Finite element analysis of flexural strengthening of timber beams with Carbon Fibre-Reinforced Polymers

Engineering Structures 101 (2015) 364–375

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Finite element analysis of flexural strengthening of timber beams with Carbon Fibre-Reinforced Polymers M. Khelifa a,⇑, S. Auchet a, P.-J. Méausoone a, A. Celzard b a b

University of Lorraine, LERMAB – EA 4370, 27 rue Philippe Séguin, CS 60036, 88026 Epinal, France University of Lorraine, Institut Jean Lamour – UMR CNRS 7198, 27 rue Philippe Séguin, CS 60036, 88026 Epinal Cedex, France

a r t i c l e

i n f o

Article history: Received 10 October 2014 Revised 17 June 2015 Accepted 27 July 2015

Keywords: Reinforced timber beams CFRP Non-linear finite element analysis Interface element Flexural behaviour Elasto-plastic behaviour

a b s t r a c t This study focuses on the flexural behaviour of timber beams externally reinforced using Carbon Fibre-Reinforced Plastics (CFRP). A non-linear finite element analysis was proposed, and was validated with respect to experimental tests carried out on seven beams. All the beams had the same square cross-section geometry and were loaded under four-point bending, but had different numbers of CFRP layers. The Abaqus software was used, and different material models were evaluated with respect to their ability to describe the behaviour of the solid timber beams. Elasto-plastic behaviour with damage effect was assumed for the timber material, linear elastic isotropic model was used for the CFRP, and a cohesive model was used to represent the interaction between two adherent surfaces (CFRP and timber). These two surfaces were paired and, taking into account the presence of an adhesive layer, one of them was defined as the master surface whilst the other was the slave surface. Predicted and measured load– mid-span deflection responses and failure modes were compared. The increases of flexural strength for the two different reinforcement schemes with 2 and 3 layers of CFRP composite sheets were 41.82% and 60.24%, respectively, with respect to the unreinforced timber beams. The maximal difference between calculated and experimental ultimate load-bearing capacity for reinforced solid timber beams with 2-layers of CFRP was around 1.2%. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The recent years have shown that technologies from the composite industry are becoming more and more utilised in civil engineering and in mechanical industries. The introduction of new construction materials and related technical know-how indeed allows achieving desired engineering properties. Timber and timber products are among the most important construction materials, and are also becoming more and more used in the form of composite materials with the introduction of the so called ‘‘highly engineered wood products’’. Timber is generally used in buildings, frames, truss bridges, and many other applications. Besides, wood is a renewable, environment-friendly and nontoxic material, available in large quantities almost all around the world. Additionally, it has a high thermal efficiency, is a net carbon absorber, and can be easily recycled. Mechanical wood properties are often inappropriate for heavy loads in buildings. Improving the structural behaviour of building

⇑ Corresponding author. Tel.: +33 329 29 61 18; fax: +33 329 29 61 38. E-mail address: [email protected] (M. Khelifa). http://dx.doi.org/10.1016/j.engstruct.2015.07.046 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

units by combining reinforcements with conventional building materials is already an old concept, extensively used in construction. For example, in the twentieth century, the combination of reinforcement steel and concrete has been the basis of a number of structural systems used in construction. Designers of products and material experts continue to develop new composite materials that might be used with conventional building materials like steel, concrete and timber, and which are expected to lead to the development of stronger, larger, more durable, energy-efficient and aesthetic structures. Timber material has adequate strength both in tension and in compression. But this high strength is often accompanied with low stiffness, so that the design is controlled by deflection limitations. Strengthening timber beams aims at achieving higher stiffness. Increased stiffness without need of increasing the thickness of the beam may result in substantial space and material savings. Timber presents damage and failure modes due to transversal tension. Besides, its natural origin gives timber a variable and heterogeneous character, and its mechanical behaviour is affected by the presence of defects such as knots, splits and slope of grain. These flaws are more detrimental in the tensile zone than in the compressive zone, since under tension they tend to develop into

