Experimental determination of the limiting flexibility of eucalyptus wood for axially compressed elements
 Authors: CajamarcaZuniga D.^{1}^{,2}, Carrasco C.^{1}, Molina B.^{1}

Affiliations:
 Catholic University of Cuenca
 Peoples’ Friendship University of Russia (RUDN University)
 Issue: Vol 16, No 3 (2020)
 Pages: 226232
 Section: Experimental researches
 URL: https://journals.rudn.ru/structuralmechanics/article/view/23962
 DOI: https://doi.org/10.22363/181552352020163226232
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Abstract
Relevance. Wood is one of the most widely used building materials throughout history, and because of its physicalmechanical properties it mainly has been used in flexed and compressed elements. Eucalyptus was introduced to Latin America in the mid19th century and nowadays is one of the most used woods for construction in the Andean region of Ecuador. To designing slender structural elements under axial loading engineers usually use the Euler formula, but it is applicable only if the compression stress does not exceed the proportional limit. One way to determine if the compression stress will be below the proportional limit is by comparing of the slenderness of the element with the limiting flexibility of its material which allows knowing if the buckling will occur in the elastic zone where Euler formula applies. The aim of the work  determine the magnitude of the limiting flexibility of eucalyptus, since this wood has been the subject of some investigations, however, no information about the limiting flexibility magnitude for the calculation of axially compressed elements. Methods. The laboratory tests to determine the magnitudes of the modulus of elasticity, proportional limit, admissible compression stress and limiting flexibility was carried out. Results. This experimental investigation shows that the magnitude of the limiting flexibility or socalled critical slenderness ratio for eucalyptus globulus is 59.
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Introduction[1]
Eucalyptus is a fast growing diffuseporous hardwood genus of trees from the Myrtle family [1], there are about 500 species in the world, most of which are originally from Australia. The genus eucalyptus first was described by the French botanist C.L. L'Héritier in 1788 [2]. After the introduction of the eucalyptus to Ecuador, a large variety of forests of this tree appeared in highlands where the spice eucalyptus globulus do
minates among others [3]. The Eucalyptus is a widely used wood in the construction sector in the Andean region of Ecuador [4], it can reach an height of 20 m and a diameter of 0.25 m at the age of 5 to 10 years, while at an older age it can reach a height of 60 m.
This investigation aims to determine the magnitude of the flexibility of Eucalyptus since this wood has not been the subject of investigations like other woods [5] and the existing investigations about it [5–11] don’t give information about the limiting flexibility (critical slenderness ratio) for the calculation of axially compressed elements.
In this article are presented the results of laboratory tests on the definition of the magnitude of admissible compressive stress parallel to grain, modulus of elasticity and proportional stress with the purpose of the definition of the value of limiting flexibility (λ_{lim}) of this material.
For the calculation of axially compressed elements, the magnitude of the critical force usually is determined with the Euler formula, however, it is applicable only when the axial compression stresses do not exceed the stress of proportionality. Therefore, is necessary to determine if the compression stresses will be below the proportional range, by comparing the magnitude of the slenderness λ (geometric characteristic of the element) with the magnitude of the limitingflexibility λ_{lim} (mechanical characteristic of the material), this comparison allows to determine if the element has great or intermediate slenderness, to proceed to the calculation of the element considering the risk of loss of stability (buckling).
If the element has great slenderness, the buckling would occur in the elastic zone and Euler formula can be applied to determine the magnitude of the critical load, but if the element has “intermediate” slenderness the loss of stability would occur in the plastic zone and the Euler formula is no more applicable, here empirical formulas such as F.S. Yasinskiy (1895) [12] could be used, however, the problem is determining if the element has “great” or “intermediate” slenderness, and the answer to this question is in the magnitude of limiting flexibility.
1. Methodology
First, a bending test was performed to verify the obtained results by comparing them with the information available about the bending mechanical properties of eucalyptus, then the compression test was performed on 21 varioussizes samples to analyse their behaviour according to the variation of slenderness. The experimental analysis was based on norms and standards of the “Pan American Standards Commission” (COPANT 461, COPANT 464, COPANT 555), “American Society for Testing and Materials” ASTM D14394 (2000) and “International Organization for Standardization” ISO 1306117:2017 [13–17], which describe the materials, equipment and procedures to obtain: the stresses for ultimate bending and compression parallel to the grain, the average modulus of elasticity, the maximum breaking load and the stressstrain graph, that permit determine the mechanical properties of the wood.
