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Analytical Modelling of Timber Beams Strengthened with Composite Materials

16 December, 2020 11 min reading
Based on the thesis by: Cyrus Hamed Hoseinpour, MSc, P.Eng, Civil & Structural Engineer, Infrastructure Specialist

Analytical Modelling of Timber Beams Strengthened with Composite Materials

The main aim of this thesis, by the author Hamed Hoseinpour, is to evaluate the experimental work of timber beams strengthened with composite materials with analytical modelling. 


The necessity of strengthening wooden structures

Wood has always been considered as a material for construction works, but during the time it needs to be improved by strengthening through other materials such as steel or FRP. Design of intervention of the timbers by the other materials needs to be studied in experimental and theoretical ways.


Historical buildings are encountered to the problems coming from degradation in mechanical properties of the material due to the time, especially for wood elements. Wooden structures are always subjected to damage by earthquakes and decaying by other environmental factors. Structural elements in the wooden buildings can be strengthened or replaced with other kinds of materials like concrete, steel, but based on the codes and charters; damaged members should be strengthened, repaired, rehabilitated or replaced with the same material.


Wood advantages (that makes it attractive to use in construction Industry):

  • A light material;
  • Good performance in both compression and tension;
  • Easy to work;
  • Aesthetic appearance;
  • Environmental friendly.


Wood disadvantages:

  • Unpredictable stiffness and strength in the same type of the wood due to the natural defects;
  • Slope of grains;
  • durability as after a while it should be replaced or reinforced with other materials like steel, concrete and recently composite materials to resist more for bending or shear loads.


Techniques of reinforcing timber structures

Steel and aluminium bars or plates were the traditional techniques of reinforcing timber structures which have their own disadvantages like high dead load and low compatibility. So reinforcing with fibre strips can be an alternative to replace the old ways. 

Use of FRP materials in the field of construction goes back to the beginning of 1990s when FRP was being used widely for the strengthening of concrete and from then on it was started to use FRP for masonry, wood and steel structures.


Recently natural fabrics (NFRPs) are readily available and more economical, which should be added to the other characters like not toxic or pollutant, recycling and less energy-consuming. The most important categories of natural composites (which can be found in the nature) are:

  • flax;
  • bamboo;
  • basalt fibres,
  • etc.


One problem in developing natural fibres is lack of natural resins with mechanical characteristics similar to artificial epoxy resins. In all carried out studies, epoxy resin is used as glue. 



Developing the use of FRPs in all aspects of the industry have made it low cost producing and easier to find than before. That is why nowadays it is coming to the wood industry more and more because of compatibility in terms of material which can improve wood mechanical properties, durability, reduce the effect of variability in wood, easy handling and high strength-to-weight ratio.


Most of the research carried on until now focused more on the strengthening of concrete and masonry structural elements and less than that for the timbers. Most of the studies in this regard have focused on the reinforcement of timber as a structural element by synthetic FRP, and some other works with natural fibres.


One of the first research on FRP using to improve mechanical properties of the structural members was done by, (Wangaard, 1964), with studying the elastic deflection of wood-fibreglass composite beams investigated with experimental and theoretical values. 


A considerable effort on the way of retrofitting structures in North America is around timber bridges which recently due to the advances in the strengthening of concrete by fibre composite materials, there are a lot of studies focusing on the strengthening of timbers through FRP materials. 


The mechanical behaviour of timber under shear and bending effects has been investigated by several researchers:

  • Through analytical works to study different failure mode of different reinforcement;
  • By bonding the external sheets on the elements to increase the strength and stiffness;
  • Taking to contribute the FRP strips with or without specific strengthening at the top;
  • By studied the effects of environmental condition and moisture;
  • By the application of bars and strips of CFRP, GFRP and AFRP; 
  • By reinforcing through steel fibres glued to the tension zone;
  • Retrofitting timber beams with CFRP (Carbon fibres) with different number of layers;


Experimental Campaign

A wide experimental work was carried out on the timber elements reinforced with different patterns of FRPs and matrix. Tests specimens reinforced by composite fibres were put under a four-point loading test which is a standard way based on the famous codes. In each case, the predominant failure mode was considered as an important factor in analyzing the tests specimens.


    1.Material characteristics

Timber specimens were made of Austrian Spruce timber (Picea abies) which was characterized as grade C35 according to (UNI EN ISO 527-5, 2009).