M. Khelifa et al. / Engineering Structures 101 (2015) 364–375

cracks. Deterioration can be stopped by reinforcing the exposed surfaces of the wood with composite materials, and the resultant strengthening is one way of improving the mechanical behaviour of timber. Recently, composite materials such as Carbon Fibre-Reinforced Plastics (CFRP) have been tested for structural strengthening and repair of timber elements, due to their attractive characteristics [1–21]. Repairing damaged timber members by CFRP is a viable alternative to the replacement of other conventional engineering materials (for example steel plates) due to its many advantages such as corrosion resistance, light weight and flexibility, ease of cutting, high elastic modulus and high strength to environmental aggressive factors. Usually, the behaviour of timber beams is governed by flexion, hence the need to strengthen them in bending. In addition, the load-bearing capacity of the timber beam specimens might be significantly improved with shorter lengths of CFRP. Several experimental programs have investigated the performance of Reinforced Timber (RT) beams with CFRP, and have shown the effectiveness of using externally-bonded CFRP materials to improve the flexural capacity of RT members [1–14]. However, most of the previous studies were carried out using CFRP laminates externally attached to the soffit of solid wood beams along the full span length. To characterise the stiffness of timber beams in flexural, ductility and strength response of FRP-wood beams, an experimental programme based on a four-point bending test configuration was carried out by Borri et al. [1]. Experimental tests of the reinforced wood showed that external bonding of FRP materials may increase the flexural stiffness and the load-bearing capacity. The study presented by De la Rosa García et al. [7] showed that the strengthening systems of pine timber beams with basalt and carbon FRP gave rise to structures having higher stiffness and carrying capacity than the initial ones. The recent work carried out by Raftery et al. [14] also described an experimental test programme which examined the strengthening of glulam in flexion using bonded-in glass fibre reinforced polymer (GFRP) rods. It was found that the geometrical arrangement of the routed out grooves played an important role in the mechanical performance of the strengthened glulam. The effect of having shorter lengths of CFRP sheet did not receive much attention and the literature lacks information on the behaviour of RT beams strengthened with shorter lengths of CFRP sheets. One of the few studies that considered the length of CFRP was realised by de Jesus et al. [8], who investigated experimentally, numerically and analytically the effect of distinct lengths of CFRP laminates on the strengthening of RT beams in flexion. It was concluded that the interfacial stress peaks decrease with the increase of strengthening length, and that the length of the strengthening does not influence significantly the stiffness of RT beams [8–10]. Several numerical works have investigated the performance of strengthened RT members using the Finite Element (FE) method [15–21]. Khelifa et al. [15] developed a numerical procedure to simulate the flexural behaviour of CFRP-strengthened timber beams wherein the kinematic continuity was assumed between adhesive and wood. This is justified by the fact that the shear modulus of wood is low, and failure modes do not occur between wood and adhesive. The obtained results showed that the proposed formulation can efficiently capture the load–displacement response with acceptable accuracy. Nowak et al. [19] developed a 3D nonlinear FE model that used shell and spatial elements to predict the mechanical behaviour of historic timber beams reinforced with CFRP strips. Shell elements were used to model the CFRP. Wood and adhesive were modelled using solid elements. In [19], the proposed FE model of wood was somewhat simplified by treating the latter as a homogenous orthotropic material. It was concluded that the use of simplified 3D FE model can accurately predict the response of the strengthened specimens.

365

In another study [20], a FE model was developed in order to analyse the flexural strengthening of timber beams reinforced with CFRP strips. The authors used cohesion elements to model the adhesive placed between wood and CFRP. It was found that the use of cohesive elements can accurately predict the response and the failure mode of the strengthened specimens. The work of Nowak et al. [19] was very interesting but neglected the interaction between elasto-plasticity and damage. It was therefore the aim of our previous study [20] to analyse the mechanical flexural behaviour in wood within the framework of continuum damage mechanics and plasticity. The aim of the present work was to numerically predict the performance of RT beams in flexion after strengthening by use of CFRP sheets of distinct lengths, externally attached to the soffit of timber beams. Such a numerical study was carried out within the framework of continuum damage mechanics and plasticity, as none of these developments are yet to be used in the modelling of the flexural behaviour of RT beams with flexure damage. No studies dedicated to this topic, having many technological, economical and environmental advantages but whose modelling is very complex, have been offered so far in the literature. A 3D nonlinear FE model was thus developed here, using the Abaqus finite element simulation code [22]. The present model considered the different material constitutive laws for mechanical orthotropic timber, isotropic adhesive and isotropic CFRP behaviours. The model was validated by comparing the predicted load–displacement curves, the ultimate load capacity and the failure modes with the measured experimental data obtained by Borri et al. [1]. A parametric study was also designed and performed. The parametric study varied the size of the tension reinforcement as well as arrangements of the CFRP sheets, and the effect of the number of CFRP layers was investigated. The FE modelling of such a problem, if simulated correctly, might be used as a numerical tool for predicting the flexural behaviour of RT beams externally strengthened with distinct lengths of CFRP sheets and anchorage systems. 2. Experimental work No experimental work was carried out in the present paper, which was only based on the experimental data reported in [1]. The present section therefore aims at presenting how the tests were performed, and what are the materials characteristics and the main assumptions used in our simulations. 2.1. Characteristics of materials 2.1.1. Wood The experimentation considered 7 beams classified in the second category according to the classification of wood reported in Ref. [1]. The moisture content was 10.88%, and the average timber density was 453.6 kg/m3 as reported in [1]. 2.1.2. CFRP Table 1 gives the main characteristics of the CFRP used to reinforce the timber beams, taken from the manufacturer’s data sheets reported in [1], measured according to the ASTM D 3039 [23] standard. The knowledge of these mechanical properties is a precondition for the sustainable application of CFRP.

