Experimental and Numerical Modeling of Bending Characteristics of Fir and Black Pine Wood from Different Forest Regions in Türkiye

Abstract

The bending test is a good indicator of wood characterization, including compression, tension, and shear stresses. Therefore, many studies have been published on bending tests for wood. Its heterogeneous structure, anisotropic behavior during the physical and mechanical tests, and anatomical and chemical differences due to species and growing area make the characterization difficult. Accordingly, research has focused on mathematical models and simulation programs for predicting material characteristics. More data on using many wood species in such models or programs still need to be collected. This study aimed to eliminate these deficiencies for two softwood species grown in Türkiye. In this context, three- and four-point bending tests were performed on 5656 samples from fir (Abies spp.) and black pine (Pinus nigra) species that were collected from 13 regions. A nonlinear material model was generated from the load-deformation data for both species, and the results were found in agreement regarding the bending features of both softwood species. The results show that nonlinear numerical modeling could predict the bending results with significant rates (min. 94%). Additionally, bending characteristics such as limits of proportionality (LOP), and elastic potential were found and compared between and within the species, regions, and test methods. Load-deformation curves showed that the LOP ranged between 40% and 60% of the maximum load, which was higher than the theoretical approach in the standards.

1. Introduction

Wood is a natural and sustainable engineering material. However, natural disasters, unconscious material use, and uncontrolled wood industry development are putting forests under pressure. Therefore, it is necessary to use wood more effectively [1]. It is necessary to know the physical and mechanical properties of the wood in the usage area to achieve this effect [2,3]. Wood has variations in physical and mechanical properties because of anisotropy, defects, biological characteristics, and growing origin factors. These properties and the correlations between them vary from species to species [4,5,6,7].

Bending is one of the most important mechanical properties of wood. It is defined as a torque that acts on a material perpendicular to the cross-section, which results in normal and tangential stress that causes stress through bending or twisting [8]. This stress includes tensile, compression, and shear stress, which cause the deformation and shape change. It is often used for material characterization because each material behaves differently with bending. Therefore, it is interpreted as a load-deformation curve graph with other variables such as modulus of elasticity (MOE), modulus of rupture (MOR), and density [9] (Figure 1). In a bending test, the load-deformation graph is generally divided into three areas. At the beginning of the load (Area A in Figure 1), linear elasticity is observed in many solids used in construction, including wood, when subjected to moderate loads (below the stress level referred to as the proportional limit). At higher loads, the stress distribution is non-uniform across the sample cross-section. If the load exceeds the elastic (proportional) limit (Fe in Figure 1), the permanent deflection (plastic deformation, Area B in Figure 1) starts on the material. This is due to the differences in stress–strain behavior and the significant difference in tensile and compressive strength [10].

The bending strength can be obtained by different test methods: three- and four-point bending [11,12]. In Europe, the three-point bending method determines bending properties on small clear wood samples [13,14], while structural size samples are usually measured using a four-point method [15]. Hein and Brancheriau [16] found that the modulus of rupture (MOR) from three-point bending tests was 5.2% higher than that from four-point bending tests in eucalyptus wood specimens. Sales [17] and Babiak et al. [18] compared the MOE obtained in three-point and four-point bending tests using softwood, hardwood, and tropical wood species. Their experimental results show that the MOE obtained from four-point bending is always higher than that obtained from three-point bending. Although the results between MOR values obtained from three- and four-point bending tests in wood were reported in a few previous works [16,19,20], the comparison between these static tests remains unclear. Because, unlike four-point bending, the three-point bending standard does not involve a shearing effect. In addition, Babiak et al. [18], who used bending equations according to wood-based panels, suggested a quantitative assessment; adding a four-point shear stress test with a varying distance of the acting forces would be necessary. Therefore, the shear effect must be calculated to compare the methods accurately. The MOE is calculated using the slope of the linear region of the load-deformation curve; the value calculated is the apparent modulus, not the true modulus, because these test methods include shear and bending stresses [21]. In another comparison study from Brancheriau et al. [11], the authors found under-estimation of about 19% in the MOE in three-point bending when compared to a four-point bending test. The under-estimations were found, respectively, at 8% and 11% when the shear effect is neglected in the three-point bending test.

As the demand for complex structures increases, researchers and civil engineers increasingly need more detailed material characteristics to predict material behaviors under different loads. This need is underscored by the limited studies investigating detailed characteristics such as the limit of proportionality and elastic potential for bending [22]. These characteristics, affected by species, moisture content, and bending methods, are crucial for accurately predicting material behaviors [2,18,23]. Due to structural and testing complexities, many prediction methods exist for wood properties. In recent years, artificial neural network, regression, probabilistic models [24,25] or non-destructive test methods [26], near-infrared spectrometry techniques [27], and the finite element method (FEM) for simulation models [28] have been used as prediction methods.

