1. Introduction
Wood is a popular building material for a wide range of lightweight structures. Further to its good mechanical properties and simplicity of manufacturing, its aesthetic appearance also makes it a favourable construction material.
The increase of loads or material deterioration in timber structures may sometimes be responsible for the destruction of existing buildings. However, it is often less costly and more time-effective to reinforce damaged wood pieces or structural members rather than replace them entirely. Carbon-fibre-reinforced polymers (CFRP) are currently strongly suggested for structural retrofitting. They have many benefits, including high elastic modulus, strength, corrosion resistance, durability, adequate bonding to wood, and low density. Compared with other types of fibres—e.g., glass or basalt—carbon fibre is more expensive to produce and tends to show brittle failure, though these drawbacks are well counterweighed by numerous advantages. It possesses high stiffness and strength, making it an appropriate alternative to steel in structural reinforcement or completely replacing it, e.g., high-strength steel bridge cables [
1]. Its good corrosion resistance makes it a suitable application in extreme environmental circumstances, such as high temperature or pressure [
2]. It performs better than fibreglass in low-velocity impact tests as well [
3].
Fibre-reinforced polymers are formed by embedding high-strength fibres (carbon, glass, etc.) in a matrix (epoxy, etc.) and can take a range of shapes and sizes (lamella, strip, rod, pultruded elements, etc.), as well as can be prepared on-site or prefabricated. Attachment to wood can usually be externally bonded, near-surface mounted, glued-in, etc. Due to its high modulus and strength, increases in stiffness, capacity, and ductility of reinforced timber structural elements is expected.
The prediction of mechanical behaviour is heavily affected by the uncertainty and variability of material properties of both timber and reinforcement. On the one hand, in the case of CFRP, the manufacturer specifies the properties of fibres. While factory-made composites are available with expected quality and specifications, in situ preparation of the reinforcement always involves uncertainty to a degree. On the other hand, timber quality widely varies due to its natural origin, which is further increased by factors such as moisture content, age, density, species, etc. Measured mechanical properties may show significant differences between samples even from the same species and are frequently only informative [
4,
5].
The adhesive’s performance directly influences the bond strength and effectiveness of the composite. It is crucial for achieving a robust CFRP-to-wood bond interface that can successfully transmit shear and normal stresses from the timber to the composite material [
6]. The bond strength can be affected by a variety of factors (e.g., the properties of wood [
7] and the moisture content [
8]). Nonetheless, regardless of the characteristics of the matrix, the failure of the timber element was identified as the primary failure mechanism in another investigation, showing that the qualities of the wood, rather than the adhesive, may affect the bonding behaviour [
9].
Several experimental studies were carried out to evaluate the effect of CFRP on the flexural behaviour of wood structural components focusing on stiffness, capacity, and ductility. With a good interface between the wood and the CFRP, a significant improvement (maximum of a 60% increase in capacity and a 30% increase in stiffness) was achieved [
10]. Two different reinforcement lengths were studied by [
11] in three-point bending tests finding that shorter reinforcements enhance stiffness by 23% and ultimate capacity by 12%, compared with longer ones that increase stiffness by 34% and load-bearing capacity by 28%. Sawn Norway spruce beams reinforced with varied quantities of CFRP were experimentally investigated by [
12], and they found a 30% improvement in load-bearing capacity and a nearly 16% increase in elastic stiffness. In a similar work, theoretical and experimental analysis on two distinct wood species retrofitted with different amounts of CFRP sheets was performed [
13], reporting an increase in load-bearing capability of at least 39%.