Table 1 Properties of CFRP (from [1]). Tensile strength Tensile modulus of elasticity Ultimate strain Equivalent thickness

3.388 GPa 41.76 GPa 1.00% 0.165 mm

M. Khelifa et al. / Engineering Structures 101 (2015) 364–375

2.1.3. Adhesive The application of CFRP to the tensile region of RT beams was done using an epoxy resin that could transfer the stresses with perfect adhesion between CFRP and timber. Moreover, it is well known that two-component thixotropic epoxy adhesive (A and B see Table 2) shows appropriate performances in civil engineering, as indicated by former comparative studies [24–26]. The relevant mechanical properties of the two-component epoxy resin, as certified from the manufacturer, are given in Table 2. The adhesive was prepared according to the manufacturer’s recommendations as reported in [1]. 2.2. Test configurations The experimental tests were carried out under four-point bending. This loading arrangement was chosen because it gives constant maximum moment, zero shear in the section between the loads, and constant maximum shear force between supports and loads. The simply supported span between the two bearings, made of two semi-cylindrical steel elements (diameter 609 mm), was 3600 mm, the overall length of each beam was 4000 mm and the cross-section was 200  200 mm. Each of these beams was reinforced with longitudinal laminates on the tension face as shown in Table 3. Three un-strengthened control timber (CT) beams were tested in order to find their flexural capacity (beams CT#1, CT#2 and CT#3, see Table 3a). Two different reinforcement schemes with CFRP sheets were performed in the experimental program: (b) two CFRP sheets bonded centrally to the tension zone of the beams

(beams RT#4 and RT#5, see Table 3b); and (c) three CFRP sheets bonded centrally to the tension zone of the beams (beams RT#6 and RT#7, see Table 3c). During loading, two linear variable differential transducers (LVDT) were used to measure the deflection at mid-span. The stiffness was calculated for each loading cycle using the average values of the mid-span deflection registered by the two LVDT at each side of each beam as reported in [1]. 2.3. Test results The load–deflection curves in bending of some tested beams are shown in Fig. 1, based on mid-span deflection. Each curve comprises linear and nonlinear parts. For example, for CT#1 beam, the overall response is linear up to a displacement of 28.1 mm corresponding to a force of 32.8 kN. Beyond this point, the response becomes nonlinear until the force reaches 51.4 kN at a displacement of 72.54 mm. Reinforced and unreinforced test beams are compared in Fig. 1. It is clear that the reinforcement leads to higher stiffness values. Only RT#4 beam did not lead to improvements in the ultimate loads, compared to the unreinforced CT#3 beam response. This is probably due either to the fact that the RT#4

Table 2 Properties of epoxy (from the manufacturer, after [1]). Specific gravity of part A Specific gravity of part B Brookfield viscosity of part A Brookfield viscosity of part B Mixing ratio: Part A: Part B Specific gravity of mix Brookfield viscosity of mix Compressive strength (ASTM C 579) Bending tensile strength (ISO 178) Compressive modulus of elasticity (ASTM C 579) Bending modulus of elasticity (ISO 178)

1.45 g/cm3 1.02 g/cm3 18000 MPa s 1500 MPa s 3:1 1.35 g/cm3 4500 MPa s 50 MPa 35 MPa 3500 MPa 2500 MPa

140 120 100

Load [kN]

366

CT#1 CT#2 CT#3 RT#4 RT#5 RT#6 RT#7

80 60 40 20 0 0

20

40

60

80

100

120

140

Mid-span deflection [mm] Fig. 1. Load–mid-span deflection curves for CT and RT beams.

Table 3 Geometry of unreinforced and reinforced timber beams in groups CT and RT. Beam Types and description (a). Unreinforced (control) beams (CT) – Not to Scale; – Dimensions in mm; – Beam series: CT#1, CT#2, CT#3

(b). 2-layers reinforced beams (RT) – Beam series: RT#4, RT#5

(c). 3-layers reinforced beams (RT) – Beam series: RT#6, RT#7

Details of a typical beam

Cross section

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M. Khelifa et al. / Engineering Structures 101 (2015) 364–375 Table 4 Ultimate load-bearing capacity and deflection at mid-span of tested beams. Specimens

Fu (kN)

Fu,mean (kN)

DFu (%) –

CT#1 CT#2 CT#3

51.4 69.1 95.9

72.2

RT#4 RT#5

91.6 113.2

102.4

RT#6 RT#7

111.2 120.5

115.7

w (mm)

wmean (mm)

(1): longitudinal Dw (%)