Figure 1. Compression test
Figure 2. Flexure test
The compression test (Figure 1) was carried out according to standards in samples with a crosssection of 5×5 cm and 20 cm of length [14], the flexure test (Figure 2) was carried out in samples with a cross section of 5×5 cm and 75 cm between supports, however an alternative crosssection of 2×2×30 cm could be used for samples from trees of small diameter (30 cm or less) and also when long samples cannot be obtained due to a bent tree, tilting of grains, knots or other defects [15].
2. Bending test
The flexural testing was carried out on 20 samples, in accordance with [13; 15–17], the information about deflextions and loads were recorded, and then the corresponding magnitude of bending moment and normal stress under flexure were calculated in order to obatin the stressstrain curve and the modulus of elasticity.
The results of the flexure test are shown in the Figure 3. According to the results of this experiment, the modulus of elasticity in bending for the eucaliptus globulus is E = 104 180.68 kgf/cm^{2}, the ultimate strength is 800.46 kgf/cm^{2}, and the proportional limit is 517.47 kgf/cm^{2}.
Figure 3. Stressstrain curve for bending, kgf/cm^{2}
Table 1
Flexural mechanical properties
Flexural mechanical properties of eucalyptus globulus  
 kgf/cm^{2}  mPa 
Modulus of elasticity  104 180.68  10 220.12 
Stress of the proportional limit  517.47  50.76 
Ultimate strength  800.46  78.53 
The obtained results are according with the existing information about the flexure mechanical properties of eucalyptus, some of which is shown in [1; 6–9; 18–21].
3. Compression parallel to the grain test
The experiments were performed on 21 specimens based on the standards [14; 16; 17], taking into account that the COPANT464 standard specifies that the crosssection of the specimen for the compression test parallel to the grain is 5×5×20 cm [14]. The standard dimensions, both in height and in the crosssection, were varied to test different slendernesses. The dimensions of the analysed samples have shown in Table 2.
The compression test was performed until the failure of each sample, and both the longitudinal deformation and the corresponding compression load were registered to calculate the compression stresses according to the crosssections of each sample and to get the stressstrain curves (Figure 4).
Table 2
Dimensions of specimens for compression test
ID.N^{o}.  Length, cm  a_{max}, cm  a_{min}, cm  Mass, kg  Area, cm^{2}  I_{min}, cm^{4}  i_{min}, cm  λ  Specific weight, kgf/m^{3}  
1  13c  4.10  4.10  3.90  0.070  15.990  20.267  1.13  3.64  1067.74 
2  13a  5.15  4.10  3.90  0.080  15.990  20.267  1.13  4.57  971.48 
3  13b  6.60  4.10  3.90  0.100  15.990  20.267  1.13  5.86  947.56 
4  8  9.59  3.95  3.95  0.160  15.603  20.287  1.14  8.41  1069.32 
5  20  10.07  2.95  2.92  0.090  8.614  6.121  0.84  11.95  1037.55 
6  19  10.16  2.93  2.93  0.090  8.585  6.142  0.85  12.01  1031.84 
7  9  10.20  3.99  3.96  0.170  15.800  20.648  1.14  8.92  1054.83 
8  18  10.20  2.14  2.06  0.060  4.408  1.559  0.59  17.15  1334.35 
9  7  12.30  3.91  3.89  0.220  15.210  19.180  1.12  10.95  1175.96 
10  5  14.57  4.00  3.93  0.230  15.720  20.233  1.13  12.84  1004.19 
11  4  15.10  3.93  3.52  0.240  13.834  14.284  1.02  14.86  1148.94 
12  1  15.14  3.97  3.97  0.240  15.761  20.700  1.15  13.21  1005.78 
13  2  15.26  4.00  3.91  0.240  15.640  19.925  1.13  13.52  1005.59 
14  3  15.30  3.90  3.77  0.230  14.703  17.414  1.09  14.06  1022.42 
15  17  19.45  3.99  3.95  0.340  15.761  20.492  1.14  17.06  1109.15 
16  10  19.75  3.97  3.96  0.430  15.721  20.544  1.14  17.28  1384.89 
17  12  19.90  3.98  3.96  0.390  15.761  20.596  1.14  17.41  1243.46 
18  14  19.90  4.04  3.93  0.380  15.877  20.435  1.13  17.54  1202.70 
19  16  19.98  3.90  3.45  0.320  13.455  13.346  1.00  20.06  1190.34 
20  21  20.05  2.95  2.90  0.180  8.555  5.996  0.84  23.95  1049.39 
21  15  20.10  4.04  3.92  0.350  15.837  20.280  1.13  17.76  1099.52 
Figure 4. Stressstrain curves for compression parallel to the grain
Compression tests parallel to the grain indicate three characteristic behaviours as a function of the modulus of elasticity. For slenderness less than λ = 20.06, 42.85% of the samples presented a minimum modulus of elasticity MOE_{min }= 42 210.42 kgf/cm^{2} (Figure 4, a), 33.33% of the samples an intermediate magnitude of MOE_{int }= 79 811.58 kgf/cm^{2} (Figure 4, b) and 23.82% a maximum value of MOE_{max }= 117 781.86 kgf/cm^{2} (Figure 4, c). The results are summarized in the Table 3 and Figure 5.