The finite element method (FEM) is widely used to solve solid mechanical problems. Numerical studies focused previously on the mechanical characterization of wood such as the bending strength and stiffness [29,30]. However, it is complicated to analyze the mechanical properties of wood with the FEM by regarding it as an anisotropic material. In order to make it adjust to computer modeling, it is usually regarded as an orthotropic material. Yoshihara and Kubojima [31] evaluated Japanese red pine (Pinus densiflora D. Don) and Japanese ash (Fraxinus spaethiana Lingelsh.) tree species according to the results of physical-mechanical tests and then performed a four-point bending resistance analysis with the finite element method. Hasanagić et al. [32] evaluated wooden beam samples from silver fir (Abies alba) with experimental four-point bending and carried out a numerical analysis by the finite element method. They indicated that numerical model can predict the bending of beams of different sizes. Fajdiga et al. [33] evaluated Norway spruce (Picea abies) samples both with a radial and a tangential orientation with three-point bending and compared them with a computational model which was an orthotropic material model with damage evolution. They found that the displacement response from the computational and experimental force had a very good correlation. Kurul et al. [34] adopted a simplified nonlinear material model to capture both elastic and plastic zones for modeling the out-of-plane bending behavior of black pine and fir boards in structural applications. Related to this approach, experimental and numerical model results were found more compatible for the bending strength and deformation capacity, especially in the better visual grade classes (1 and class 2) in the Turkish structural softwood grading standard [35]. In addition, Edgars et al. [36] performed design, prototyping, and experimental tests of plywood sandwich panels and compared these tests with the finite element analysis results. The results of the four-point experimental test performed on many materials, both wood and non-wood, were compared with the finite element analysis method, yielding similar results. Apart from in the wood industry, there are studies on this subject in the literature on construction, automotive, etc. Müller et al. [37] compared a material model which was validated against experimental data in tension, compression, and three-point and four-point bending tests, customarily used to model FRPs (fiber reinforced plastics) and three hardwood species. Results showed that FEM (finite element method) simulation correlated very well with the experimental data. The fabricated specimens, which are welded pipes, were tested using three- and four-point bending tests and a universal testing machine.

This study aimed to determine and compare the differences in bending properties, especially strength, deformation, elastic limit, and shear effect on the three- and four-point test methods for Turkish fir and black pine small clear specimens. Furthermore, a simplified nonlinear material model was proposed for the numerical modeling of these specimens for both methods in general usage. Therefore, the generated model can be used in simulation models such as FEM programs for the species studied.

2. Materials and Methods

2.1. Materials

Black pine (Pinus nigra Arnold. subsp. pallasiana) and fir (Abies spp.) kiln dried lumbers were provided according to ISO 3129 [38] from different regions in Türkiye. The regions were chosen according to top-selling regional forest directorates for each species (Figure 2).

A total of 5656 small-clear specimens with 20 mm × 20 mm × 320 mm (height × width × length) were obtained from the lumbers, including 2640 fir and 3016 black pine from a different region of Türkiye as described in

Table 1. Sample distribution according to regions and species.

Species	Regional Forest Directorates	Number of Specimens	Species	Regional Forest Directorates	Number of Specimens
Fir	Artvin	545	Black pine	Balıkesir	545
	Bilecik	357		Bolu	339
	Bolu	704		Çanakkale	374
	Kastamonu	343		Çorum	390
	Mersin	323		Kahramanmaraş	444
	Zonguldak	368		Kastamonu	458
				Muğla	466
	Subtotal	2640		Subtotal	3016
Total 5656

. Before the tests, all specimens were conditioned at 20 °C temperature and 65% relative humidity until reaching air-dry moisture content.

2.2. Methods

In this study, we used three- and four-point bending methods according to European standards. Tests were carried out with a universal testing machine (LS-100, Lloyd Instruments, Bognor Regis, UK), which was equipped with a load cell of 100 kN. All specimens were tested in the tangential direction, and the deformations (w) were measured at the center of the span. The applied standards specified that the load was applied continuously at a constant rate of loading such that the test piece was broken between 0.5 min and 5 min from the beginning of loading. The deflections were recorded from the loading head moving distance.