66.4 74.45 90.8

77.2

–

41.82

64.3 88.4

76.35

1.1

60.24

77.3 111.8

97.75

26.6

(2): radial

(3): tangenal

Grain orientaon beam presents defects or to the debonding of the fibre under loading. Table 4 shows the values of ultimate load (Fu) for the tested beams, the mean ultimate load values (Fu,mean), the increase in load-bearing capacity (DFu) with respect to the CT beams series, the corresponding mid-span deflection (w), the mean deflection (wmean) and its increase with respect to the unreinforced model (Dwmean). The load-bearing capacity of the tested beams reinforced with CFRP sheets increased by 41.82% for the RT#4 and RT#5 beam series, and by 60.24% for the RT#6 and RT#7 beam series. Differences in load-bearing capacity are probably due to the different reinforcement schemes and/or to timber defects. The test results showed that the strengthening of solid timber members under bending with CFRP sheets had a beneficial effect on the load-bearing capacity (Table 4) and on the rigidity of the reinforced specimens (Fig. 1). The scattering of the experimental results for the different loading schemes illustrated in Table 3, observed for all tested specimens (control and reinforced) used in [1], is probably due to the fact that the inherent variability of the wood material was not taken into account. 3. Behaviour of materials and constitutive modelling The proposed model has been implemented as a Vumat subroutine in Abaqus/Explicit only for defining the mechanical constitutive equations of the wood material in finite element calculations. The analysis of the adhesive behaviour was based on the Cohesive Zone Model (CZM) available in Abaqus software, allowing an accurate description of the progressive damage of the interaction between two adherent surfaces (within the finger-joint on one hand, and at the timber-CFRP interface on the other hand) until failure. 3.1. Wood behaviour 3.1.1. Elasto-plastic model without damage Timber is an orthotropic material. Its properties depend on the tree trunk’s orientations (Fig. 2). The three main directions of timber were assigned to the three principal axes: (1) along timber grain; (2) along the radial direction (i.e. across grain orientations and perpendicularly to annual rings); (3) along the transversal direction (i.e. across fibre direction and parallel to annual rings). In structural timber element, one can easily determine direction (1), but it is practically impossible to identify orientations (2) and (3). Nonetheless, it was assumed that the behaviour in the radial direction (2) is similar to that observed in the transverse direction (3), which translates into [15,20,27–31]:

E2 ¼ E3 ;

G23 ¼ G32 ;

G12 ¼ G13 ;

m23 ¼ m32 ; m12 ¼ m13

ð1Þ

The wood plasticity model uses the Hill’s stress potential for anisotropic behaviour [32]. The Hill’s stress function is:

Fig. 2. Local coordinate system for timber.

f ðrÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 Hðrx  ry Þ þ Gðrx  rz Þ þ Fðry  rz Þ þ 2Ns2xy þ 2Ms2xz þ 2Ls2yz  R  ryield ð2Þ

where F, G, H, L, M and N are constants obtained by tests of wood material in different orientations, R is the isotropic hardening stress and ryield is the limit yield stress. The constitutive equations are given as follows: (a) State relations

r ¼ K : ee ðElastic lawÞ

ð3Þ

R¼Q r

ð4Þ

where r represents the Cauchy stress tensor, e represents the tensor of elastic strain, r represents the isotropic hardening, Q is the isotropic hardening modulus and K is the fourth order symmetric elastic properties tensor. e

(b) Complementary equations

_ e_ p ¼ kn

ð5Þ

_  brÞ r_ ¼ kð1

ð6Þ

p where e_ is the plastic strains rate tensor, k_ is the plastic multiplier, n is the normal to the Hill’s stress function f (Eq. (2)), r_ is the isotropic hardening strain rate and b is the non-linear isotropic hardening parameter.

3.1.2. Elasto-plastic model with damage In our previous works [20,27,28], the damaged finite elasto-plastic behaviour was described in the framework of the thermodynamics of irreversible processes with state variables. We limited ourselves to the fully coupled isothermal formulation using a single yield surface for both plasticity and ductile isotropic damage (the mechanical dissipation was described using the theory which considers only one potential of dissipation). The formulation using a single yield surface controls both the plastic flow and the evolution of the damage, with one plastic multiplier. In this framework, the damage follows the evolution of the plastic flow. The large plastic strains were observed in the compressive zone corresponding to load zone application where the damage increases rapidly and the first crack initiates. This is why a numerical failure is often observed in the compressive zone, which is a major drawback. For avoiding this, the wood damaged plasticity model was selected in the present study as it can completely

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account for the inelastic behaviour of wood, both in tension and in compression, including damage characteristics. The damaged plasticity model assumes that the two main failure mechanisms in wood are the tensile cracking and the compressive crushing. This is a significant advance compared to our previous works [27,28], in which these assumptions were not required. Three couples of internal state variables were used for this isothermal case: ðee ; rÞ for the plastic flow, (r, R) for the isotropic hardening and (D, Y) for the isotropic ductile damage. The constitutive equations are given as follows: (a) State relations

r ¼ ð1  DÞK : ee Cauchy stress tensor

ð7Þ

R ¼ ð1  DÞ  Q  r

Isotropic hardening stress

ð8Þ

Isotropic damage driving force

ð9Þ

Y¼

1 e e : K : ee 2

(b) Complementary equations

H:r e_ ¼ k_ pffiffiffiffiffiffiffiffiffiffiffiffi

ð10Þ

  1 Isotropic hardening strain rate r_ ¼ k_ pffiffiffiffiffiffiffiffiffiffiffiffi  br 1D

ð11Þ

 s Y D_ ¼ k_ S

Isotropic ductile damage rate

ð12Þ

with krk ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r : H : r Norm of the Cauchy stress tensor

p

k_ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi n Plastic strain rate tensor 1  Dkrk 1D

