Research Paper
Determination of the influence of size and position of knots on load capacity and stress distribution in timber beams of Pinus sylvestris using finite element model

https://doi.org/10.1016/j.biosystemseng.2012.12.010Get rights and content

The finite element method was used to investigate the influence of size and position of cylindrical knots on load capacity considering the elastic–plastic constitutive law of Scots pine timber. A finite element model for a four points bending test was generated considering four different knot conditions in the beams: without knot; knot as a hole; live knot and spring contact between the knot and the beam. For knots placed in the compression side, the live-knot-model best simulates real behaviour; however, when located in the tension side, the hole-model was most reliable. The bending strength of the beam, including different sizes and positions of knots, were presented in simplified diagrams and compared with clear timber strength. The results showed the influence of knots and their local grain deviation on stress distribution. The model allowed the ranking of bending strength of the beams caused by knots as a combination of three quantified indexes: tension parallel and perpendicular to the grain and shear.

Highlights

► Numerical simulation of timber beams containing of knots and local grain deviation around them. ► Determination of influence of size and position of knots on load capacity in bending of timber beam. ► Influence of knots and their local grain deviation on stress distribution. ► Analysis of the influence of normal and shear stresses in the rupture criterion.

Introduction

The mechanical properties of timber are different depending on the direction of the grain, namely axial, radial and tangential, which are perpendicular to each other. In addition, the modulus of elasticity differs in tension and in compression. In tension there is only one linear phase, while in compression, there are three phases of behaviour. During the first phase of behaviour of a bending timber beam, the stress–strain curve in compression is linear and elastic. Once the yield stress in compression (fc,0,y) is reached, a curve is drawn in an elastic–plastic phase, before the ultimate stress in compression parallel to the grain (fc,0,u) occurs. Figure 1 shows the stress–strain curve for Scots pine from Spain and their associated values of ultimate strength and yield stress (Baño, Argüelles-Bustillo, Regueira, & Guaita, 2012).

Usually, the calculation of structural timber in elastic behaviour considers the following: 1) two directions for the mechanical properties: longitudinal (along the axis) and transversal; 2) an apparent modulus of elasticity in bending, taken as a combination of the modulus of elasticity in compression and in tension. To taking into account the curve of the elastic–plastic behaviour in compression and the difference between tension and compression, it is necessary to develop an appropriate constitutive model using the finite element method (FEM).

A finite element (FE) model showed the stress distribution through the height of the beam, assuming both the elastic and elastic–plastic behaviour of the material. It also presented the downward shift of the neutral axis as the compressed part of the beam becomes more plastic (Baño et al., 2012).

Timber beams usually contain defects, such as knots, fissures, wanes, deviation of grain and holes. The most relevant singularities are knots and associated grain deviations, which affect the strength and stiffness of timber beams (Kretschmann, 2010, pp. 5.26–5.28 and 5.1–5.3). These defects can be the cause of stress concentration around them. The placement of holes and their relative size were studied with respect to beam depth by Danielsson et Gustafsson (2010) for glulam beams with cylindrical holes. Williams, Fridley, Cofer, and Falk (2000) used FEM to study the stress concentrations around a hole drilled in a beam and demonstrated that when a beam presents a hole, the initial points of failure start at the stress concentrations points which develop around the top and bottom of the hole. However, the authors pointed out that these stresses are different to stress concentrations found around a natural knot in a beam which present local grain deviation and a relatively smooth flow of stress.

When a knot is located in the tension side of a beam the deviation of grain around it produces stresses of tension parallel and perpendicular to the grain (Argüelles, Arriaga, & Martínez, 2003). Studies on the consideration of the effect of knots and deviation of grain around them on the strength of timber beams have been carried out in experimental tests on Nordic spruce specimens by Mitsuhashi, Poussa, and Puttonen (2008). The combination of FE techniques and the theory of strength of materials have been used by many authors in order to predict the ultimate load capacity of solid sawn timber with knots and local grain deviation (Cramer, 1996; Itagaki, Mihashi, Ninomiya, Yoshida, & Esashi, 1999; Pellicane & Franco, 1994a; Pellicane & Franco, 1994b; Zandbergs & Smith, 1988). Nagai, Murata, and Nakano (2009) also used FEM to detect the location of the knots in beams at which failures initiate.

For cylindrical knots, perpendicular to the face, located in the tension side of a beam, a two-dimensional model considering the knot as a hole and the grain deviation around it was a good predictor of the rupture failure. A comparison of the results of the analytical model with experimental behaviour demonstrated that the FE model generated accurate results. These results were found for Scots pine (Pinus sylvestris L.) of 45 × 145 mm of cross section and 3000 mm long (Baño, Arriaga, Soilán, & Guaita, 2011).

The present work was carried out as a continuation of the work of Baño et al. (2011), using the validated FE model to design 165 virtual beams of sawn Scots pine timber. These beams contained live cylindrical knots of different diameters perpendicular to the face, which were located in several positions throughout the height of the beam. The main objective was to analyse the influence on bending strength (MOR) of the different sizes and positions of the knots and their associated grain deviation. The effect of the presence of knots in the stress distribution was studied and the values of normal and shear stresses in rupture were presented.

Section snippets

Definition of the beams and the mechanical properties of timber considered for the numerical simulation

A two dimensional (2D) numerical simulation of the four points bending test of a timber beam was made using the software ANSYS 12.0, Canonsburg, PA, USA. The geometry of the timber beams was created by rectangular areas and a 4-node quadrilateral elements plane stress (Plane 42) from the ANSYS library, using uniform meshing of 10 mm of size. Steel plates were modelled in the supports to avoid local stress concentrations. The timber beams were modelled with a cross section of 45 × 145 mm, length

Study of the variation of bending strength for different types of knots modelled

From the value of maximum load in bending, the modulus of rupture (MOR) was calculated, Eq. (3).MOR=a·Fmax2·Wwhere, Fmax is the maximum load of each beam (N); a is the distance from support to the nearest load point in a bending test (mm) and W is the section modulus (mm3).

Table 2 shows the percentage of bending strength for type, diameter and position of knots. As an example, Fig. 3 shows the results of MOR for the three models, considering a knot of 30 mm diameter. The vertical line

Conclusions

The numerical model predicted the bending strength of sawn timber beams, considering different positions and sizes of cylindrical knots perpendicular to the face of the beam and the local grain deviation around them. In addition, the model produced a graphic representation of the bending strength distribution for several diameters of knot located throughout the height of the beam.

The model for the prediction of bending strength of a beam with knots in the tension side, which considered the

Acknowledgements

The work was founded through Research Project AGL2009-11331 from Ministerio de Ciencia e Innovación entitled: “Methodological approach to the calculation and testing of joint types recently implemented in timber structures and the influence of knots on the load capacity”. We thank to A. Dieste and R. Lendrum for their helpful comments and the proofreading of the document.

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