2.2.1. Three-Point Bending

A total of 1064 fir and 1525 black pine samples with dimensions of 20 mm × 20 mm × 320 mm (height × width × length) were tested in a three-point bending test setup which was adjusted according to ISO 13061-3 [13] and 13061-4 [14] with a 280 mm span length (Figure 3). Thereby, the apparent global MOE_a and MOR were determined by “Equations (2) and (3)” respectively as in the standards. Furthermore, the true MOE was determined by Equation (1) according to F. Mujika [39].

$$\mathrm{M}\mathrm{O}\mathrm{E}={\displaystyle \frac{\left({\mathrm{F}}_2-{\mathrm{F}}_1\right)·{\mathrm{l}}^3}{4·\mathrm{b}·{\mathrm{h}}^3·\left(\left({\mathrm{w}}_2-{\mathrm{w}}_1\right)-\frac{3·\left({\mathrm{F}}_2-{\mathrm{F}}_1\right)·\mathrm{l}}{10·\mathrm{G}·\mathrm{b}·\mathrm{h}}\right)}}$$

(1)

$${\mathrm{M}\mathrm{O}\mathrm{E}}_{\mathrm{a}}={\displaystyle \frac{({\mathrm{F}}_2-{\mathrm{F}}_1)·{\mathrm{l}}^3}{4·\mathrm{b}·{\mathrm{h}}^3·({\mathrm{w}}_2-{\mathrm{w}}_1)}}$$

(2)

$$\mathrm{M}\mathrm{O}\mathrm{R}={\displaystyle \frac{3·{\mathrm{F}}_{\mathrm{m}\mathrm{a}\mathrm{x}}·\mathrm{a}}{\mathrm{b}·{\mathrm{h}}^2}}$$

(3)

2.2.2. Four-Point Bending

A total of 1579 fir and 1491 black pine samples with dimensions of 20 mm × 20 mm × 320 mm (height × width × length) were tested in a four-point bending test setup which was adjusted according to EN 408 [15] with a 300 mm span length (Figure 3). Thereby, the true global modulus of the MOE and MOR was determined by “Equations (3) and (4)” as in the standards. Furthermore, the apparent MOE_a was determined by Equation (5) according to Ross [40].

$$\mathrm{M}\mathrm{O}\mathrm{E}={\displaystyle \frac{3·\mathrm{a}·{\mathrm{l}}^2-{4·\mathrm{a}}^3}{2·\mathrm{b}·{\mathrm{h}}^3·(2·\frac{{\mathrm{w}}_2-{\mathrm{w}}_1}{{\mathrm{F}}_2-{\mathrm{F}}_1}-\frac{6·\mathrm{a}}{5·\mathrm{G}·\mathrm{b}·\mathrm{h}})}}$$

(4)

$${\mathrm{M}\mathrm{O}\mathrm{E}}_{\mathrm{a}}={\displaystyle \frac{3·\mathrm{a}·{\mathrm{l}}^2-{4·\mathrm{a}}^3}{2·\mathrm{b}·{\mathrm{h}}^3·(2·\frac{{\mathrm{w}}_2-{\mathrm{w}}_1}{{\mathrm{F}}_2-{\mathrm{F}}_1})}}$$

(5)

where the MOE is the true global modulus of elasticity (MPa), the MOE_a is the apparent global modulus of elasticity, the MOR is the modulus of rupture (MPa), b is the width (mm) and h is the height of samples (mm), l is the span distance (mm), a is the distance between a loading position and the nearest support in a bending test (mm), w₂ − w₁ is the deflection difference at 10% and 40% of the maximum load (mm), G is the shear modulus assumed as 650 MPa (according to EN 408 standard), and F₂ − F₁ is the load difference at 10% and 40% of the maximum load (N).

Load-deformation data and diagrams from the testing machine were used. The density of each specimen at the moisture content during the test was determined from a piece of the sample cut that was as close as the location of the break area after the tests. Then, the pieces were weighed with an accuracy of 0.01 g, and their dimensions were measured with an accuracy of 0.01 mm according to ISO 13061-2 [41]. The moisture contents were found according to ISO 13061-1 [42] standard. The density of specimens and experimental values for the MOR and MOE were adjusted to 12% by using the following formulas:

$${\mathrm{M}\mathrm{O}\mathrm{R}}_{12}={\mathrm{M}\mathrm{O}\mathrm{R}}_{\mathrm{m}}·\left[1+\mathsf{\alpha }·\left(\mathrm{M}-12\right)\right]$$

