Empirical wood material model for three dimensional finite element analysis of a nail joint

Jung-Pyo Hong Ph.D. Candidate

Department of Wood Science, University of British Columbia Vancouver, Canada

J. David Barrett

Professor Emeritus Department of Wood Science, University of British Columbia

Vancouver, Canada

Summary Development of a wood material model for three-dimensional finite element analysis of a nail joint was presented. Anisotropic plastic material theory and the nail beam and wood foundation method were used to determine the material constants of the model. Wood behaviour under a nail was modeled by a three-dimensional wood foundation model. From nail embedment tests the required three-dimensional foundation modulus was determined. A feasibility study of a statistical approach to the material model was conducted. With simple experiments the material model developed showed a very efficient simulation on wood compression behaviour. An example of the application of the material model in three-dimensional finite element analysis of a nail joint was presented. A consecutive paper includes an extensive study on the finite element analysis of nail joints.

1. Introduction

Information from uniaxial loading tests of a wood connection is not often enough for design of a connection or development of new connections. Knowledge on strength behavior inside a connection might be more important. For example, to make a connection more ductile a design should be focused on avoiding the high stress concentration or, understanding the stress distribution inside mortise and tenon joint might be essential to study the optimization of its size and shape. To this end, a stress analysis of the connections in three directions should be required. As an efficient tool for this purpose three-dimensional finite element (3D FE) methods have been preferred. Provided that mathematically rigorous material model of wood was established the analysis could be readily performed by 3D FE model with solid elements. This is one of advantages of 3D FE analysis. It does not need any member theories such as beam, plate and shell. However, such a wood material model is yet far from numerical theories of material available currently. Alternatively, as the most plausible material model of wood, orthotropic material theory has worked well for a numerical model of wood in the elastic range (Bodig 1982). For a more advanced model accounting for nonlinear behavior of wood anisotropic plasticity theory could be used (Moses 2000, Chen et al. 2003 and Kharouf et al. 2005). These material models for 3D FE analysis of a wood connection have provided acceptable results only in limited applications when the elastic behavior or mild nonlinear behavior of wood was concerned (Patton-Mallory 1996, Moses 2000 and Guan et al. 2001). To date only bolted connections were explored by 3D FE analysis. Any attempts with a nail joint which shows highly nonlinear behavior have never been made using 3D FE method.

In general, wood strength properties required for 3D FE material model were determined according to standard test procedures. The model with “the standard strength properties” represented only the average strength behavior of wood in macroscopic level. The model proved to be appropriated for bolted connections which have comparatively large contact area between bolt and wood. However, in a nail joint, intricacies of wood behavior under a nail shank would not follow the average

standard strength behavior. Though this discrepancy of wood strength behaviour cannot be clearly explained it is thought that in microscopic level the high anisotropicity of wood and the porous wood fiber are related. Therefore, the “beam on a foundation” theory has been the most highlighted approach to model wood behaviour under a nail (Foschi et al. 1977 and Ni 1997).

The objective of this study was to develop a wood material model for three-dimensional finite element analysis of a nail joint. Using the anisotropic plastic material theory and the wood foundation theory a method to construct the 3D wood material model and the 3D wood foundation model was presented. An example of the application of the material model to 3D FE model of a nail joint was illustrated. An extensive study on 3D FE analysis of nail joints was reported in the accompanying paper.

