# Bending Stiffness, Load-Bearing Capacity and Flexural Rigidity of Slender Hybrid Wood-Based Beams

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}K, whereas thermal transmittance of the glazing is almost half this value and varies from 0.5–0.8 W/m

^{2}K [1]. To improve thermal transmittance of the window we need to either improve thermal transmittance of the frame or increase the surface of the glazing, compared to the whole window area. The architectural trends prefer the open view through windows, with as little visible window frame as possible. Therefore, for this study small cross sectioned (i.e., slender) profiles were analysed, to explore the limits of improving their bending stiffness and strength, when being reinforced with stiffer materials.

## 2. Materials and Methods

^{3}. The mechanical characteristics of the spruce, that correspond to structural grade C30 [23], used in window production, compared to those of the reinforcing materials, are presented in Table 1. The shear strength was determined experimentally, according to standard EN 205 [24].

- The moment of inertia of the cross section should be maximized by proper orientation of the reinforcements and the provision of a sufficient number of them.
- An aesthetic requirement—no reinforcement should be visible when the window is completed.
- The dimensions of the reinforcements should correspond to those of standard products available on the market, in order to decrease production costs.
- The production feasibility aspect—the basic thickness of wooden lamellas, to produce window scantlings, is 24 mm.
- Cost-effectiveness of the hybrid beams.

_{e}) = 84 mm × 68 mm × 1950 mm. Basic wooden beams for specimens were manufactured from lamellas with thicknesses from 8 to 28 mm. The type A specimens were unreinforced (i.e., reference specimens). The type B specimens had two horizontal reinforcements (with width × height dimensions of 80 mm × 4 mm) placed in the pre-milled grooves of the contact regions between the two neighbouring wooden lamellas. The type C specimens had twelve vertical reinforcements (with width × height dimensions of 3 mm × 20 mm), which were located in each of the two outside wooden lamellas (six in each lamella). Type D specimens were prepared with two vertical reinforcements with dimensions of 68 × 4 mm. All the pre-milled grooves were in width and height 1 mm bigger than reinforcements, leaving 0.5 mm space for the adhesive around the reinforcements. The adhesive was applied to the grooves and the reinforcements were then manually inserted into them. Specimens C had the highest number of reinforcements with its highest area (12.6%), while specimens D had smallest number of reinforcements with the smallest area (9.5%) (Table 3).

_{1}was calculated as the average value of the deflections measured at the midspan by the IND 1 and IND 2. All the physical properties (i.e., the displacements, the strains, and the load P) were measured and recorded by a Dewesoft DEWE 2500 data acquisition system (DEWESoft d.o.o., Trbovlje, Slovenija).

_{eff}was calculated on the basis of the average absolute midspan deflection w

_{1}, using an expression of technical mechanics (Equation (1)) that corresponds to the four-point bending test load arrangement. The equation neglects a small part of deflections caused by shear stresses.

_{eff}= 0.33 P

_{u}·a·(3L

^{2}− 4a

^{2})/48·w

_{1}

_{u}) is the load where the failure occurred, a is the distance between the support and the load, L is the span length and w

_{1}is the absolute midspan deflection.

## 3. Results

_{u}); at the limit of proportionality (P

_{p}), which represents the end of the region of linear behaviour; and at the ultimate load (P

_{u}) (Table 4). Beside the corresponding load, the deflection occurring at midspan (w

_{1,i}) and the bending stiffness (k

_{1,i}) are given at each loading step (i = 1/3 for the loading step 0.33 P

_{u}, i = p for the loading step P

_{p}and i = u for the loading step P

_{u}). The bending stiffness was calculated as k

_{1,i}= P

_{i}/w

_{1},

_{i}. The flexural rigidity was calculated from the displacement w

_{1}, and was the highest at C specimens and the lowest at A specimens (Table 4).

_{u}) and bending stiffness (k

_{i}) were the highest in the case of the type C beam specimens, which had the highest percentage of reinforcements, whereas it was the lowest, as expected, in the case of the type A beam specimens, which had no reinforcement (Figure 7). The comparisons of the bending stiffnesses k

_{1,p}and k

_{1,u}show the plasticity capacity of the tested beams. In the case of the type B beam specimens, both parameters have almost equal values, which corresponds to an almost linear load-deflection relationship up to failure (Figure 7). The biggest difference between these two parameters (i.e., 34%) occurred in the case of the beams D1–D5, which indicates their nonlinear behaviour between the limit of the proportionality and the ultimate load.

