EVALUATION OF OFF-AXIS WOOD COMPRESSION STRENGTH
Nilson Tadeu Mascia
School of Civil Engineering, Architecture and Urbanism-State University of Campinas
Caixa postal 6021-Unicamp-Campinas-SP-Brazil
[email protected]
Elias Antonio Nicolas
[email protected]
Rodrigo Todeschini
[email protected]
ABSTRACT
This research focuses on the study of anisotropic material failure criteria, specifically Tsai-Wu tensor failure criterion, Tsai and
Wu, with theoretical and experimental applications for wood.For materials as wood, when applying a compression or tension
load, in an inclined direction of a small angle in relation to its grain, a great reduction of the strength occurs. This reduction is
related directly to the wood anisotropic nature. Hence, it is necessary to use for wood a criterion of strength that considers this
anisotropic behavior and the asymmetry of strength, or either, different compression and tension strengths in one same
direction. Due to the complexity of the phenomena of wood rupture and its composites, the predictions using failure theories
had still not been developed completely. Consequently, empirical methods are used, which under some circumstances are
reasonable accurate. Hankinson´s formula is one of these methods and it has been applied frequently for wood, particularly in
the compression in directions inclined in relation to grain. According to Liu ,Bodig and Jayne, Hankinson´s formula is well
adjusted to predict the off-axis wood strength under compression or tension.This present research had been carried through
o
compression tests in wood specimens, of the Brazilian species Goupia glabra, in the directions inclined in relation to grain (0 ,
o
o
o
o
o
o
15 , 30 , 45 , 60 , 75 and 90 ), having as main purpose to compare the results of tests with the values obtained by
Hankinson´s formula and the Tsai-Wu criterion. Observe to use this criterion, it was necessary to perform tension, shear and
biaxial tests too. In general, the most important conclusions that were drawn can be summarized as follows: the prediction of
Tsai-Wu criterion was close of Hankinson’s formula and also the compression tests results; Tsai-Wu criterion presents good
results, facility in utilizing when comparing to other criteria due to especially its tensor form, and can be applied to evaluation
wood strength and other anisotropic materials; Hankinson´s formula is more practical and even empirical presented the fitting
next to the values obtained in compression tests.
Introduction
In the whole, the use of strength criteria has been firstly initiated for homogeneous and isotropic materials. To other materials
as non-homogeneous and anisotropic that have not exhibited simple characteristics was necessary to develop specific criteria.
In general, for fibrous materials as wood it had not available in technical literature a criterion that took into account the
anisotropic nature and non-symmetrical strengths. In other words, that considered the fiber directions as an important
characteristic for analyzing the wood strengths. Hakinson´s formula, even being an empirical procedure and also restricts to an
axial stress field, could be considered the first successful criterion for wood.
With the development of new materials by the modern engineering, general strength criteria were necessary to be established.
Tsai-Wu criterion was an important of them especially by considering the difference in strengths, different material
symmetries and by satisfying the invariant requirements of coordinate transformation.
The aim of this paper is to compare experimental results of off-axes compression tests with both the Tsai-Wu criterion and
Hankinson´s formula. To achieve this, compression, biaxial compression, tension and shear tests were necessary to be
performed.
Theoretical Analysis
The main objective of a strength criterion is to interpret the combined loadings ( bi and tridimensional stresses) that conduct to
an eventual failure. As for instance, whereas the tension bar safety is defined by comparing the strength determined in tension
tests with the current tension stress applied in the bar, on the other hand, to judge the safety of a structure subjected by
complex loadings would be needed and inconvenient to perform for each solicitation a respect experimental test. For these
cases, a strength criterion is intended to be developed by taking into consideration a limited number of material parameters.
Each criterion is a hypothesis of working. Firstly, we identify arbitrarily the phenomenon responsible for the failure, after
several mechanical analyses in relation to possible combinations of loadings and finally the validity of the criterion is verified
by comparing the real behaviour in the complex combination of solicitations. This implies that the establishment of a criterion
is incoherent due to the variety of materials used in engineering.