(6)

$${\mathrm{M}\mathrm{O}\mathrm{E}}_{12}={\displaystyle \frac{{\mathrm{M}\mathrm{O}\mathrm{E}}_{\mathrm{m}}}{1-\mathsf{\alpha }·\left(\mathrm{M}-12\right)}}$$

(7)

$${\mathsf{\rho }}_{12}={\mathsf{\rho }}_{\mathrm{m}}·(1-{\displaystyle \frac{(1-\mathrm{K})·(\mathrm{m}-12)}{100}})$$

(8)

where M is the moisture content of the sample determined according to ISO 13061-1 (%), α is the correction factor 0.04 for the MOR and 0.02 for the MOE according to the moisture content, and K is the coefficient of volumetric shrinkage for a change in moisture content of 1%. The K value can be taken as equal to $\left(0.85{·10}^{-3}{·\mathsf{\rho }}_{\mathrm{m}}\right)$ when the density is expressed in kg/m³.

2.2.3. Determination of Other Bending Characteristics

Other bending characteristics such as elastic limit (proportional limit, F_e or red point in Figure 1) and elastic potential were determined from the load–deflection curve and its raw data. The EN 408 [15] standard describes finding the longest portion of this section that gives a correlation of 0.99 or better on the 0.1–0.4 F_max (maximum load) for a regression analysis. However, some differences were observed during the application. The standard describes the finding ‘area’, while the elastic limit is this area’s ‘endpoint’. Additionally, 0.4 of the F_max is a theoretical value, and most of the Fe points could be higher than 0.4 of the F_max, as observed in this study’s results. Therefore, the transition point must be determined to find the LOP and PE correctly.

The transition point (red point) defines the start of the plastic region. It was found for each specimen by detecting the change in the slope of the load-deformation curves. It was detected after exceeding the linear load limit beyond 40% of the maximum load with a correlation coefficient of 0.99 (Figure 1). In addition, visual confirmation was made on the graphs for each sample. Thereby, the LOP was determined by Equation (9) for three- and four-point bending tests as in the standards [13,15]. Furthermore, the PE was determined by Equation (10) according to [2] for both methods.

$$\mathrm{L}\mathrm{O}\mathrm{P}={\displaystyle \frac{3·{\mathrm{F}}_{\mathrm{e}}·\mathrm{a}}{\mathrm{b}·{\mathrm{h}}^2}}$$

(9)

$$\mathrm{P}\mathrm{E}={\displaystyle \frac{3·{\mathrm{F}}_{\mathrm{e}}·{\mathrm{w}}_{\mathrm{e}}}{2·\mathrm{b}·\mathrm{h}·\mathrm{l}}}$$

(10)

where LOP is the limit of proportionality (MPa), PE is the elastic potential (MPa), l is the span distance (mm), a is the distance between a loading position and the nearest support in a bending test (mm), Fe is the force at the transition point (N), “we” is the deflection at the transition point (mm), b is the with (mm), and h is the height of samples (mm).

2.2.4. Numerical Modeling

In the study, it was aimed to model a three- and four-point bending test for both species using the design program to determine how accurately the behavior can be represented through a finite element structural analysis program and an experimental (real) one. For this purpose, simple structural analysis was applied to the samples and supports prepared with the SolidWorks design program using the ANSYS Workbench 21 FEM analysis program. This section consists of several stages.

In the first stage, the design and assembly parts were prepared. In this stage, the samples for three-point and four-point bending setups were prepared with exactly the same size, thickness, and features, and fixed frictionless supports were designed in accordance with the tests and positioned in accordance with the system as in Figure 4.

Afterwards, the values (MOR, MOE in bending, Poisson constant, density) obtained as a result of physical mechanical tests given in

Table 2. Descriptive statistics and comparisons of all values for fir ¹.