ð13Þ where n is the normal to the loading surface f (Eq. (14)), s and S describe the isotropic ductile damage evolution, H is the classical Hill anisotropic tensor defining the initial plastic anisotropy, which is function of the six parameters F, G, H, L, M, and N. The plastic flow f accounting for the isotropic hardening and the isotropic ductile damage can be expressed as follows:

krk  R f ¼ pffiffiffiffiffiffiffiffiffiffiffiffi  ryield 6 0 1D

ð14Þ

These Eqs. (7)–(14) give the classical elasto-plastic constitutive equations defined in Section 3.1.1 if the damage vanishes. To predict brittle failure, an attempt was made by means of an option in the Vumat subroutine for decreasing to zero the stresses in the element once the failure criterion was satisfied. The verification for the failure criterion was carried out as follows. For tension perpendicular and parallel to the grain, the principal stresses (rI; rII) in the elements in high stress areas were checked at every iteration against the critical tensile stresses (ft0; ft90), which were activated individually for the two directions 0° and 90°. The wood used in this study was assumed C40 class, where values of ft0 = 24 MPa and ft90 = 0.6 MPa were taken according to EC5 (Eurocode 5: Design of timber structures – Part 1–1: General – common rules and rules for buildings. Brussels, 2010).

longitudinal modulus is the parameter of highest importance. This is the reason why an isotropic model was considered suitable here. The elastic modulus in the fibre direction of the unidirectional CFRP material used was obtained by Borri et al. [1] as ECFRP = 41.76 GPa. This value for ECFRP and m = 0.3 were used for the isotropic model [19]. 3.3. Cohesive model In Abaqus software [22], there are two ways of modelling the adhesive interaction, using either: (1) cohesive element; or (2) surface-based cohesive behaviour, specifying the cohesive behaviour between the surfaces. These two methods are very similar in terms of their function, but surface-based cohesive behaviour is easier to use since there is no need to create additional elements. Besides, it can be used in a broader range of interactions, such as two sticky surfaces coming into contact during simulation. In this work, the thickness of an adhesive layer is negligibly small. Therefore, it is more appropriate to simulate the composite material (wood-CFRP) by specifying surface-based cohesive behaviour because it can significantly reduce computational cost. The mode of failure between CFRP and wood can be modelled using surface-based cohesive behaviour, with a traction-separation description [22]. This cohesive model assumes a linear elastic behaviour followed by damage initiation and evolution. The linear elastic behaviour of the initial response (Fig. 3), before damage initiation, can be written as:

frg ¼



rn rt



 ¼

Kn

0

0

Kt



dn dt

 ¼ ½Kfdg

where rn and rt are the normal and the shear stresses; dn and dt are the corresponding separations in the normal and tangential directions, respectively. Kn and Kt are the initial material stiffnesses, and can be specified as material properties. From Fig. 3, the initial material stiffnesses Kn and Kt are simply obtained by:

Kn ¼

rI0;n dI0;n

Kt ¼

;

rII0;t dII0;t

ð16Þ

The Abaqus software ‘‘traction-separation’’ law is based on a quadratic stress criterion to deal with damage initiation, and can be expressed as:

"

rIn rI0;n

#2

" þ

rIIt rII0;t

#2 ¼1

ð17Þ

Based on the notation used for the Abaqus software cohesive model, the adhesive parameters are the critical cohesive strengths (rI0;n ; rII0;t ), the initial stiffness (Kn, Kt) and the maximum displacements (dI;max ; dII;max ). Generally, the cohesive parameters in pure n t mode I are calculated based on a modified Double Cantilever

3.2. CFRP behaviour In simulations, there are two ways of modelling the mechanical behaviour of the CFRP: (1) either the CFRP material is considered as linear elastic isotropic, or (2) the CFRP is modelled as a linear elastic orthotropic material. In bending tests, during which the CFRP is primarily stressed along the grain orientation, it is likely that the

ð15Þ

Fig. 3. Typical traction-separation response of a cohesive layer.