According to Bodig[1], the strength criteria were developed to isotropic and homogeneous materials, possessing a linear
mechanical behaviour, or better, linear stress- strain relationship until the failure. Consequently, when these criteria were
applied to wood and wood composites the results are not adequate because we have to consider the non-homogeneity and
non-elastic behaviour of this material. Bodig observes that empirical methods can be used to wood and wood composites and
in several circumstances are accurate. Hankinson´s formula constitutes an excellent example that has been frequently applied
to wood, especially in compression stresses, and can be written by:
σθ =
f0 ⋅ f90
n
(1)
n
f0 ⋅ sin θ + f90 ⋅ cos θ
Where: σ θ : off-axis compression stress; f0 : parallel to fiber strength ; f90 : perpendicular to fiber stress; θ : fiber direction
orientation,; n : exponent.
Tsai and Wu [2] developed a general strength criterion to evaluate anisotropic materials. Equation 2 is the mathematical way in
tensor form to describe this criterion:
Fi ⋅ σ i + Fij ⋅ σ i ⋅ σ j = 1 (i, j =1 to 6)
(2)
Where Fi and Fij are strength tensor of second and fourth rank, and σ i are stresses.
Considering now an orthotropic material and a plane stress field in which 1 and 2 are the principal directions, Equation 2 can
be rewritten by:
F1 ⋅ σ 1 + F2 ⋅ σ 2 + F11 ⋅ σ 12 + F22 ⋅ σ 22 + 2F12 ⋅ σ 1 ⋅ σ 2 + F44 ⋅ σ 42 = 1
Where: σ1, σ2 are the normal stress, σ4 is the shear stress, F1 =
F44 =
1
fv24
, F11 =
1
1
, F22 =
with
f t 1 ⋅ f c1
ft 2 ⋅ fc 2
(3)
1
1
1
1
−
, F2 =
−
,
f t 1 f c1
ft 2 fc 2
f t : tension strength,
fc :compression strength, fv : shear strength and
F12 < ± F11 ⋅ F22 . The strength coefficient F12 is the interaction coefficient, which is obtained by biaxial test, Wu [3] and
Mullner et al. [4].
According to Tsai-Wu [2] certain stability conditions are associated to the strength tensors and the magnitude of interaction
terms are related to the following inequality:
− F11 ⋅ F22 < F12 < F11 ⋅ F22
(4)
Geometrically, this inequality insures that the failure surface will be ellipsoidal in tridimensional stress field and an elliptic in
plane field. The shape of this surface cannot be open-ended like a hyperboloid or a hyperbole. Figure 1 shows these surfaces.
Figure 1- Closed and open-ended surfaces
Table 1 presents some values of the the strength tensors of the second and fourth rank. All these terms are determined by
experimental tests of compression, tension and shear except F12 that it is needed biaxial tests.
Table 1 – Strength Tensors
1
1
F2 =
−
ft 2 fc 2
1
1
F1 =
−
ft 1 fc1
F11 =
1
ft 1 ⋅ fc1
F22 =
1
ft 2 ⋅ fc 2
F44 =
1
fv24
F12 < ± F11 ⋅ F22
Where ft means tension strength, fc compression strength and fv shear strength.
Literature Overview
On the basis of the Tsai-Wu strength criterion, Hasebe and Usuki [5] carried out a strength criterion applied to wood species
Japanese Cedar.
This way, taking into account Equation 2, the authors considered the influence of the stress in direction 2, adopted as
tangential or radial axis, less significant in relation to direction 1, adopted as longitudinal axis and proposed the following
equation to express the model:
(5)
F1 ⋅ σ 1 + F11 ⋅ σ 12 + F44 ⋅ σ 42 = 1
Where the following terms F1 ,F11 and F44 were determined by experimental tests .
LIU [6] used the mechanical properties of the Sitka spruce wood species showed in Table 2 to determine the strength tensors
presented in Table 3 in order to apply the Tsai-Wu criterion. By working with the stability condition F12 was estimated as
lim = F11 ⋅ F22 = 5,65 ⋅ 10 −3 MPa −2 .
Table 2 – Mechanical properties of “Sitka spruce”.
Properties
Values (MPa)
ft 0
79,30
f t 90
2,55
fc 0
38,68
f c 90
4,00
fv
7,93
Table 3 – Strength tensors.
Parameters
Values
F1
-1,32 x 10-2 MPa-1
F11
3,26 x 10-4 Mpa-2
F2
1,42 x 10-1 Mpa-1
F22
9,80 x 10-2 Mpa-2
F12
− lim ≤ F12 ≤ lim
F44
1,59 x 10-2 MPa-2
Off-axis compression stress (MPa)
With three valous of F12 it was performed curves, accordingly Figure 2, to compare the best fit with Hankinson´s formula. As a
result Liu concluded that F12 equals zero was the most accurate fitting.
