengthened fibre-plaster boards, Int.J. Solids Struct. 41 (24/25) (2004) 70357048] special approximate mathematical models have been developed. The modelsenable simultaneously to consider the influence of inserted CFRP strips, flexibility of mechanical fasteners in the connect-ing areas and any appearing of tensile cracks in the coating boards.

2007 Elsevier Inc. All rights reserved.

Keywords: Timber structures; Walls; CFRP strips; Mathematical modelling

1. Introduction

There is an increasing tendency worldwide to build multi-level (four and more) prefabricated timber struc-tures with timber frame wall panels as the main bearing capacity elements. The treated wall is a compositeelement consisting of framed panels made from sheets of board-material fixed by mechanical fasteners toone or both sides of the timber frame (Fig. 1). There are many types of panel products available which

may have some structural capacity such as wood-based materials (plywood, oriented strand board, hardboard,particleboard, etc.) or plaster and fibre-plaster boards (FPB), recently the most frequently used in Central Eur-ope. One of the most important reasons for an increased application of these types of gypsum products is theirrelatively good fire protection. Additionally, gypsum is a healthy natural material and is consequently partic-ularly desired for residential buildings. In the presented research we will limit our attention to FPB.

0307-904X/$ - see front matter 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2007.02.009

* Corresponding author. Tel.: +386 2 22 94 347; fax: +386 2 25 24 179.E-mail address:[email protected](M. Premrov).

Available online at www.sciencedirect.com

Applied Mathematical Modelling 32 (2008) 725737

www.elsevier.com/locate/apm
mailto:[email protected]:[email protected]


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In structural analysis panel walls for design purposes can be regarded separately as vertical cantileverbeams with the horizontal force (FHFH;tot=nacting at the top (Fig. 1), as can be found for example in Fah-erty and Williamson[1], Hoyle and Woeste[2]and Eurocode 5 [3].

Described walls can be treated as composite elements. Distribution of the horizontal force by a compositetreatment of the element depends on the proportion of the stiffness. Because the tensile strength of FPB isapproximately 10-times lower than the compressive one, and evidently smaller than the wood strength ofall members in the timber frame, the FPB are usually a weaker part of the presented composite system. Thus,

especially in multi-level buildings located in seismic or windy areas, cracks in FPB usually appear. In thesecases the FPB lose their stiffness and therefore their resistance should not be considered at all. Stresses inthe timber frame under a horizontal loads are usually not critical.

There are several possibilities to reinforce panel walls in order to avoid cracks in FPB:

by using additional boards. The boards are usually doubled: symmetrically (on both sides of a timber frame), non-symmetrically (on one side of a timber frame),

by reinforcing boards with steel diagonals, by reinforcing boards with carbon or high-strength synthetic fibres (FRP, CFRP, etc.).

In Dobrila and Premrov[4]experimental results using additional FPBare presented. The test samples dem-onstrated higher elasticity, whilst bearing capacity and especially ductility were not improved in the desiredrange.

With the intention to improve the resistance and especially the ductility of the walls it is therefore moreconvenient to insert classicaldiagonal steel strips, which have to be fixed to the timber frame. In this case onlya part of the horizontal force is shifted from boards over the tensile steel diagonal to the timber frame after theappearance of the first crack in the tensile zone of FPB (Dobrila and Premrov[4]).

2. Reinforcing with diagonal CFRP strips

Literature provides few investigations on wood-based panels strengthened with high-strength fibres (HSF).

The use of HSF sheathing material does not increase the bearing capacity much if mechanical fasteners are

FH,tot FH

x

h

CFRP strips

bd

b b

nby

n

FF tot,HH= zt

timber frame

coating board

Fig. 1. Static design and cross-section of the timber-framed wall.

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applied to connect the wood-based sheets to the timber frame. Kent and Tingley [5] presented experimentalresults for high-strength synthetic fiber reinforced panels bonded to hollow beams. Test experiments per-formed in EMPA on wood-based panels reinforced with Sika CarboDur strips demonstrated an essentialincrease in bending resistance by 43% (Zagar[6]).