Tests	Method	Region						Total
		Artvin	Bilecik	Bolu	Kastamonu	Mersin	Zonguldak
Sample Number (pcs.)	3p	205	140	225	162	150	179	1061
	4p	340	217	479	181	173	189	1579
ρ12 (kg/m3)	-	392	426	421	418	443	417	417.4
MC (%)	-	11.6	11.3	11.0	11.8	12.0	11.8	11.5
MOR (MPa)	3p	55.2 a (14.0)	63.2 cd (12.9)	58.6 b (12.7)	59.4 b (14.0)	64.0 d (16.5)	61.8 c (15.4)	60.0 1 (15.1)
	4p	58.6 a (14.2)	66.5 d (14.5)	62.7 c (14.2)	57.9 a (14.2)	62.3 c (18.2)	60.4 b (15.5)	61.5 2 (15.2)
MOE (MPa)	3p	7640 a (20.1)	9113 d (15.0)	8066 b (17.0)	8048 b (17.0)	8603 c (21.2)	8646 c (16.7)	8293 1 (18.7)
	4p	10,475 b (20.0)	12,303 d (18.0)	11,549 c (16.0)	9139 a (16.7)	10,434 b (21.8)	10,359 b (20.5)	10,880 2 (20.2)
MOEa (MPa)	3p	7096 a (18.8)	8350 d (13.9)	7470 b (15.8)	7458 b (15.5)	7867 c (20)	7964 c (15.3)	7651 1 (17.4)
	4p	9776 b (18.7)	11,340 d (16.4)	10,706 c (14.9)	8607 a (15.8)	9740 b (20.3)	9709 b (19.2)	10,127 2 (18.8)
Fe (N)	3p	541.6 a (19.5)	598.1 c (17.4)	550.1 a (16.8)	564.6 ab (18.0)	596.5 c (19.8)	579.7 bc (17.5)	568.6 1 (18.5)
	4p	784.9 b (18.7)	878.9 d (17.1)	817.6 c (16.1)	747.6 a (17.1)	795.7 bc (19.9)	788.9 b (18.0)	805.1 2 (18.1)
We (mm)	3p	2.77 cd (17.9)	2.51 a (13.6)	2.63 b (15.1)	2.72 bc (14.0)	2.83 d (16.8)	2.63 b (15.8)	2.68 1 (16.1)
	4p	3.01 b (15.0)	2.80 a (15.4)	2.85 a (13.4)	3.15 c (15.8)	3.05 b (17.3)	3.07 b (15.7)	2.96 2 (15.6)
LOP (MPa)	3p	27.8 a (19.4)	29.9 c (16.2)	28.2 b (16.4)	28.6 ab (16.4)	30.4 c (19.8)	29.6 bc (17.7)	28.9 1 (18.2)
	4p	28.7 b (18.7)	31.6 d (17.1)	29.8 c (15.9)	27.0 a (17.1)	28.9 bc (20.0)	28.7 b (18.7)	29.2 1 (18.0)
PE (MPa)	3p	0.0202 a (37.8)	0.0194 a (28.8)	0.0192 a (35.2)	0.0202 a (32.8)	0.0227 b (32.5)	0.0200 a (34.7)	0.0202 1 (34.5)
	4p	0.0295 a (30.3)	0.0305 a (29.2)	0.0287 a (28.4)	0.0291 a (31.6)	0.0301 a (31.8)	0.0303 a (31.8)	0.0295 2 (30.2)

¹ The same small letter on the mean values in the same row and the small number on the mean values in the same column (the latest column) show that there is no difference in group homogeneity for each parameter (p < 0.05). The highest value was shown in bold and the lowest value in italics for each test. ρ₁₂: Air-dry density, MC: Moisture Content, MOR: Modulus of rupture, MOE: True global modulus of elasticity, MOE_a: Apparent global modulus of elasticity, F_e:: Force at the transition point, W_e:: Deflection at the transition point, LOP: Limit of proportionality, PE: Elastic potential. The values in the parenthesis are the coefficient of variation.

and

Table 3. Descriptive statistics and comparisons of all values for black pine ¹.