Figure 5. Stressstrain curves and MOE of eucalyptus in compression parallel to the grain
Table 3
Grainparallel compression mechanical properties
Mechanical properties under compression parallel to the grain  
Modulus of elasticity, kgf/cm^{2}  Proportional limit, kgf/cm^{2}  
Minimum (E_{min})  42 210.42  277.61 
Intermediate (E_{int})  79 442.47  296.30 
Maximum (E_{max})  117 781.86  329.13 
Mean value (E_{m})  79 811.58  301.01 
In this test case, the obtained results are according with the existing information about the compression parallel to the grain mechanical properties of eucalyptus, some of which is shown in [8; 9; 19; 20].
4. Analysis of results
The buckling of a compressed slenderelement can lead to a sudden failure of a structure, and as a result, special attention must be given to the design of these elements so they can safely support their intended loadings without buckling. The wellknown Euler formula (1) usually is used by engineers to designing slenderstructural elements under axial loading, but this formula is applicable only if the compression stress does not exceed the proportional limit.
${F}_{\text{cr}}=\frac{{\text{\pi}}^{2}\xb7E\xb7{I}_{\text{min}}}{{\left(\text{\mu}\xb7L\right)}^{2}}$. (1)
${\text{\sigma}}_{\text{cr}}=\frac{{F}_{\text{cr}}}{A}=\frac{{\text{\pi}}^{2}\xb7E\xb7{I}_{\text{min}}}{{\left(\text{\mu}\xb7L\right)}^{2}\xb7A}=\frac{{\text{\pi}}^{2}\xb7E\xb7{I}_{\text{min}}}{{\text{\mu}}^{2}\xb7{L}^{2}\xb7A}$ . (2)
${i}_{\text{min}}=\sqrt{\frac{{I}_{\text{min}}}{A}}$. (3)
In the formula (2) the length L, the area moment of inertia I_{min} and area of crosssection A are geometric characteristics of the element, then the expression (2) can be written like this:
${\text{\sigma}}_{\text{cr}}=\frac{{\text{\pi}}^{2}\xb7E\xb7{i}_{\text{min}}^{2}}{{\text{\mu}}^{2}\xb7{L}^{2}}$. (4)
In the formula (4) the only geometric characteristics of the element are the radius of gyration i_{min} and the length L, while the effectivelength factor is µ. The effectiveslenderness ratio of the element as a function of geometric characteristics is shown in the formula (5), in consequence the slenderness can be considered as a geometric characteristic of the element.
$\text{\lambda}=\frac{\text{\mu}\xb7L}{{i}_{\text{min}}}$ (5)
Then, the expression (2) could be been wrote like (6).
${\text{\sigma}}_{\text{cr}}=\frac{{\text{\pi}}^{2}\xb7E}{{\text{\lambda}}^{2}}\le {\text{\sigma}}_{\text{pl}}$. (6)
Here we can see that the critical stress is a function of both a physicalmechanical property of the material E and a geometric characteristic of the element λ, where E represents the constant of proportionality and is applicable under stresses that not exceed the proportional limit, for this reason, Euler formula is applicable only for stresses that do not exceed the proportional limit of the material.
From the expression (6) it is possible to obtain the inequality (7), whose right side represents a physicalmechanical characteristic of the material known as “limiting flexibility” (λ_{lim}).
. (7)
Therefore, if we compare the slenderness of the element with the limiting flexibility of its material and when λ is higher than λ_{lim}, then the buckling will occur in the elastic region and the Euler formula is applicable, otherwise buckling will have a plastic behaviour [22].