Since the tensile strength of FPB is obviously lower than the compressive one and corresponding capacity

of timber frame, the treated wall elements tend to fail because the cracks are forming in the tensile area of theFPB, therefore this tensile area could be reinforced with high-strength materials. This strengthening concept issuch that the composites would contribute to tensile capacity when the tensile strength of FPB is exceeded.Experimental results obtained on timber-framed walls strengthened with diagonal CFRP strips (Premrovand Dobrila[7]), which were glued on FPB, demonstrate some important facts, which should to be consideredby mathematical modelling of the wall elements:

(a) There was no essential influence on the element stiffness of any reinforcement before cracks appeared intensile area of un-strengthened FPB.

(b) The elastic resistance (force forming the first crack) essentially increased for all kinds of CFRP strength-ened test samples.

(c) After the first cracks in un-strengthened FPB appeared, the test samples proved an important distinction

in behaviour in timber frame-fibreboard connecting area dependant on the boundary conditions betweeninserted CFRP strips and timber frame. If the strips were additionally glued to the timber frame the fas-teners produced substantially smaller slip in the connecting area, which never exceeded 1 mm when thefirst tensile cracks in FPB appeared. Therefore it can be assumed that the yield point of the fasteners wasnot achieved before cracks appeared at all and the elements tend to fail because of cracks appearing inFPB. On the other hand, in the case where the CFRP diagonals were unconnected to the timber frame,the slip between the FPB and the timber frame was evidently higher and the walls tend to fail because offastener yielding.

(d) It has been shown that the inclusion of CFRP diagonal strip reinforcement on the load-carrying capacitycan be quite high and that it is maximized when the carbon strips were additionally glued to the timberframe.

3. Mathematical modelling of diagonally reinforced wall elements

We will focus our research on numerical stress and deformation analysis of the prescribed wall element sub-jected to a horizontal force acting at the top of the cantilevered panel wall (Fig. 1). The wall element actuallybehaves like a composite deep beam, however in engineering praxis a simplified mathematical shear model isusually used. We will shortly describe the shear model, but more attention will be dedicated to the compositemodel developed by ourselves.

3.1. Shear model

Many design models have been proposed in order to analyse and predict the behaviour of wall diaphragmssubjected to lateral loads. Kallsner [8] and Akerlund [9] proposed an agreeable approach to determine theload-carrying capacity of the wall unit, based on the following key assumptions:

behaviour of the joints between the sheet and the frame members is assumed to be linear-elastic untilfailure,

the frame members and the sheets are assumed to be rigid and hinged to each other.

Two simplified computational methods are given in the final draft of Eurocode 5 [3] in order to determinethe load-carrying capacity of the wall diaphragm.

The first Method A, is identical to the Lower bound plastic method, presented by Kallsner and Lam

[10]. It should be provided that:

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the spacing of fasteners (s) is constant along the perimeter of every coating board (sheet), the width (b) of each sheet is at least h/4.

This method defines the walls shear resistance (Fv,Rd) as a sum of all the fasteners shear resistances (Ff,Rd)along the loaded edges using an assumption that the timber frame members and the sheets are rigid and hinged

to each other:

Fv;RdX

Ff;Rdbsc 1

c1 for bP b0b

b0for b 6 b0

8 FH;cr:AbAbc2txII22:54140:819207:442 cm2

GAs

eff 120207:44263201:6 1

1:237 593:84 kN

k12sin2 24:50cos 24:5023100300:1226 026:83 kN; k2 137593:84

kN1

FH;CFRPk1k2FH26026:8337593:84

FH0:693FH

46

5. Conclusions

A special approximate analytical model for composite timber-frame wall elements has been developedbased on fundamental assumptions that the element behaves like a composite beam and that the cracks in

FPB appear before the yield point of the fasteners is achieved. To assure the second assumption the CFRP

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strips should be additionally glued to the timber frame. The recommended model with the fictive enlargedthickness (t*) of the FPB simultaneously considers the influence of the inserted CFRP diagonal strips, the flex-

ibility of the mechanical fasteners between the boards and the timber frame and the appearance of cracks in atensile area of the FPB (see Table 2).The presented numerical results for the force forming the first crack (FH;crand for the crushing force (FH;u

show relatively good agreement with the measurement performed on the test samples [7]. The accuracy isapproximately at 3.7% forFH;crand at about 5.5% forFH;u. The part of the acting horizontal force which is takenover the CFRP diagonal strip strongly depends on cracks appearing in FPB. It is evident from the numericalresults that the CFRP contribution is almost two-times higher after the crack appearance. It is also presentedthat compressive stresses in timber and in FPB are tolerably under the yield points, therefore our assumptionsof elastic behaviour of both materials after the appearance of the first crack in FPB are quite acceptable.

