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A finite element model is developed to analyse, as a function of volume fraction, the effects of reinforcement geometry and arrangement within a timber beam. The model is directly validated against experimental equivalents and found to never be mismatched by more than 8% in respect to yield strength predictions. Yield strength increases linearly as a function of increasing reinforcement volume fraction, while the flexural modulus follows more closely a power law regression fit. Reinforcement geometry and location of reinforcement are found to impact both the flexural properties of timbersteel composite beams and the changes due to an increase in volume fraction.
Timber beams used in buildings that are subject to flexural deformation can benefit from the application of internal reinforcements such as flitch plates, and rectangular or solid cylinder rods inserted into routed groves. Such rods are used for repairing timber beams but also have significant potential for upgrading fast grown, low quality softwoods. Mild steel is most commonly used as good strength and stiffness properties can be obtained at much lower prices than fibre reinforced plastic (FRP) equivalents based on glass and carbon fibre.
Reinforcements are often applied to enhance the load bearing capabilities of structural timber. Flitch beams are a classic example of a steel reinforced timber beam which has been researched in detail during the early half of the 1970 [
Timber structures are normally designed so as to remain well within their elastic capabilities. Occasionally however, timber structures are underdesigned and the onset of plastic yielding may endanger the lifetime of the structure, which is in turn a considerable safety hazard. A theoretical determination of the proportional limit is therefore paramount and is possible using an elasticplastic analysis method. The objectives herein are to develop a nonlinear finite element model to simulate the yield strength of four different reinforcement configurations, and to use the model to determine which reinforcement configuration is most effective in resisting load as a function of volume fraction. The reinforcing geometries and composites modelled are commonly used for repairing damaged structures [
The models were developed in ANSYS 5.7. The validity of the model predictions were tested against experimental samples tested to destruction. This section begins by briefly describing the experimental series against which model predictions were compared.
Composite beam reinforcement configurations for series I, II, III and IV.
Composite beams were subjected to loading under conditions of fourpoint bending at a crosshead rate of 2 mmmin^{−1}. The beams were tested under service class 2 conditions [
Schematic of fourpoint bending test set up.
Reinforcing timber is a means by which low grade timbers can be upgraded, or, by which timber structures can be repaired [
Series IIV beams were constructed according to the cross sectional dimensions shown in
Node sharing contact between the beam components was specified for each beam. This assumed a theoretically perfect interface, which is infinitely thin and inseparable, even beyond the maximum bond stress capabilities of the adhesivesteel or adhesiveLVL interface. This approximation is acceptable provided there is no debonding between any of the adhesive interfaces throughout the duration of monitored loading (
Series IIV beam components were discretised using 1st order threedimensional solid elements. Each element comprises eight nodes and three degrees of freedom exist at each node in the directions perpendicular and parallel to the LVL grain axis, which coincide with the orthogonal Cartesian axes x, y and z. The element edge lengths were defined according to the geometric requirements of each component within each model. The final mesh patterns for the cross sections of series IIV composites are shown in
Cross sectional dimensions and layout used in finite element models for beams from series IIV (labelled). In each series, the laminated veneer lumber (LVL) is textured brown, the adhesive is black and the steel reinforcements are shaded grey.
Mesh patterns of cross sections for composite beams in series IIV.
Stressstrain characteristics of steel are adequately represented using bilinear plots where the onset of yield is defined and work hardening is taken to 15% strain. The steel is assumed to be an isotropic solid with a density of 7900 kgm^{−3}, a shear modulus of 83 GPa, an elastic modulus of 210 GPa and a Poisson’s ratio of 0.27. Work hardening in the steel begins at the critical vonMises stress,
The tensile and compressive characteristics of Rotafix CB10TSS adhesive were approximated to elasticperfectlyplastic systems. Tests were performed on clear samples of CB10TSS. Tensile tests were performed on three samples of CB10TSS, which had a thickness of 4 mm and a width of 10 mm. An extensometer was used to measure the strain as a function of loading. The values for the compressive properties of CB10TSS adhesive were provided by the manufacturer, Rotafix Ltd. The elastic moduli of CB10TSS adhesive are the same in tension and compression, however, the compressive strength of this adhesive is superior to its strength in tension.