M. Khelifa et al. / Engineering Structures 101 (2015) 364–375

Beam (DCB) test, similar to that presented by Fortino et al. [33], whereas the cohesive parameters in pure mode II are identified based on the pure shear test, which is commonly used to evaluate the bond performance of adhesive joints as reported by Yashida et al. [34]. In the absence of experimental data for different modes I and II, the adhesive constants were chosen arbitrary in the simulations. From a qualitative point of view, the agreement between experimental and numerical values in terms of load–displacement results confirms that the modelling works well. More details about the identification procedure of adhesive properties can be found in references [35,36]. 3.4. Failure modes of timber beams in bending Timber beams in bending test were submitted to tensile stresses on one side of the neutral axis, whereas the other side of the beams’ neutral axis endured compressive stress. Although wood is weaker in compression than in tension, timber beams may fail according to various modes as reported elsewhere [37]. The failure mode is related to the physical characteristics of each individual beam. High-density timbers often fail in tensile zone, see Fig. 4b, whereas beams oriented across the grain fail in tension at an angle parallel to the angle of the cross grain, as shown in Fig. 4c. Splintering tensile failure occurs in dry wood, see Fig. 4d, and brash tension fracture may signify molecular level abnormality in timber, see Fig. 4e. Compressive failure can be observed in clear, low-density timber, see Fig. 4f, and horizontal shear failure near the neutral axis, as shown in Fig. 4g, often occurs in timbers that have sharp changes in growth zones. Strengthening flexural solid timber beams with other materials such as CFRP can influence the mechanical properties and can be used to control the mechanism of failure. 4. Numerical modelling A 3D FE model was used to perform a failure analysis and to predict the flexural behaviour of the rectangular solid timber beams strengthened by CFRP.

(a)

369

For reasons of symmetry in both geometry and loading, only one half of beam was discretised using a FE available in Abaqus [22], as shown in Fig. 5. Fig. 5a schematically illustrates the loading and the boundary conditions. An 8-node solid element with reduced integration (three degrees of freedom per node) was used to model the timber beams. The FE mesh of timber beam consisted of 3430 brick elements type C3D8R (Fig. 5b). Because the fibre-reinforced plastics are relatively thin compared to the timber beam, they were modelled by a 4-node shell element with reduced integration (six degrees of freedom per node). In the simulation model, the CFRP sheet was modelled with 500 shell elements type S4R (Fig. 5c). The CFRP shell elements were directly attached to the bottom surface of the timber beam, and the interaction between CFRP and timber was modelled using surface-based cohesive behaviour, as shown in Fig. 5. The loading was applied as an imposed displacement at a rate of 1 mm min1. The main material properties for wood are listed in Table 5, and the properties for CFRP and adhesive are given in Table 6. 5. Results and discussion 5.1. Equivalent stress distribution for CT beams without damage effect Fig. 6 shows the contour of the equivalent stress at different displacements. The CT beam shows a typical stress distribution of a homogeneous beam, where the neutral axis is clearly visible (Fig. 6a). At a displacement level of about w = 15 mm, the equivalent stress reaches a value of about r = 15.23 MPa (Fig. 6a). It can be noticed that the stress contours widen as the displacement increases. The CT reaches a maximum equivalent stress of r = 74.35 MPa at a maximum vertical displacement of about w = 107 mm (Fig. 6b). 5.2. Equivalent stress and damage distributions for CT beams Fig. 8a and b show the stress contours and the damage distributions, respectively, at the final displacement w = 102 mm. It can be noticed that the stress field is not uniform. The first damaged zone occurs in the tensioned fibres in the vicinity of the loading region. When the damage reaches the maximum value D = 1 (Fig. 7b), the equivalent stress tends towards zero (r = 0) in the cracked zones (Fig. 7a), and the finite elements corresponding to the damaged zones are removed from the numerical model as shown in Fig. 7. 5.3. Analysis at the global scale for CT beams

(b) (c) (d) (e) (f) (g) Fig. 4. (a) Bending test. Failure modes in timber beams subjected to flexion, caused by: (b) tensile, (c) cross-grain tensile, (d) splintering tensile, (e) brash tensile, (f) compressive and (g) horizontal shear.

The load–deflection curves obtained for CT beams from experiments and from FE analysis are shown in Fig. 8. The simulation was carried out using two different values of elastic modulus (E1 = 12 GPa and E1 = 14.5 GPa). The elastic parameters for wood material are unknown and the actual elastic response is expected to be orthotropic. To help choosing orthotropic elastic properties, it is well known that the linear part of the load–mid-span deflection curve is highly sensitive to elastic parameters. The good agreement found here indicates that the constitutive models using E1 = 14.5 GPa for wood can account for the fracture behaviour very well. The predicted results match closely the experimental ones. For an elastic modulus E1 = 14.5 GPa, the overall response is linear up to a displacement around 45 mm corresponding to a force of 58.7 kN. Then it becomes nonlinear up to a maximum displacement w = 102 mm corresponding to a maximum force of 92.7 kN (with damage) and of 109.8 kN (without damage). From a qualitative point of view, the predicted results obtained by the damaged

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M. Khelifa et al. / Engineering Structures 101 (2015) 364–375

Timber solid Load

CFRP shell element

Plane of symmetry

(a)

(b)

CFRP shell element

(c) Fig. 5. FE model of timber beams: (a) loading and the boundary conditions; (b) CT and (c) RT beams.