It was also shown that, in most cases, strengthening timber elements with CFRP had only a minor influence on stiffness [
14,
15,
16], and the increase is generally not more than 30%. Wood beams from an ancient bridge were strengthened using bidirectional carbon fabrics and laminate strips, resulting in a substantial improvement in shear and bending capacity (40–53%) but only a modest increase in stiffness (17–27%) [
17]. A research study employing GFRP dowel bars to strengthen wood stringers found a 70% improvement in ultimate capacity and ductility with a minor increase in elastic stiffness [
18]. However, the amount of fibres might be proportionate to the improvement in stiffness [
14]. FRP lamella was used for in situ reinforcing to restore and strengthen existing wood parts, reporting a 6% increase in bending stiffness and a 25% increase in flexural strength [
19]. A comparable study achieved a 58% increase in load capacity and a minor improvement in beam elastic stiffness [
20]. Another work also revealed a 21.5% rise in strength and a slight improvement in stiffness of 4.69% [
21]. A 10% increase in stiffness and a 44–60% increase in capacity were observed by [
22]. Moreover, the use of CFRP sheets for wood strengthening resulted in a stiffness increase of 15–60% [
23], and an increase of 20% in another study [
24]. The introduction of carbon plates and sheets leads to a stiffness increase varying from 20.2% to 29.6% [
25].
It is apparent that results are scattered in a wide range and sometimes contradict expectations. A review comparison between the measured stiffness increase and the theoretical values obtained from calculations using classical beam theory was shown in [
26] based on several research works processed. In some studies, the increase agreed with the theoretical estimates, e.g., [
17,
27,
28], while the results contradicted the theoretical expectations in many other studies, e.g., [
23,
29,
30].
Several numerical and analytical investigations addressed the question. They attributed the differences to an individual factor such as the boundary conditions, the adhesive stiffness, the FRP-to-wood ratio, or the wood material characteristics to resolve the disparities from a technical perspective. For instance, ref. [
31] observed a considerable error of 25% between the experimental and the numerical stiffness increase. Ref. [
11] restricted the issue to the boundary conditions and parametrically adjusted the penetration coefficient constant, introduced in the FEA software (ANSYS), between the timber and the support plates to anticipate the same experimental stiffness increase. This approach does not resolve the problem of having to set different values in each individual case. Ref. [
32] stated that as the fibre percentages increase, the stiffness of the material will most certainly increase. The adhesive materials were also considered the only factor responsible for the differences between the experimental and analytical/numerical applications [
33]. The authors suggested considering the cohesive stiffness of the adhesive in the numerical method. As for the wood, it was stated that the CFRP could improve the load-carrying capabilities of a timber beam with a low elastic bending modulus [
29].
All the methods mentioned above are strategies to resolve the problem from a technical point of view without explaining the cause of the problem, whereas this work investigates the underlying mechanical behaviour of the materials of CFRP-reinforced timber beams and gives an explanation as to the observed differences between measurements and analysis. For this purpose, a number of sawn Norway spruce beams were tested. Opposed to conventional testing programs, in this work, the bending stiffness properties and enhancement were not obtained as statistics on samples but each specimen was measured with and without unidirectional carbon-fibre fabric reinforcement attached to the tension side in order to obtain a series of specimen-specific data on mechanical properties. Analysis by both classical beam theory and three-dimensional finite element modelling was performed, also capable of predicting the non-linear behaviour by appropriate settings of material parameters. Recommendations concluded by this study on timber modelling enable more adequate assessment of reinforced timber beams in terms of stiffness, preventing possible overestimation of actual stiffness and failing to comply with design objectives.
The outline of the paper is as follows.
Section 2 presents the bending tests performed on the sample of beams together with the axial compression test used later for verification of the numerical analysis.
Section 3 elaborates on the methods used in this study, including the formulation for the classical beam theory for inhomogeneous beams and the finite element modelling applied as an independent control for the analysis.
Section 4 presents the results divided into appropriate parts, including the stiffness values, timber parallel-to-grain moduli of elasticity, observations on capacity and failure, and numerical simulations. The last section summarises the conclusions of this work.
3. Methods
3.1. Overview
Data obtained from the experiments form the basis of analytical and numerical investigations. The overall purpose is to evaluate the effect of the CFRP on the bending stiffness of timber beams to make an adequate assessment of the material properties and to provide an accurate prediction for the stiffness increase using analytical and numerical approaches. The individual steps are as follows.