Tests	Method	Region							Total
		Balıkesir	Bolu	Çanakkale	Çorum	K.Maraş	Kastamonu	Muğla
Sample Number (pcs.)	3p	275	174	189	195	228	234	230	1525
	4p	270	165	185	195	216	224	236	1491
ρ12 (kg/m3)	-	532	570	563	580	505	530	586	550.4
MC (%)	-	11.3	11.3	11.5	11.6	11.0	11.4	11.3	11.3
MOR (MPa)	3p	83.3 c (13.2)	78.2 b (16.1)	79.0 b (17.6)	82.8 c (13.4)	71.8 a (15.6)	77.7 b (14.2)	83.2 c (18.5)	79.5 1 (16.3)
	4p	83.2 c (12.9)	74.2 b (17.1)	76.4 b (12.9)	83.3 c (12.9)	70.3 a (15.4)	74.1 b (14.9)	84.1 c (20.0)	78.3 1 (17.5)
MOE (MPa)	3p	11,428 d (16.5)	9919 b (22.1)	10,449 c (25.8)	10,362 c (21.2)	9330 a (19.4)	10,293 bc (17.6)	11,238 d (22.6)	10,481 1 (21.6)
	4p	13,358 d (16.2)	11,008 b (23.0)	11,921 c (25.2)	12,169 c (19.9)	10,600 a (21.2)	11,657 c (18.7)	13,199 d (22.8)	12,084 2 (22.3)
MOEa (MPa)	3p	10,293 d (14.9)	9044 b (20.3)	9480 c (23.6)	9411 c (19.3)	8551 a (17.7)	9362 bc (16.0)	10,129 d (20.5)	9509 1 (19.7)
	4p	12,272 d (14.9)	10,255 b (21.5)	11,034 c (23.5)	11,257 c (18.4)	9892 a (19.7)	10,814 c (17.4)	12,123 d (21.1)	11,175 2 (20.7)
Fe (N)	3p	729.5 c (12.4)	676.3 b (18.9)	677.9 b (19.7)	715.4 c (13.4)	649.7 a (17.8)	690.4 b (16.1)	723.0 c (20.3)	696.3 1 (17.4)
	4p	1049.7 cd (16.9)	900.0 a (20.7)	923.8 a (21.5)	1048.6 c (16.6)	899.8 a (17.1)	962.1 b (16.5)	1080.6 d (21.5)	987.4 2 (19.9)
We (mm)	3p	2.67 a (12.1)	2.79 bc (14.5)	2.76 bc (16.3)	2.88 d (15.0)	2.84 cd (15.5)	2.79 bc (13.0)	2.74 ab (15.9)	2.77 1 (14.8)
	4p	3.24 a (10.1)	3.46 cd (15.8)	3.32 ab (15.5)	3.54 d (14.7)	3.43 c (15.9)	3.38 bc (12.8)	3.45 cd (14.3)	3.39 2 (14.4)
LOP (MPa)	3p	38.6 c (12.3)	36.0 b (18.3)	36.4 b (19.4)	37.9 c (12.7)	34.6 a (17.7)	36.7 b (15.8)	38.3 c (20.2)	37.0 1 (17.0)
	4p	39.6 de (16.0)	34.1 ab (20.3)	35.2 bc (20.8)	39.7 d (15.9)	34.1 a (16.9)	36.3 c (16.0)	40.8 e (18.1)	37.4 1 (19.5)
PE (MPa)	3p	0.0263 a (22.1)	0.0260 a (27.4)	0.0252 a (30.8)	0.0282 b (23.2)	0.0253 a (29.7)	0.0264 a (25.5)	0.0268 a (32.3)	0.0263 1 (27.5)
	4p	0.0430 c (22.3)	0.0397 ab (31.0)	0.0390 ab (27.9)	0.0469 d (25.4)	0.0393 a (27.3)	0.0411 bc (24.9)	0.0472 d (30.4)	0.0425 2 (27.9)

¹ The same small letter on the mean values in the same row and the small number on the mean values in the same column (the latest column) show that there is no difference in group homogeneity for each parameter (p < 0.05). The highest value was shown in bold and the lowest value in italics for each test. ρ₁₂: Air-dry density, MC: Moisture Content, MOR: Modulus of rupture, MOE: True global modulus of elasticity, MOE_a: Apparent global modulus of elasticity, F_e: Force at the transition point, W_e: Deflection at the transition point, LOP: Limit of proportionality, PE: Elastic potential. The values in the parenthesis are the coefficient of variation.

were defined separately as fir and black pine in the isotropic material definition section. The aim was to ensure that the sample properties applied in real experiments are the same in the program and in the same properties. In the second stage, the samples whose designs were created and material definitions were made were taken to the ANSYS Workbench 21 program and the relations between them (contact) were defined and before reaching the solution stage, the mesh (a finite element mesh size of 12 mm was selected for the model, resulting in a total of 1322 nodes and 682 elements) definition was made it.

In the final phase of the experiment, the wood specimens were subjected to both three-point and four-point bending test configurations. For the four-point bending test, the loading system consisted of two equal point loads applied at one-third intervals along the length of the wood, thus dividing the beam into three equal segments. In contrast, the three-point bending test setup involved two identical point loads applied at 1 s intervals, effectively dividing the beam into two equal halves. This setup is designed to simulate real-life loading conditions and to analyze the wood material flexural performance, including reaction forces at supports, stress distribution, deflection, and potential failure points under the applied loads.

2.2.5. Statistical Analysis

Three- and four-point bending test results were used for the analysis of variance (ANOVA) to obtain possible statistically significant differences between variables especially considering the region of the specimens with the significance level of p < 0.05. Average values for all variables including moisture content (MC) and air-dry density (ρ₁₂) were given separately for each method and region in Table 2 for the fir and Table 3 for the black pine species. Furthermore, the coefficient of variation values were given in parentheses. A Duncan’s multiple range test was applied to the detected variables. Furthermore, the largest value was displayed in bold and the smallest value in italics for each test.