References

[1] K.F. Faherty, T.G. Williamson, Wood Engineering and Construction Handbook, McGraw-Hill Publishing Company, 1989.[2] R.J. Hoyle, F.E. Woeste, Wood Technology in the Design of Structures, Iowa State University Press, Ames, Iowa, 1989.[3] CEN/TC 250/SC5 N173, Eurocode 5: Design of Timber Structures, Part 1-1. General rules and rules for buildings, Final draft prEN

1995-1-1, Brussels, 2003.[4] P. Dobrila, M. Premrov, Reinforcing methods for composite timber frame-fiberboard wall panels, Eng. Struct. 25 (11) (2003) 1369

1376.[5] S. Kent, D. Tingley, Structural Evaluation of Fiber Reinforced Hollow Wood Beams, in: Proc. of Innovative Wooden Structures and

Bridges, IABSE Conf., Lahti, 2001.[6] Z. Zagar, Timber Structures Part II, Modelling of Timber Structures, Udzbenici Sveucilistva u Zagrebu, Zagreb, 1999.[7] M. Premrov, P. Dobrila, B.S. Bedenik, Analysis of timber-framed walls coated with CFRP strips strengthened fibre-plaster boards,

Int. J. Solids Struct. 41 (24/25) (2004) 70357048.[8] B. Kallsner, Panels as wind-bracing elements in timber-framed walls, Swedish Institute for Wood Technology Research, Report 56,

Stockholm, 1984.[9] S. Akerlund, Enkel berakningsmodell for skivor paregelstomme (Simple calculation model for sheets on a timber frame), Bygg &

Teknik, No. 1, 1984.[10] B. Kallsner, F. Lam, Diaphragms and shear walls, Holzbauwerke: Grundlagen, Entwicklungen, Erganzungen nach Eurocode 5, Step

3 15/1-17, Fachverlag Holz, Dusseldorf, 1995.[11] C. Chou, A. Polensek, Damping and stiffness of nailed joints: response to drying, Wood Fiber Sci. 19 (1) (1987) 4858.[12] A. Polensek, K.M. Bastendorf, Damping in nailed joints of light-frame wood building, Wood Fiber Sci. 19 (2) (1987) 110125.[13] W.J. Van Wyk, The strength, stiffness and durability of glued, nail-glued and screw-glued timber joints, South Afr. Forest. J. 138

(1986) 4144.[14] M. Premrov, P. Dobrila Peter, B.S. Bedenik, Approximate analytical solutions for diagonal reinforced timber-framed walls with fibre-

plaster coating material, Construct. Build. Mater. 18 (10) (2004) 727735.[15] Knauf Gipsfaserplatten Vidivall/Vidifloor, 2002.[16] Sika, Sicher bauen mit System. Technische Merkblatter. Ausgabe 5, 2003.[17] European Committee for Standardization, EN 338:2003 E: Structural timber Strength classes, Brussels, 2003.[18] H. Bruninghoff, et al., Eine Ausfuhrliche Erlauterung zu DIN 1052, Teil 1 bis Teil 3, Beuth Kommentare, Beuth Bauverlag, Berlin,

1988.

Table 2Numerical results for stresses, force acting on one fastener, slip and neutral axis

FH(kN)

Tension inFPB rtb;max(N/mm2)

Comp. inFPB rcb;max(N/mm2)

Tension intimberrtt;max(N/mm2)

Comp. intimberrct;max(N/mm2)

F1 (N) D (mm) xII (cm)

10.0 1.069 1.069 1.452 1.452 64.271 0.218 62.500

20.0 2.138 2.138 2.905 2.905 128.541 0.435 62.50023.39=Fcr 2.560 7.110 3.763 146.397 0.496 40.81925.0 2.793 7.600 4.020 156.474 0.530 40.81930.0 3.351 9.118 4.826 187.769 0.636 40.81935.0 3.922 10.646 5.616 221.377 0.769 40.88240.0 4.504 12.180 6.389 245.299 0.889 40.98842.68=Fu 4.811 13.0=ft,0,k 6.810 259.918 0.952 41.011

Measured[7]: Fu;meas40:33 kN.

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