Stiffness of all tested beams are determined by linear curve fitting to the linear sections of the load–deflection diagrams.
Considering the classical Euler beam model, equilibrium equations of a homogeneous cross-section (for a non-reinforced beam) and of an inhomogeneous (composite) cross-section (reinforced beam) are formulated in terms of geometric data and the modulus of elasticity for timber and fibre reinforcement. In each measurement pair (same timber beam with and without reinforcement), modulus for timber can be computed from the stiffness of the non-reinforced one, and then the modulus for reinforcement from the other once the modulus for timber is known.
As in several cases, the obtained modulus for reinforcement is beyond any acceptable tolerance for the factory data provided by the manufacturer, an improved model involving different moduli for tension and compression is introduced, and the equations are reformulated. With the assumption of the factory data for reinforcement, the timber moduli are computed from the equations.
In order to cross-check the validity of computations based on the Euler beam, a three-dimensional finite element model was constructed involving a fully orthotropic material model and non-linear analysis. The two models are compared for each beam.
The methods are detailed in the following.
3.2. Analytical Considerations
All load–deflection curves recorded in the bending tests can be evaluated to obtain bending stiffness characterising linearly elastic behaviour. In the case of loading in the linearly elastic range, the entire curves are used for linear fitting, whereas in the case of total loading (until failure) only the section between and of the ultimate load is considered, where linearity holds. Slope S of the line fitted with linear regression relates the increment of applied force F and of displacement e of the actuator as .
Considering the classical Euler beam model, some straightforward analytical derivation yields the stiffness of a homogeneous beam under the given test arrangement as , where E is the modulus of elasticity (customarily denoted by the acronym MOE), I is the moment of inertia related to the axis of bending, and L is the span. This allows the computation of the modulus of elasticity from the measurements. It is important to note that the modulus of elasticity is unquestionably the most determining of the material properties of timber beams in terms of displacements. The properties of natural materials vary in a large range, even within the same species. This implies that the calculation of the actual values of MOE for non-reinforced timber beams enables a more accurate evaluation of the same beams with reinforcement than the adoption of statistical averages from literature data or other samples.
Values of modulus of elasticity determined for specimens in group F in the non-reinforced stage are incorporated in the subsequent analysis of reinforcements. The increase in the bending stiffness of reinforced beams is attributed to the contribution of CFRP to the composite structure. Stresses in the inhomogeneous cross-section ought to satisfy conditions
where
N,
M, and
are the normal force, bending moment, and curvature in the cross-section, respectively;
and
refer to cross-sectional domains of wood and reinforcement, respectively;
and
are the moduli of elasticity of wood and reinforcement, respectively;
denotes the axial strain; and
z is the vertical coordinate of points of the domain in a suitable coordinate system. Strains
are to be expressed parametrically in terms of the curvature and the position of the neutral axis in order to solve the equation system for an effective modulus of elasticity
of the reinforcement, which is equivalent to the bending stiffness (
) obtained from the measurements.
The analytically predicted behaviour may differ from the observations, which is readily reflected in possible discrepancies between calculated and nominal moduli of the CFRP. Differences may be attributable to the effectiveness of the preparation of in situ reinforcement with accidental damages or imperfections introduced during the procedure or inaccurate assessment of timber material properties.
Several authors reported a low or moderate increase in stiffness resulting from FRP reinforcement. In a number of cases, the measured and predicted values significantly differ, as shown in a review article [
26]. Therefore, we propose to improve the modelling by considering different moduli for tension (
) and compression (
) in timber. (Note that different moduli were considered previously, e.g., by [
35,
36], for finite element simulations for knotted beams, as well as for the analysis of load-bearing capacities by [
37].)
By tension and compression zones distinguished in terms of material properties, the resulting inhomogeneous cross-section should satisfy conditions
where, further to the previously introduced notation, subscripts
t and
c refer to tension and compression in timber, respectively, and
E is the modulus of elasticity of an equivalent homogeneous cross-section. Tension and compression domains and strains are to be expressed parametrically with the position of the neutral axis. The solution of the equation system yields the formula
which relates the tension and compression moduli via MOE.