40
35
30
25
20
15
10
5
0
0
15
30
45
60
75
90
Fiber Orientation (degrees)
Hankinson
F12 = - lim
F12 = 0
F12 = + lim
Figure 2- Tsai- Wu criterion,Hankinson´s formula according to Liu´s study.
Eberhardsteiner [7] presented results of biaxial tests using several specimens with fiber orientation in different angles (0o, 7,5o,
15o, 30o and 45o) in relation to the loading. These experimental data were compared to the Tsai-Wu strength criterion. Figure 3
shows some results in which we can observe the shape of the dot points leads to an ellipsis.
Figure 3 – Biaxial test results.
Experimental Procedure
Uniaxial Compression Test
Off-axis compression tests in the following directions 15o, 30o, 45o, 60o and 75o using a Brazilian wood specimens were
performed in this work. For each angle it was tested 12 specimens. The dimensions s of specimen (5 cmx5cmx15cm) followed
the Brazilian Code NBR 7190 [8]. The specific gravity was 0,7 Kgf/cm3 and moisture content reached 12 %.
Biaxial Compression Test
In order to carry out biaxial compression test a special equipment was developed as we can see in Figure 4. This equipment
was connected with a Universal test machine, EMIC, and, where a vertical and other, horizontal, loads were applied to surface
of the specimen to accomplish a biaxial stress field.
Figure 4– Biaxial device and biaxial compression test.
The specimen that we adopted in this research had the following dimensions: 4 cmx4cmx4cm (See Figure 5) and the specific
gravity, the moisture content and the species were the same used in uniaxial test. A total of 62 specimens was tested.
Figure 5– Wood specimen to biaxial compression test (Dimensions: 4cmx4cmx4cm).
Results and Discussion
From the experimental results we can demonstrate for this current wood sample that:
-the exponent n equals 1,5 of Hankinson´s formula was the best adjust with the experimental data. Figure 6 exhibits this and
shows two other curves with n = 2 and n =2,5.
Off-axis compression stress (MPa)
70
60
50
40
30
20
10
0
0
15
30
45
60
75
90
grain angles (degrees)
test
n=1,5
n=2,0
n=2,5
Figure 6 – Hankinson´s formula with different exponents and experimental data.
-the strength tensor F12 equals + lim ( maximum value) was the best fit for the Tsai-Wu strength criterion. Similar to Figure 5 ,
the next figure gives the fitting between experimental results and the Tsai-Wu criterion for three cases: F12 = -lim (minimum
value) ,F12 = 0 and F12 = + lim.
Off-axis compression stress (MPa)
100
90
80
70
60
50
40
30
20
10
0
0
15
30
45
60
75
90
grain angles (degrees)
test
F12 = - limit
F12 = 0
F12 = + limit
Figure 7 – Tsai-Wu criterion with different F12 and experimental data.
To summarize the above considerations, we plotted Figure 8 in which we can observe the best fits between the two procedure
and the experimental results.
Off-axiscompression stress (MPa)
70
60
50
40
30
20
10
0
0
15
30
45
60
75
90
grain angles (degrees)
test
n=1,5
F12 = + limit
Figure 8 – Hankinson´s formula, Tsai-Wu criterion and experimental data.
Conclusions
A comparison between experimental results from off-axis compression tests of Brazilian wood species and two strength criteria
was performed in this paper. Hankinson´s formula even an empirical criterion and restricts to uniaxial stress field was the first
criterion employed and the Tsai-Wu strength criterion used for both anisotropic materials and multi-axial stress case the
second.
Hankinson´s formula fitted well when the exponent was 1,5. In general, we found in technical literature n = 2 as usual value for
Hankinson´s formula in compression.
The Tsai-Wu criterion presented good approach when the coefficient of interaction F12 was the high positive value ( + lim).
We also noted that to determine F12 was necessary to construct a special equipment in order to measure two dimensional
loads.
As a final conclusion, we observe that Hankinson´s formula is still a good procedure to evaluate off-axis wood compression
strength and it is possible to adopt the Tsai-Wu criterion also. This criterion that is based on a theoretical formulation is
considered to be a general strength procedure, even being more complex to be defined and applied as we presented in this
article.
Acknowledgements
The authors gratefully acknowledge CNPq (n. 301504/2004-0) and FAPESP(n. 2004/15481-5), Brazilian Foundations for the
financial supporting of this research.
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