We experimentally have determined the physicalmechanical characteristics of eucalyptus, including the value of the “limiting flexibility”. The results are shown in the Table 4.
Table 4
Experimental results for limiting flexibility of eucalyptus
Mechanical properties of eucalyptus wood  
 Modulus of  Proportional  Limiting 
E, kgf/cm^{2}  , kgf/cm^{2}  λ_{lim}  
Minimum  42 210.42  277.61  39 
Intermediate  79 442.47  296.30  52 
Maximum  117 781.86  329.13  59 
The Euler formula for “intermediateslenderness” elements will predict very high values of critical force that do not reflect the failure load seen in practice. To account for this, a correction curve is used for these elements. The J.B. Johnson formula has been shown to correlate well with real buckling failures [23], and is given by the equation (8) [24; 25].
${\text{\sigma}}_{\text{cr},J}={\text{\sigma}}_{adm}\frac{1}{E}\xb7{\left(\frac{{\text{\sigma}}_{adm}}{2\text{\pi}}\xb7\text{\lambda}\right)}^{2}$. (8)
Based on the results of the laboratory tests, the stressslenderness graph (Figure 6) has been prepared, where the Euler and Johnson curves are indicated for the minimum, intermediate, maximum and mean value of the MOE. Consequently, from the graph the magnitude of the limiting flexibility of eucalyptus has been determined as λ_{lim} = 57.5.
Figure 6. Stressslenderness graph for eucalyptus wood
Conclusion
According to [10; 11] the mechanical properties (MOE and MOR) of eucalyptus wood under compression parallel to the grain showed a strong correlation with the basic density and the linear and volumetric contractions of the tree.
A total of 41 samples were tested. Based on the tests and the calculations different magnitudes of limiting slenderness have been determined, i.e. from formula (7) we have three values 39, 52 and 59 and 57.5 from the Figure 6 for the average value of the MOE. As we know, the Euler formula for “intermediateslenderness” elements will predict highvalues of critical force. For this reason and safety, we consider that the greater value of limiting slenderness should be assumed. This is 59.
In this experimental investigation, the following mechanical properties of eucalyptus globulus were obtained (Table 5).
Table 5
Mechanical properties of eucalyptus globulus wood
Static bending  
Modulus of elasticity  E  104 180.68  kgf/cm^{2} 
Proportional limit  ${\text{\sigma}}_{\text{pl}}$  517.47  kgf/cm^{2} 
Ultimate strength  ${\text{\sigma}}_{u}$  800.46  kgf/cm^{2} 
Compression parallel to the grain  
Modulus of elasticity  E  79 811.58  kgf/cm^{2} 
Proportional limit  ${\text{\sigma}}_{\text{pl}}$  301.01  kgf/cm^{2} 
Admissible stress  ${\text{\sigma}}_{adm}$  431.90  kgf/cm^{2} 
Limiting flexibility  λ_{lim}  59 

Although some research has been performed on relevant eucalyptus species, the mechanical behaviour of Eucalyptus wood is far less known compared to other woods [5], e.g. pine or bamboo [22; 26–32].
In this investigation was determined a novel data about the limiting flexibility or socalled “critical slenderness ratio” for eucalyptus globulus, and it could be considered λ_{lim} = 59. However, there are required more researches about the physicalmechanical properties of the eucalyptus.
About the authors
David CajamarcaZuniga
Catholic University of Cuenca; Peoples’ Friendship University of Russia (RUDN University)
Author for correspondence.
Email: cajamarca.zuniga@gmail.com
Master of Science, PhD postgraduate student, Department of Civil Engineering, Engineering Academy of RUDN University; Docent of the Department of Civil Engineering at CUC
Av. De las Americas & Humboldt, Cuenca, 010101, Republic of Ecuador; 6 MiklukhoMaklaya St, Moscow, 117198, Russian FederationCristhian Carrasco
Catholic University of Cuenca
Email: cajamarca.zuniga@gmail.com
Civil Engineer graduate, Department of Civil Engineering
Av. De las Americas & Humboldt, Cuenca, 010101, Republic of EcuadorBelen Molina
Catholic University of Cuenca
Email: cajamarca.zuniga@gmail.com
Civil Engineer graduate, Department of Civil Engineering
Av. De las Americas & Humboldt, Cuenca, 010101, Republic of EcuadorReferences
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