Stress resultants in the cross-section are to be modified to account for the different moduli such that in Equations (1a) and (1b), the wood domain is divided into tension and compression zones:
Assuming the modulus of the reinforcement to be nominal, the solution of the equation system (Equations (
3), (
4a), and (4b)) yields the wood moduli, which can constitute a more adequate representation of the composite system than the assumption of the homogeneous wood section. The obtained compression moduli can be compared with values obtained from experiments.
Mechanical behaviour beyond the linear range is dependent on the non-linearity of timber in compression. Considering the commonly used constitutive model consisting of linear elasticity and perfect plasticity (
Figure 3a), the stress–strain field in a cross-section can be reconstructed in terms of the curvature
of the axis line of the beam (
Figure 3b). The stress resultant yields the bending moment, establishing a bending moment–curvature relationship (
Figure 3c), which then enables the simulation of the load–deflection diagram (
Figure 3d) based on the applied beam configuration and model.
3.3. Finite Element Analysis
Finite element simulations were carried out to verify the analytical derivations based on the one-dimensional beam model. The three-dimensional FE models for both test groups (0 and F) were created in the general purpose software ANSYS (Canonsburg, PA, USA). The finite element mesh of a reinforced beam is shown in
Figure 4. Supports and loadings are defined at appropriate nodes indicated by turquoise triangles. Particularly, to create symmetric simple supports, nodes along the support lines are constrained vertically and along one of them also transversely. Longitudinal support is provided by constraining a node centred on the top face in the longitudinal direction.
In structural modelling, wood is considered a linearly elastic orthotropic material at low-stress levels. While it is typically assumed that elasticity is preserved until brittle failure in tension, some kind of non-linear behaviour is adopted beyond elastic limit stress or yield stress in compression. This can be perfect plasticity, multilinearity, higher-order stress–strain relationships, etc.
In this study, elastic behaviour is characterised by the general form of Hooke’s law with nine orthotropic constants in the material stiffness matrix. Moduli parallel to grain were taken from the computation of this work, while the rest of the constants were adopted from literature [
4,
5,
38,
39].
Non-linearity was effectuated by the generalised anisotropic Hill potential model incorporated in ANSYS, which allows the definition of distinct yield strengths in tension and compression and different behaviour in the principal material directions. An anisotropic work-hardening rule and an associated flow rule are included in the model. The material behaviour is characterised by the stress–strain curves in the three orthogonal and shear directions. For each direction, a bilinear response is considered. Plasticity is represented by zero tangent modulus at the compression yield stress and the absence of plasticity in tension by sufficiently large values for yield stress. The elastic moduli of the material are the same for compression and tension. In order to implement different values, the timber domain was divided into a tension part and a compression part such that the appropriate modulus is defined in each.
The composite reinforcement is a mixture of epoxy matrix and fibres forming a transversely isotropic material. Epoxy is a linearly elastic isotropic material with modulus
GPa and Poisson’s ratio
. The modulus of elasticity for the fibres is provided by the manufacturer. Assuming that the fabric has a maximum possible compactness at its nominal thickness (volume ratio
), the longitudinal (
) and transverse (
) moduli of the composite are obtained by the rule of the mixture and the inverse rule of mixture, respectively, as
where subscripts
f and
m refer to fibre and matrix, respectively. The shear modulus
and the Poisson’s ratio
are approximated with the rules as
and
Poisson’s ratio follows from the symmetry of the material matrices.
Three-dimensional solid elements (SOLID45) were employed to represent the wood material. The eight-node element has three translational degrees of freedom per node and is compatible with the defined Hill anisotropic model. A fine mesh was achieved by dividing the beam into 200 elements longitudinally (x-direction), and 12 and 6 elements in the other two directions (y and z, respectively). By constraining the displacements of respective nodes, conditions of the test arrangement were realised (simple supports at the bottom face and two lines of loads at the top).