_{1}—calculated as the mean values of IND 1 and 2) of the beam specimens of types A, B, and C, of Series S1, were 32.2 mm, 21.5 mm, and 19.8 mm, respectively (Figure 8). The bending stiffness improvements of the beam specimens of types B and C when compared to the type A reference specimens, were 33% and 39%, respectively. The average midspan deflections of the Series S2 beams of types A, B, C and D amounted to 42.0 mm, 30.1 mm, 30.0 mm, and 30.6 mm, respectively. The improvement in bending stiffness of the hybrid beams of types B, C, and D, compared to that of type A beams, amounted to 28%, 29%, and 27%, respectively. In both series of tests, the midspan deflection of the reinforced hybrid beams was decreased compared to the reference specimens of type A. The largest standard deviation of the midspan deflection was observed with the type A beams, in the case of both series. This was expected, since these specimens were made solely of wood, which is a non-homogeneous material, with significant variations in density and different grain orientations.

_{eff}was calculated on the basis of the average absolute midspan deflection w

_{1}, using a well-known expression of technical mechanics (Equation (1)) that corresponds to the four-point bending test load arrangement. The equation neglects a small part of deflections caused by shear stresses.

_{0}) and the shear modulus (G) of wood (i.e., E

_{0}/G = 16, Table 1) and a relatively high span and specimen height (L/h) ratio (which amounts to L/h = 25.7 and L/h = 27.2 for Series S1 and Series S2, respectively), the deflection caused by shear stresses should not be neglected. Kretschmann [30] suggested that the flexural rigidity should be increased by 10% if the deflection due to the shear stresses is neglected, as it is in the case of Equation (1). The results of a study reported by Eierle and Bös [31] confirmed, that the shear deflection depends on the length and height ratio (i.e., L/h) of the beam, and on the ratio between the shear and elastic moduli (i.e., G/E

_{0}). The smaller this ratio is, the larger is the effect of shear stresses on the deflection. At rectangular shaped simply supported beams with L/h ratio above 25, the shear deflection should not represent more than 2% of the total deflection.

_{1}) was measured in dependence of the load (P), for all beam types of Series S1 and Series S2 (Figure 12 and Figure 13). The angle of the linear part of the curves from the abscissa is defining the bending stiffness; the higher the angle the higher the bending stiffness. Load-bearing capacity is defined by the load at which the curve ends from the exception of specimens marked with * sign, which are specimens, where the full range of at least one LVDT was reached, before the failure occurred.

_{u}(Series S2) for only 12% compared to ultimate load of specimens of Series S1. The reason lies in a prevailing failure mechanism, which was with Series S2 specimens a typical tensile failure.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Schematic diagram of the four-point bending test field configuration for the specimens of Series S1.

**Figure 4.**Schematic diagram of the four-point bending test field configuration for the specimens of Series S2.

**Figure 7.**Ultimate load-bearing capacity and stiffness coefficient of the tested specimens of types A, B, C, and D.

**Figure 8.**Average midspan deflection (w

_{1}) of the beam specimens A, B, C, D at the load P = 10 kN and the bending stiffness improvement (%) of the beams of types B, C, D compared to the type A beams.

**Figure 9.**Average load-bearing capacity (P

_{u}) of the beams of type A, B, C, and D, and the improvement in this capacity of the beams of types B, C, and D compared to the type A beams.

**Figure 12.**Load-midspan deflection relationship of Series S1. * B4 was excluded from further analysis due to the change in cross section dimension.

**Figure 15.**Example of window profile with horizontally and vertically positioned reinforcements (measurements in mm).

Norway Spruce | Aluminium | GFRP | CFRP | |
---|---|---|---|---|

Density ρ (kg/m^{3}) | 460 | 2700 | 2000 | 1420 |

Poisson’s ratio ν | 0.37 | 0.30 | 0.27 | 0.3 |

Modulus of elasticity E_{0} (MPa) | 12,000 | 70,000 | 38,000 | 120,000 |

Shear modulus G (MPa) | 750 | 25,000 | 5000 | 4000 |

Tensile strength f_{t} (MPa) | 19 | 195 | / | / |

Compressive strength f_{c} (MPa) | 24 | / | / | / |

Shear strength f_{v} (MPa) | 8.4 * | / | / | / |

Thermal conductivity λ (W/mK) | 0.11 | 210 | 0.3 | 5 |

Price factors compared to wood (per m^{3}) | 1 | 28 | 71 | 180 |

Brand Name | Type | Mean Shear Strength (MPa) | Standard Deviation of Shear Strength (MPa) | Technical Sheet Shear Strength (MPa) |
---|---|---|---|---|

Körapox 565/GB | 2K-EP | 9.9 | 0.96 | 24 |

Permabond ET515 | 2K-EP | 4.4 | 0.53 | 10 |

COSMO EP-200.110 | 2K-EP | 6.1 | 0.75 | 18 |

Körapur 790/30 | 2K-PU | 8.9 | 1.13 | 18 |

Novasil P-SP 6944 | 2K-PU | 4.0 | 0.96 | / |

COSMO PU-200.280 | 2K-PU | 11.2 | 0.75 | 18 |

Specimen | Number of Specimens (-) | Series | Dimensions of Reinforcement b × h (mm) h | Number of Reinforcements (-) | A_{real} Percentage of Reinforcement (%) |
---|---|---|---|---|---|