3. Results and Discussion

According to Table 2 and Table 3, significant differences were found in all tests. Bilecik and Balıkesir came to the forefront with the highest values for fir and black pine, respectively, for both methods.

Additionally, the values showed that some Fe (elastic transition) limit points differed from the 0.4 F_max value (Figure 5). As seen in Table 2 and Table 3, most of the Fe points are mainly distributed between 40% and 50% ratios of the maximum loads, and they were similarly found on both species and bending methods. A total of 40% of the maximum force is a theoretical approach, which can vary depending on the test results. Otherwise, the elastic limit is related to the material’s elastic properties, while the maximum load is related to the strength properties. Different factors at different levels affect a material’s strength and elasticity characteristics. When the differences are considered, knots and cracks affect the strength more than the elasticity, while the micro-fibril angle affects the elasticity more than the strength [10].

Tests in both methods for all species were compared and are shown in Figure 6. The four-point method gave higher values than the three-point method except for MOR values. For MOR values, the three-point method was 1.5% higher than the four-point method in black pine. On the contrary, the four-point method was 2.5% higher in fir. Hein and Brancheriau [16] found it 5.2% higher in the three-point bending method. The MOE was found about 8.5% (for fir 7.7% and black pine 9.3%) higher in the three-point method and about 7.2% (fir 6.9% and black pine 7.5%) higher in the four-point method than the MOE_a. The shear effect was determined as 8.4% and 10.2% in fir and black pine in the three-point bending tests, respectively, and 7.4% and 8.1% in the four-point tests. These results are consistent with other studies in the literature that focused on determining the shear effect. In three-point bending, Kollmann and Côté Jr [43] found about a 9.5%, Brancheriau et al. [11] about a 9%, and Guerrin [44] about a 8.0% shear effect. Furthermore, Brancheriau et al. [11] found about a 6% shear effect in four-point bending.

F_e and W_e values were found higher in the four-point method than in the three-point method (Figure 7). Black pine’s value was higher than fir according to both methods. The transition point was determined at about 47.9% of the maximum load (for fir 49.8% and black pine 46.0%) for the three-point bending method. Also, it determined about 48.2% of the maximum load (for fir 49.1% and black pine 47.3%) for the four-point bending method. Although the bending method was changed, the transition points were found very close to each other. Maximum deformation within the elastic limit was found to be 10.5% and 22.4% higher in the four-point method compared to the three-point method for fir and black pine, respectively.

LOP values were found higher in the four-point method than in the three-point method but it was insignificant at the level of p < 0.05 (Figure 8). Black pine’s value was 28% higher than fir according to both methods. These results are very similar to those of the study by Gaff et al. [2], who found the effect of wood density on the limit of proportionality and confirmed the effect by citing other studies.

The values of elastic potential are statistically significantly influenced by the wood species and test methods. Black pine was 46.0% and 61.6% higher than fir for the three-point and four-point methods, respectively. Similarly, Babiak et al. [18] found higher elastic potential for the four-point bending method in oak and spruce wood at an 8% moisture content.

3.1. Regression Matrices for Fir and Black Pine

The frequency histograms of specimens and regression matrix of mechanical properties were plotted using binary combinations of the measured values (MOR, MOE, LOP, PE) for each bending method. Linear regression matrices for MOR, MOE, LOP, and PE were shown in Figure 9 and Figure 10 for fir and black pine, respectively. Histograms were located on the cross-axis.

For fir, there were strong correlations in MOR-MOE (R² = 0.597 and 0.536), MOR-LOP (R² = 0.596 and 0.608), and LOP-PE (R² = 0.771 and 0.737) for three-point and four-point bending tests, respectively. Furthermore, moderate correlations were found in MOE-LOP (R² = 0.399 and 0.390), MOR-PE (R² = 0.282 and 0.259) and a weak correlation was found in MOE-PE (R² = 0.091 and 0.042). Similarly, for black pine, there were strong correlations in MOR-MOE (R² = 0.692 and 0.655), MOR-LOP (R² = 0.643 and 0.696), LOP-PE (R² = 0.667 and 0.750), and MOE-LOP (R² = 0.480 and 0.537). In addition, moderate correlations were found in MOR-PE (R² = 0.193 and R² = 0.354) and a weak correlation was found in MOE-PE (R² = 0.034 and 0.109).