Composite materials were fibre strips from Carbon (CFRP) and flax (FFRP) fibres in various numbers of layers with two types of matrixes as Epoxy and Vinyl. The problem of durability of reinforcing fibres is more connected to the matrix. Epoxy has good resistance in degradation, including corrosion and it doesn't absorb water. The Vinyl matrix was applied only to the samples reinforced with natural fibres (FFRP(V)) which was the first time of application of this kind of material for this purpose.

Samples with flax fibres reinforcement made by one, three and five layers pattern in which the epoxy or vinyl matrixes were applied between all layers and also it was made sure that the matrixes fill the space between fibre yarns.


    2.Tests configuration


To measure the increase in the strength of the samples, it was decided to carry out four-point bending tests on 12 samples, including the ones which were used as the pilot test. The arrangement of the tests was one set for the pilot tests (without any reinforcement), CFRP tests with one layer of FRP strip and epoxy resin, NFRP tests with three and five layers of natural FRP strips, one set with epoxy resin and another set with vinyl resin. Two beams for each set were tested Fig. 1.

Fig. 1. Preparation of the specimens (Nardon, 2014)


Based on the configuration of the tests, the cross-section of the beam was (height x width) 135 x 115 mm2 and the length between 2,115 and 2,420. A close view of the different type of FRP attachment is provided in Fig. 2.

Fig. 2. Left, reinforcement by natural fibres with three and five layers and right, using CFRP in one layer (Nardon, 2014)


All tests were performed according to (UNI EN 408, 2010). A hydraulic jack and several displacement transducers (LVDT) were employed and installed symmetrically to measure displacement along with the specimen. To be sure about free rotation and horizontal displacement of the specimens, two steel cylinders were used while a steel plate was transferring the loads from timber to the cylinder. This was the case also for the load transferring from hydraulic jack to the specimen. A loading pattern as a four-cycle of loading and unloading was applied to the system Fig. 3.


Fig. 3. Experimental work configuration and schematic model (Nardon, 2014)

Reviewing the results of CFRP reinforcement, they had the best performance (46% respect to unreinforced condition) between the other types but, the results for the natural fibres were also notable as this values for them were about 31% and 24%, computed for 3 layers of FFRP and 5 layers of FFRP(V), respectively. In Fig. 4, a rough comparison of an increase in the capacity of each case can be found.

Fig. 4. Load–displacement curves (midspan measures) (Valluzzi, et al., 2015)


In all cases, the failure modes were brittle due to the collapse of the timber beam and rapture of the reinforcement which was coming after wood failure. For the beams reinforced with FFRP(V) ductility increased a lot and showed better performance in terms of structural behaviour. Because of the brittle nature of the woods, most of the beams fails suddenly without any precaution before failure and it was started from a defect (e.g. knot) in the maximum bending moment zone. After starting the cracks, it was propagated along the timber grain toward the supports.


Following, the failure mode of some cases is shown, Fig. 5.

Fig. 5. (a) Wood fracture in tension and compression zone (NR-02) (Nardon, 2014)


Fig. 5. (e) Rapture of the FRP because of tensile collapse in wood (FFRP-EP-3-01) (Nardon, 2014)



Fig. 5. (f) Rapture of the FRP because the tensile collapse in wood (FFRP-EP-5-01) (Nardon, 2014)


Analysis Methodology

Considering the configuration of the experimental bending tests on the timber specimens, mentioned in the previous section, an analytical model based on the well-known beam theories so-called Bernoulli and Timoshenko, was developed and calibrated with the results coming from pilot samples and then it was used to anticipate the behaviour of the reinforced samples with a different type of used FRP, matrix and different patterns.


  • Material properties

To develop the analytical model, one needs to know about the properties of the employed materials in the tests, which in this case, is coming from the direct tests or literature and technical datasheets.

For wood material, all these values are for the starting step of modelling. In the next step, they will have resulted from the calibration process.


For FRP material, these values will be used until the end of the analysis process. 

Keeping in mind the truth that the materials working as reinforcement are a combination of FRP strips and matrix, it should be noticed that all the intended geometrical and mechanical characteristics brought in Table 1 are for the combined material and the part for wood will be used as the initial estimation of the mechanical parameters to start the calibration process. 



  • Constitutive models

Behaviour in tension: One of the most common models used for wood in tension is linear behaviour up to failure so stress and strain can easily result once having another one assuming a constant value for E (modulus of elasticity). This model was taken into account in this study.