A1–A5 | 5 | S1 | - | - | - |

* B1–B5 | 4 | S1 | 80 × 4 | 2 | 11.2 |

C1–C5 | 5 | S1 | 3 × 20 | 12 | 12.6 |

A6–A10 | 5 | S2 | - | - | - |

B6–B10 | 5 | S2 | 80 × 4 | 2 | 11.2 |

C6–C10 | 5 | S2 | 3 × 20 | 12 | 12.6 |

D1–D5 | 5 | S2 | 4 × 68 | 2 | 9.5 |

**Table 4.**Summary of the experimental results of the specimens of both series of tests (i.e., S1 and S2). The loads (P

_{i}) and midspan deflections (w

_{1,i}) are presented as averaged measured values of all specimens, of the same type and series, and the bending stiffness was calculated as k

_{i}= P

_{i}/w

_{i}. In the last six columns comparisons of the different loads (P

_{i}), the midspan deflections (w

_{1,i}), the bending stiffnesses, and flexural rigidity are presented.

At 33% of Ultimate Load | At the Proportional Limit | At Ultimate Load | Comparisons | Flexural Rigidity | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Specimens No. | P_{1/3} (kN) | w_{1,1/3} (mm) | k_{1,1/3} (kN/mm) | P_{p} (kN) | w_{1,p} (mm) | k_{1,p} (kN/mm) | P_{u} (kN) | w_{1,u} (mm) | k_{1,u} (kN/mm) | P_{u,(B,C,D)}/P_{u,A} (-) | P_{p}/P_{u} (-) | w_{1,p}/w_{1,u} (-) | k_{1,p}/k_{1,u} (-) | k_{1,1/3}/k_{1,u} (-) | (EI)_{eff} (kN*m^{2}) | |

Series S1 | A1–A5 | 6.2 | 19.5 | 0.32 | 12.7 | 39.4 | 0.32 | 18.6 | 67.4 | 0.30 | - | 0.68 | 0.58 | 1.09 | 1.13 | 27.5 |

B1–B5 * | 5.7 | 12.1 | 0.48 | 14.3 | 30.9 | 0.46 | 17.1 | 37.5 | 0.46 | 0.92 | 0.84 | 0.82 | 1.01 | 1.04 | 38.7 | |

C1–C5 | 8.2 | 17.0 | 0.47 | 15.9 | 33.2 | 0.48 | 24.7 | 58.2 | 0.44 | 1.33 | 0.64 | 0.57 | 1.09 | 1.10 | 41.2 | |

Series S2 | A6–A10 | 4.8 | 20.0 | 0.24 | 10.3 | 43.3 | 0.24 | 14.5 | 78.6 | 0.19 | - | 0.71 | 0.55 | 1.26 | 1.29 | 27.3 |

B6–B10 | 5.7 | 17.3 | 0.33 | 14.0 | 42.5 | 0.33 | 17.2 | 54.6 | 0.32 | 1.18 | 0.82 | 0.78 | 1.04 | 1.04 | 37.5 | |

C6–C10 | 7.0 | 20.8 | 0.34 | 13.9 | 40.6 | 0.34 | 21.1 | 80.4 | 0.27 | 1.45 | 0.66 | 0.51 | 1.28 | 1.27 | 39 | |

D1–D5 | 6.5 | 20.0 | 0.32 | 11.8 | 36.2 | 0.33 | 19.5 | 81.7 | 0.24 | 1.35 | 0.60 | 0.44 | 1.35 | 1.34 | 37.6 |

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**MDPI and ACS Style**

Šubic, B.; Fajdiga, G.; Lopatič, J.
Bending Stiffness, Load-Bearing Capacity and Flexural Rigidity of Slender Hybrid Wood-Based Beams. *Forests* **2018**, *9*, 703.
https://doi.org/10.3390/f9110703

**AMA Style**

Šubic B, Fajdiga G, Lopatič J.
Bending Stiffness, Load-Bearing Capacity and Flexural Rigidity of Slender Hybrid Wood-Based Beams. *Forests*. 2018; 9(11):703.
https://doi.org/10.3390/f9110703

**Chicago/Turabian Style**

Šubic, Barbara, Gorazd Fajdiga, and Jože Lopatič.
2018. "Bending Stiffness, Load-Bearing Capacity and Flexural Rigidity of Slender Hybrid Wood-Based Beams" *Forests* 9, no. 11: 703.
https://doi.org/10.3390/f9110703