3.2. Numerical Modeling

The modeled modules were designed to replicate the conditions of the real-life experiments, and numerical solutions were obtained using the finite element method. The finite element analysis revealed that the highest stress concentrations occurred around the load application points—specifically, at the mid-span for the three-point bending test and at the 120 mm upper span for the four-point bending test, as shown in Figure 4. These regions, where fir and black pine wood materials are anchored to the base, experience the most significant load transfer between materials. The color transition from red to blue along the beam indicates a stress gradient, with stress levels decreasing toward the supports. Although fir and black pine exhibited different levels of stress, both materials demonstrated similar strain patterns, as illustrated in Figure 11.

It was revealed by the finite element method that the deformation amount of the black pine (12.5 mm) at the maximum load was higher than the fir (10.2 mm). For both materials, maximum deformation occurred at the midpoint in three-point tests. In addition, in the four-point tests, it was revealed that the deformation amount of the fir (13.2 mm) at the maximum load was greater than that of the black pine (10.3 mm).

FEM analyses were a simulation of mechanical tests in this study. For this reason, as in

Table 4. Comparison of experimental results and analysis results ¹.

Species	Point	Experimental Results		Analysis Results		% Dif. Fmax	% Dif. Wmax
		Fmax (N)	Wmax (mm)	Fmax (N)	Wmax (mm)
Fir	3	1297.2	9.5	1375	10.2	6.00	7.37
	4	1774.2	12.7	1725	13.2	−2.77	3.94
Black Pine	3	1469.5	12.6	1459	12.5	−0.71	−0.79
	4	2066.1	10.1	1908	10.3	−7.65	1.98

¹ Difference, F_max: Maximum force, W_max: Maximum deformation.

, the maximum force and deformation results of the bending test outputs performed within the scope of the project are shown in comparison with the experimental and analysis results. It was also determined that the difference between the two methods, shown as a percentage, was not more than 7.65%.

The results of the FEM analyses and experimental data performed according to three-point and four-point bending tests are shown with load-deformation graphs as in Figure 12 and Figure 13 for fir and black pine, respectively. When the graphs are examined, it is seen that they coincide with the graphs created for the experimental results. In the studies conducted on wood and wood-based samples by Kurul et al. [34], Müller et al. [37], Yoshihara and Kubojima [31], and Edgars et al. [36], the experimental results and finite element analysis results were found to be similar to this study. However, these differences may vary according to species and sample dimensions. Hasanagić et al. [32] found an average of 1.85% (with a maximum of 18.3%) for maximum deformation (displacement) in silver fir (Abies alba) wooden beams during the four-point bending.

4. Conclusions

This study reveals the bending properties, especially strength, deformation, elastic limit, and shear effect on the three- and four-point test methods for Turkish fir and black pine small clear specimens:

(1) When all the results were evaluated, the values obtained in four-point bending for both species (except the MOR for black pine and LOP for both species) were higher than those obtained in three-point bending. It is thought that the difference detected in the MOR is due to the variation in the sub-sample groups (regions). In the elastic limit (LOP), it was determined that the tree species and loading type do not cause a statistically significant difference.

(2) Contrary to the studies in the literature, the shear effect was considered in both three-point and four-point bending tests to make an accurate comparison. As a result, the shear effect was higher for both species in three-point bending. This result shows that the shear effect decreases as the top loading point increases, and it is thought to be due to the force going towards the distributed load approaching the compression force applied to the entire surface area. However, the bending deformation continues since the support points below do not increase.

(3) The LOP (elastic limit), a bending transition point, is accepted as 40% of F_max in the load-deformation graph with a theoretical approach in the standards. However, this study determined that this transition point was between 40% and 60% in three- and four-point bending tests performed on 5656 samples of two softwood species. Consequently, in studies where the elastic limit is important, it would be beneficial to use the existing raw load-deformation data or the suggested method in this paper instead of the theoretical approach.

(4) In three-point and four-point bending, a high correlation was found between all the investigated characteristics except between the MOR-PE and MOE-PE for both species.

(5) It has been shown that the result obtained with nonlinear numerical modeling can predict the results of over 5000 experiments with significant rates (92.6% for fir, 92.4% for black pine). Thus, the numerical modeling outputs can be used in more complex calculation models related to two softwood species growing in Turkey. Accordingly, other factors that may affect the bending characteristics of wood should be added to increase the prediction success rate.

(6) In general, the study shows the importance and accuracy of numerical modeling for the wood bending. Due to the fact the study was realized on small and clear specimens, the approach may be used on the structural samples. However, the researchers should consider important defects such as knots, crack, slope of grain, etc., to obtain similar accuracy.