Elastic rigidplastic stressstrain relationships for the adhesive in tension and compression.
The LVL was modelled as an orthotropic elastic–anisotropic plastic material with a density of 520 kgm^{−3}. The elastic constants are provided in
Elastic properties of Kerto S laminated veneer lumber.
Property  Direction or plane  

x  y  z  xy  yz  xz  

12.75  0.255  0.255       

      0.62  0.62  0.62 

      0.03  0.29  0.03 
Normal yield stresses in compression, σ_{cy}, were defined in the orthotropic principal axes of the LVL, which were taken to coincide with the local orthogonal Cartesian axes, the
It is incorrect to specify solely the nonlinear compressive characteristics of LVL for beams subjected to flexure. Although such an approximation may be valid for compression dominated failure, there will certainly be the development of high tensile stresses on the underside of a bending beam below the neutral axis. Plastic incompressibility restricts the use of both tensile and compressive characteristics in the same model. The problem is overcome by linearly interpolating between the data predictions of two models, one possessing the tensile characteristics of LVL and the other with the compressive characteristics. Given that there is symmetry about the axis of neutrality prior to loading, it is plausible to assume that exactly half the beam experiences tension, whilst the other half is subjected to compression. This approximation essentially violates true beam behaviour whereby the neutral axis moves closer to the tensile edge of the beam at the onset of compressive yielding. It is however an easily applicable approximation that minimises the errors associated with leaving out the characteristics of tensile failure altogether.
The normal compressive and shear yield stresses as well as the gradients describing the 1st order (linearly) proportional plastic stress against plastic strain were determined through experimental compression and shear block testing of LVL specimens with dimensions of (50 mm)^{3}. These tests were conducted at a displacement rate of 2 mmmin^{−1}. The tangent moduli for the compression block tests were measured at a 0.2% proof stress. A strain of 0.2% was chosen to ensure that the slope of the tangent corresponded with the slope of the linear plastic stressstrain gradient. The bilinear stressstrain plots for the shear planes xy, yz and xz followed the appropriate shear modulus in each plane until reaching the peak shear stress of the material, after which the plastic slope supposed a zero tangent modulus.
Stressstrain approximations and related values for LVL in each principal axis.
Bilinear stressstrain approximations and related values for LVL in xy, yz and xz shear planes.
The model was symmetrised about ½l and ½b, where l is the span and b is the width of an entire composite beam such as was tested in series IIV. Displacement controlled loading was applied via nodes attached to a line across the upper edge of the LVL segments for all beams (series IIV). The line was located at a distance of 300 mm from the centre point of the beam. Force would be transferred therefore, through the LVL to the reinforcement via the adhesive. Nodal translations at the lower edge of the beam end were restricted in every degree of freedom along the Cartesian axes at a position on the lower edge of the beam at a distance of 900 mm along the span from the centre point of the beam. Rotational freedom was left unrestricted.
Computer simulations were carried out for five different reinforcement volume fractions to predict the flexural yield strength and stiffness properties of series IIV composite beams. The width of series IIII steel plates and the diameter of series IV steel rods were adjusted to yield reinforcement volume fractions of 1%, 2%, 2.8%, 4% and 5%. The fraction of 2.8% was used as opposed to 3% since the full depth flitch beam had this fraction in the original simulations/experiments and it was deemed more efficient to bring reinforcements from the other series’ to this fraction.
Finite element predictions for series IIV composite beams are compared with the experimental loaddisplacement characteristics (grey lines) for series IIV and are shown in
Loaddisplacement curves for series I tested steelreinforced LVL samples (in grey) and estimated finite element predictions (black dots). A comparison is made between the predicted and experimental proportional limits.