Table 5 Material properties for wood. Elasticity

E1 = 14.5 GPa E2 = 1.2 GPa E3 = 1.2 GPa

m12 = 0.37 m13 = 0.43 m23 = 0.45

G12 = 590 MPa G13 = 590 MPa G23 = 73 MPa

Plasticity

ryield = 70 MPa Q = 750 MPa b = 17

F = 0.35 G = 0.4 H = 0.6

L = 1.5 M = 1.5 N = 1.45

Damage

S = 0.8 MPa

s = 1.4

Table 6 Adhesive constants.

rI0;n ¼ 27 MPa rII0;t ¼ 22 MPa

Kn = 12740 N/mm3

dI;max ¼ 0:02 mm n

Kt = 2970 N/mm3

dII;max ¼ 0:21 mm t

model agree very well with the experimental CT#3 behaviour as shown in Fig. 8. Numerical predictions from the damaged-plasticity model, shown in Fig. 8, evidence the strong effect of the rupture-induced softening, giving a final fracture of the CT beam at the vertical displacement w = 102 mm (at a load of 92.7 kN). The computed ultimate load Fu = 98.3 kN at a displacement w = 90.4 mm is almost

identical to the experimental one, 95.9 kN (CT#3, Fig. 8). The numerical results in terms of load–displacement obtained for CT beams in bending by the damaged-plasticity model are very realistic. 5.4. Equivalent stress and damage distributions for RT beams with 2-layers CFRP The distribution of the equivalent stress and damage at a final vertical displacement w = 88.8 mm (load equal to 83.5 kN) for the RT beams with two layers of CFRP in bending test are shown in Fig. 9. The parts of the RT beams which are far from the CFRP have a different stress distribution compared to those of the CT beams. This indicates that the effect of strengthening is not local but affects the stress distribution of the beam as a whole. All simulations which were used in this work gave the same indication about this. The CFRP sheets are prone to higher stresses (r = 331.4 MPa) than the solid timber (r = 73.22 MPa). The wood material deforms plastically as shown in Fig. 9. 5.5. Analysis at the global scale for RT beams with 2-layers CFRP The comparison between numerical and experimental load–deflection curves is shown in Fig. 10. A good correlation can

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371

(a)

(b) Fig. 6. Equivalent stress [MPa] distributions for CT beams without damage at different displacements: (a) w = 15 mm and (b) w = 107 mm.

be observed between numerical and experimental results. An ultimate load of 92.7 kN is given by the FE model. Beams RT#4 and RT#5 failed at 91.6 kN and 113.2 kN, respectively. From 13 kN, the model gives a lower stiffness than the experimental one. However, the maximal difference between calculated and experimental ultimate loads is 1.2%. The FE model is thus able to predict correctly the ultimate load-bearing capacity of the RT beams in flexion.

of flexural CFRP-strengthening of solid timber beams in the literature, it is difficult to obtain better values of mechanical properties for each material composing the system (wood, adhesive and CFRP), and no comparison could be made with other published works from other authors. In our opinion, it is an identification problem, and it is the main reason suspected for the discrepancy.

5.6. Analysis at the global scale for RT beams with a 3-layers CFRP

A parametric study was carried out in this section to investigate the effect on beam strengthening of the number of CFRP layers and of the length of CFRP sheets. It should be noticed that the FE model applied here used the same material constitutive laws and assumptions as those described in the previous sections.

A discrepancy between predicted and experimental results can be seen in Fig. 11. For the model, the overall response is linear up to a displacement of 51.3 mm, corresponding to a force of 70.1 kN. Beyond this point, the response becomes nonlinear until the force reaches 94.85 kN at a displacement of 85.5 mm. At the same displacement, the RF beam (RT#6) gives a force of 109.9 kN, which is 36.2% higher than the experimental one. The observed experimental (RT#6) failure took place through shear at a displacement of 115.2 mm and a corresponding force of 111.2 kN. There may have various reasons for the observed discrepancy. First, there is a significant scattering of experimental results, as shown in Fig. 1, which is directly derived from the results of Borri et al. [1]. Second, in the absence of more experimental data

5.7. Parametric study

5.7.1. Effect of the number of CFRP layers The strengthening of RT beams with a single-layer CFRP sheet is now considered. Fig. 12 shows the effect of changing the number of layers on the overall behaviour of the different strengthened beam specimens. Table 7 summarises the results of the simulated specimens in terms of predicted ultimate load-bearing capacity and associated mid-span deflection. It is clear from Fig. 12 that increasing the number of layers tends to increase the overall capacity of the beam specimens. On the other hand, the changes of deflection

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(a)

Damaged zone

(b) Fig. 7. (a) Equivalent stress [MPa] and (b) damage distributions for CT beams.

120 100

Load [kN]

80 60 CT#1 CT#2 CT#3 Model (E1=12000MPa) without damage Model (E1=14500MPa) without damage Model (E1=14500MPa) with damage

40 20 0 0

20

40

60

80

100

120

Mid-span deflection [mm] Fig. 8. Load–mid-span deflection curves for CT beams: comparison between numerical and experimental data.