Behaviour in compression: Based on the stress-strain diagram taken from the uniaxial tests, it is assumed that there are two behavioural parts. First linear behaviour up to a strain so-called yielding strain (elastic behaviour) where from that point on there is no increase in the stress and it remains constant until a failure happens which makes the second part (plastic behaviour). This is a model which in this study was considered as the constitutive model in compression, which was presented by Buchanan, Fig. 7.

Fig. 7. Constitutive model employed in this study



In the current study since the behaviour of the timber right before failure in the tensile area of the wood is considered, which is also the failure criteria of the system, there is a continuous stress distribution along with the height of cross-section and, actually, we know the height of compressive plastic part after yielding. So instead of estimating the height of the plastic part, it was tried to apply this coefficient to the width of the stress block of the plastic part as shown in Fig. 9.


Fig. 9. Amplification of the stress distribution on the plastic compressive area


Analytical modelling procedure 

The procedure of the analytical modelling is importing the initial parameters like beam dimensions, modulus of elasticity, strains… as the input values and taking out displacement along the beam as output.


The programming was done in the MATLAB code.


After giving the input parameters, a built-in function was taken to use for calculation of the place of natural axis and then moment-curvature diagram for a specific pattern of reinforcement of the cross-section.


Having the moment-curvature diagram and calculation the moment at each section of the beam by simple static rules, make it possible to put a moment of each section also deriving the corresponding curvature. It results in a curvature diagram along the beam and based on the knowledge from static and mechanics of materials by making two times integration on the curvature the diagram of the rotation and displacement can result respectively.


The assumptions in this model to calculate displacement all over the beam are the ones usually used in the beam theories.

It is assumed that strain variation in the height of the cross-section remains linear.

Both Bernoulli and Timoshenko theories were taken into account. So in each case, the assumptions of each one were considered.

  • Bernoulli: Governing equations of partial differential equations, all loading acts in the plane of symmetry of the cross-section, shear deformation of the beam is negligible: plane sections initially normal to the beam axis remain planar and normal to the axis after deformation, the shape of the cross-section remains the same after deformation and displacement is infinitesimal.
  • Timoshenko's theory: extension of Bernoulli-Euler theory and taking into account the effect of shear deformation, line segments perpendicular with the centre line in the undeformed configuration move only perpendicularly to the axis and undergo no rotation.



In conclusion, it is worth mentioning that the quality of doing the tests was high, regarding the reasonable values in deflection and good agreement between results of the theoretical modelling and experimental tests. The average value of the error was around 15 per cent which shows the precise of the performed tests.


The results of this experimental and theoretical work show that tensile strength of the wood is not enough to use the maximum capacity of the timber and non-reinforcement specimens fail mostly in the elastic area or at the beginning of the plastic area, but reinforcing of the timber in the tensile area by composite fibres is an excellent way to reach the plastic behaviour of the wood and utilize maximum capacity.


Considering the shape of deflection in the theoretical graphs, displacement of the point in the middle is more match compared to the points located on the beam sides. In theory, this area of the beam show more plastic behaviour and that can be a reason.


The results of both Bernoulli and Timoshenko theories, in this case, are the same with a smaller difference. That shows the minor effect of the shear considering the test's configuration. The most important reason is the small dimension of the cross-section as with increasing the height, results coming from Timoshenko theory are different than the Bernoulli's ones and show more displacement.


Considering the good agreement of the results between calibrated model and the other tests with different reinforcement it shows for both of the cases (CFRP and FFRP) a unique design procedure can be used during calculation for intervention.


The other conclusion coming from an agreement of the analytical and experimental data, considering the selected wood properties in the calibration process which were the only parameters derived in this step, prove this truth that properties of the combination of the FRP and matrix which were taken from the direct tests, had high accuracy.


The main result of this study was the achievement of an equation to estimate wood plastic stress distribution by a rectangular stress block, which is unique for this kind of wood and can be used in future studies. 


Further work

Since there were only two pilot tests as non-reinforced beams in which one of them was stopped before reaching failure, calibration of the model only with the other one did not show good agreement with the results of the other tests. So it was decided to make calibration with the specimen reinforced with CFRP. This can be taken into notice for the future works that the number of pilot tests should be enough to analyze either for comparing the results of the other tests with a trustable base result.


Also, the achievement of an equation to estimate wood plastic stress distribution by a rectangular stress block, which is unique for this kind of wood and can be used in future studies. 



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