Loaddisplacement curves for series II tested steelreinforced LVL samples (in grey) and estimated finite element predictions (black dots). A comparison is made between the predicted and experimental proportional limits.
Loaddisplacement curves for series III tested steelreinforced LVL samples (in grey) and estimated finite element predictions (black dots). A comparison is made between the predicted and experimental proportional limits.
Loaddisplacement curves for series IV tested steelreinforced LVL samples (in grey) and estimated finite element predictions (black dots). A comparison is made between the predicted and experimental proportional limits.
On examining the tables in
A sound estimate of the linear tangent and the force at yield is vital when designing timber structures. The aim of design is to remain within safe limits and to avoid loading above composite yield. Theoretical studies of reinforced timber composites have employed moment relationships based upon the failure modes of composite beams to predict the ultimate failure loads [
Having ascertained that the model predictions are close to the experimental equivalents, the next stage was to compare the effects of individual reinforcing geometries as a function of volume fraction, on the yield strength and stiffness properties of the composite beams.
The trends between reinforcement series IIV are the same for the flexural modulus and the yield stress. Series II yield stress nevertheless exhibit slight deviation from those general trends observed for series I, III and IV. Here it can be noted from the coefficients of determination that there is greater scatter in the series II simulations than in any of the others, which gives rise to the intersection between series II and III regression fits. This anomaly is most likely an effect of using an insufficient mesh density in series II models. The predicted flexural modulus values fit well to a nonlinear trend and confirm the experimental findings of [
The yield strain of mild steel is approximately 0.0015. Increased resistance to flexure as a function of load is possible by increasing the volume fraction of reinforcement or indeed, by employing different reinforcement geometries. It is nevertheless plausible to suggest that once the yield strain of the steel reinforcement is surpassed, the magnitude of loading will no longer increase as dramatically as it does within the Hookean region of elasticity. This is evidenced when the magnitude of difference within each series as a function of volume reinforcement fraction is compared for the elastic modulus against the yield strength. The different reinforcement geometries used in series IIV composite beams will resist higher loads to achieve a strain of 0.0015 as a result of the effectiveness of their location within the LVL, as a consequence of their local geometries and as a function of increasing volume fraction.
Finite element model predictions for the flexural modulus plotted as a function of the reinforcement volume fraction for series IIV steelLVL composite beams.
Finite element model predictions of the yield strength plotted as a function of the reinforcement volume fraction for series IIV steelLVL composite beams.
An assessment of the load bearing benefits of reinforcement geometry and location within flexural timber has been possible by developing a nonlinear finite element model. The model has been used to predict the flexural modulus and yield strength of steelLVL composites retaining equal volume fractions of reinforcement. These models have been verified by comparison to experimental equivalents and show a maximum error of 8% for yield strength predictions. Model predictions show that for volume fractions of steel reinforcement between 1% and 5%, the flexural modulus of series IIV beams can fit a power law trend relatively well, while the yield strength of series IIV beams can be represented linearly. Model predictions suggest that the series III reinforcement configuration (small vertical plates located at the outermost fibres) give the greatest stiffness enhancement for LVL and produce the highest yield strengths out of the four reinforcement configurations (series IIV).
In the design of flexion resisting members in buildings, the mechanical stability of a beam is often weighed against material cost and more recently, environmental impact. Moreover, in view of aesthetical impact, reinforcements may be positioned internally to a structural member, rather than externally. Rectangular section reinforcements located close to the outermost fibres of a beam in flexure have the greatest benefit in view of yield strength and stiffness improvement and moreover, give rise to reduced material wastage and economic loss. The yield strength is a most important design parameter as it indicates the onset of building failure and there is benefit therefore, not only in maximising this parameter, but also in predicting its value. The present study shows how this parameter can be accurately predicted and indeed, how reinforcement geometry is a significant design consideration for flexural beams.