M. Khelifa et al. / Engineering Structures 101 (2015) 364–375

(a)

(b)

(c) Fig. 9. (a) Equivalent stress [MPa] in wood, (b) equivalent stress [MPa] in CFRP and (c) damage distributions for RT beams with two layers of CFRP.

373

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M. Khelifa et al. / Engineering Structures 101 (2015) 364–375 Table 7 Effect of varying the number of layers of CFRP sheet on reinforcement.

120 100

Load [kN]

80

FE model

Fult (kN)

wult (mm)

Fexp/mean (kN) (CT#1, CT#2, CT#3)

F num F exp F exp

1-layer 2-layers 3-layers

90.3 92.7 94.4

89.8 89.1 87.4

72.2

25.06 28.40 30.75

(%)

60

RT#4 RT#5 Model

40

100

20 80 0 20

40

60

80

100

Mid-span deflection [mm] Fig. 10. Load–mid-span deflection curves for RT beams with 2-layers CFRP.

Load [kN]

0

60

Model (LR=600mm) Model (LR=1200mm) Model (LR=2000mm)

40

20

120

0

100

0

Load [kN]

20

40

60

80

100

Mid-span deflection [mm]

80

Fig. 13. Load–mid-span deflection curves for different lengths of CFRP.

60

RF#6 RF#7 Model

40 20 0 0

20

40

60

80

100

120

Mid-span deflection [mm] Fig. 11. Load–mid-span deflection curves for three-layer RT beams.

Damaged zone

(a)

100

Load [kN]

80

60

Model (1 layer) Model (2 layers) Model (3 layers)

40

20

(b)

0 0

20

40

60

80

100

Mid-span deflection [mm] Fig. 12. Load–mid-span deflection curves for different numbers of CFRP layers.

response are negligibly small. It is also clear from Table 7 that all strengthened specimens in each group (with 1, 2 and 3 layers) showed an increase of load-bearing capacity, within the range of 25–31%, with respect to the control (unreinforced) specimens.

Fig. 14. Comparison between (a) simulated and (b) experimental failures [38].

5.7.2. Effect of length of CFRP sheets Three series were simulated with reinforcements of CFRP sheets using two layers, glued to the most stressed face of the beam, with distinct lengths (LR) of 600 mm, 1200 mm and 2000 mm. The simulation with the smallest reinforcement length (LR = 600 mm) shows a load deflection curve with a lower non-linearity compared

M. Khelifa et al. / Engineering Structures 101 (2015) 364–375

to the other simulations. For different values of LR, it is seen that the load–displacement curves are almost identical, and it may be noticed that the length of CFRP sheets does not affect the load–displacement response (Fig. 13). 5.8. Failure modes The failure modes which are observed on the CFRP-strengthened beams are different from those of classical CT beams. It should be noticed that the rupture of the strengthened wood beams always occurred due to the first crack of the solid timber in the tensile region. The adhesion between wood and composite materials often failed only after timber crack as reported elsewhere [1,38]. A comparison between numerical and experimental failures is shown in Fig. 14. It can be seen that the model predicts the damages in the tensile zone, and the experimental failure indeed took place in the same zone. The most common failure mode for the wood material is often initiated in the tensile zone and propagates to the top corresponding to the compressive zone. 6. Conclusions A 3D nonlinear FE model was developed to simulate the response in flexion of un-strengthened and strengthened timber beams with CFRP sheets. The proposed model was validated with respect to the experimental data from [1]. In addition, a parametric study was carried out. The parametric study varied the lengths as well as the number of the tensile CFRP layers. Based on the results obtained in the present work, the following conclusions could be drawn: 1. Advanced numerical methodology is relevant to simulate the flexural strengthening of solid timber beams with CFRP sheets. 2. The flexural behaviour of solid timber beams can be modelled through a local approach based on the coupling of orthotropic elasticity, Hill’s plasticity anisotropic quadratic criterion and cohesive model, using surface-based cohesive behaviour between CFRP and timber. 3. The overall comparison between the ultimate load-bearing capacity and mid-span deflection at failure showed good agreement between numerical and experimental results. 4. Compared to experimental data, the numerical results are realistic and predict very well the damaged zones. 5. The proposed numerical methodology might thus be used by engineers as a helpful numerical tool to analyse bending tests of timber beams. The developed FE model can be used to further explore the flexural behaviour and predict the performance of solid wood and glued laminated timber beams using near surface mounted (NSM) CFRP technique (or externally strengthened technique with CFRP sheets) with distinct lengths and different numbers of layers. Acknowledgements The authors gratefully acknowledge the financial support of the French National Research Agency (ANR) as part of the ‘‘Investissements d’Avenir’’ program (ANR-11-LABX-0002-01, Lab of Excellence ARBRE). References [1] Borri A, Corradi M, Grazini A. A method for flexural reinforcement of old wood beams with CFRP materials. Compos Part B-Eng 2005;36:143–53.

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