Three-dimensional Layered Structural Solid Elements (SOLID185) were used to represent the unidirectional CFRP composites and epoxy. This element can be used to create multilayered thick shells or solids, and the anisotropic material characteristics are included in the element input data. The element needs to be associated with predefined shell sections for both the FRP and the epoxy with their specifications (thickness, material properties, and orientation). The CFRP is relatively thin; so, the mesh had to be generated carefully to comply with rules on aspect ratios and to avoid improperly shaped elements for adequate accuracy. The longitudinal lines within the volume representing the CFRP were divided into certain sub-lines. The vertical ones along the CFRP thickness were divided into two sub-lines and the mesh was later defined accordingly.
No constraint equations were defined to link the wood and CFRP components since both solid elements had the same degrees of freedom. Interface components and slide behaviour were not considered as an adequate bond between the wood and the CFRP was assumed.
5. Conclusions
Over the last few decades, several studies reported on the mechanical enhancement of timber by applying fibre-reinforced polymers. The increase in stiffness observed in experiments covered a wide range with a minimum near zero and often clearly contradicted expectations by analytical models. This work presented an experimental, analytical, and numerical study on timber beams reinforced with CFRP to provide better modelling for the explanation of the discrepancies. The experiments featured measurements of stiffness of beams under bending, with and without CFRP reinforcement of each producing specimen-specific data, which enabled the direct analysis of stiffness increase on an individual basis. The role of fibre-reinforced fabric in the analysis is significant because the composite structure enables the distinction between tension and compression moduli as opposed to a pure timber structure in which the moduli cannot be uniquely determined. The main findings of the work are summarised as follows:
The adequate in situ reinforcement of non-defected timber beams produced an increase in stiffness in the range to , even with a small amount ( to volume fraction) of fibre (disregarding the failed specimens). Generally, the beams exhibited lower stiffness than expected by analytical or numerical modelling, and the differences could not be attributed to faulty or imperfect reinforcement preparation.
Computations assuming different tension and compression moduli for timber showed in most cases that measured stiffness increase could be achieved using the nominal factory value for reinforcement modulus. Both the beam model and the finite element simulations using three-dimensional models showed good correspondence with the measured data with appropriately set compression yield stresses, indicating that the applied models are adequate for the description of the non-linear behaviour as well. The finite element simulations of the load–deflection curves resulted in an average error of in ultimate force and in compliance. The calculated compression moduli are verified by direct compression test results (mean values of GPa and GPa, respectively, with t-test showing statistical match at ). The yield stresses are also validated by the results of the axial compression tests (error of for the beam model and for the finite element model with respect to compression tests).
Statistical comparison of stiffness values in the reinforced group with the non-reinforced reference group showed that they come from the same sample. The average increases of in load-bearing capacity, in ultimate displacement, and in compliance were achieved. These indicate that both tension and compression (with plasticity) capacities of wood could be effectively exploited and that the reinforcing technique performed adequately (with the exception of one reported case).
The results lead to the conclusion that the differences observed between experiments and analytical considerations regarding the bending stiffness of reinforced timber beams are likely to be attributed to the use of a single modulus in timber for tension and compression, and the distinction enables adequate modelling complying with experiments.
Differences between simulated and experimentally obtained stiffness increase are in most cases in favour of the calculations, which would therefore overestimate the stiffness of the reinforced beams in a design process. This, in turn, would lead to non-compliance with the objectives on structural deflections at the expense of safety. Its effect is likely to rise if larger amounts of reinforcement (in volume ratio) are applied. Inclusion of this distinction into modelling could more efficiently describe the mechanical behaviour of timber with or without reinforcement, and enable designers to apply the right amount of reinforcement for their task.
The research can be continued with experimental investigation of beams with different dimensions and amount of reinforcement; however, attention needs to be paid to eliminate all factors that could invalidate calculations, e.g., imperfections of in situ reinforcement, insufficient bond contact if factory-product lamellae are used, defects of timber, etc.