2. Three-dimensional wood material model

2.1. Assumptions

Transverse isotropic plasticity with bilinear stress-strain relations was adopted for 3D FE material modeling of wood. Based on compressive strength properties equality of bilinear stress-strain relationships in tension and compression was assumed. Although introducing the different yield properties in tension was possible theoretically, it was abandoned because it caused trade-off’s including the solution convergence and the increase of experimental expense. Transversely averaged Poisson’s ratios taken from Wood Handbook were used (Forest Products Laboratory 1999). Yield points in the bilinear stress-strain relationships were determined by bilinear curve fitting to the load-deformation curves from wood compression tests. For shear strength properties

Table 1 Material constants for 3D FE wood material model Requisite constant Direction1 Method of determination

Elastic MOE X, Y, Z Compression test (or nail embedment test)

Elastic Shear modulus XY, YZ, XZ Ratio to Elastic MOEY, given in Wood Handbook ( //RT//LRTL E07.0G,E7.0GandG ⋅=⋅= )

Poisson’s ratio XY, YZ, XZ Average values from Wood Handbook 2 Tensile yield stress X, Y, Z Equal to compressive properties Tensile tangent modulus X, Y, Z Equal to compressive properties Compressive yield stress X, Y, Z Compression test (or nail embedment test) Compressive tangent modulus X, Y, Z 0.01 × elastic MOE 3 Shear yield stress XY,YZ,XZ Coupled bilinear constitutive model4 Shear tangent modulus XY,YZ,XZ TAN.//TAN.// E007.0andE07.0 ⋅⋅ 1 Y=longitudinal and X=Z=transverse 2 Except for the ratio in XY : ( )LTLTTL E/E⋅=νν 3For wood foundation model, 0.035 × elastic MOE or experimental value. 4Theoretical model (E.P Saliklis et al 2003)

Ex.) for yield strain in TL : ( ) ( )( )( ) TLt

0TLy G

EEE2

⊥

⊥⊥

⊥ ⋅−⋅

=σ

γ

G = shear modulus yγ = shear yield strain in bilinear shear stress-strain curve

0σ = stress intercept in bilinear normal stress-strain curve

shear moduli, shear yield points and shear tangent moduli were estimated using theoretical relationship with the parallel and perpendicular to grain strength properties (E.P Saliklis et al 2003). For details, methods of determination of the material constants were summarized in Table 1.

In this study all relevant finite element analyses were performed using the commercial software, ANSYS® 8.0 (elements; SOLID45, CONTA174 and TARGE170).

2.2. Three-dimensional wood foundation model

Foschi proposed one-dimensional exponential wood foundation model assuming wood beneath a nail as a nonlinear foundation (Foschi 1974). Based on this approach a procedure to expand the one-dimensional foundation to the three-dimensional one was devised. A certain cylindrical volume of wood where a nail shank was embedded was considered as a 3D nonlinear wood foundation. During 3D FE meshing the 3D foundation should be integrated into the final wood volume (Fig 1). It was assumed that the strength behaviour of the wood foundation follows the foundation modulus.

The foundation modulus should be determined from the nail embedment test in three-direction.

2.3. Wood foundation modulus

The material constants as defined in Table 1 were required also for the wood foundation model. Note that independent material constants obtained from experiment are only the elastic moduli and the yield stresses. The bilinear stress-strain relationships for the foundation should be induced from the load-embedment curves of nail embedment tests. Assumed that the bilinear load-embedment curves have correlations with the corresponding bilinear stress-strain curves the equations could be written as follows [1].

Lrk

AE ////F ⋅⋅

⋅=π

[1 a]

Lrk

BEF ⋅⋅⋅= ⊥

⊥ π [1 b]

yy wC ⋅=ε [1 c]

Where,

⊥F//F E,E = elastic foundation modulus in parallel/perpendicular to grain

⊥k,k // = slope of load-nail embedment curves in parallel/perpendicular to grain

yw = yield deformation in load-nail embedment curve

yε = yield strain of wood foundation

r = radius of nail shank

L = length of wood block

B,A and C = conversion factor

Fig 1 Incorporation of 3D wood foundation in a nail-embedment block

Then, the unknown values, only three conversion factors A, B and C remain. By trial-and-error run of the nail embedment FE analysis with trial inputs of the factors they should be determined when the simulated load-embedment curve matched the experimental curve. The factors are dependent on the dimension of the foundation and the size of dowel. They are independent of wood species because the factors are geometry-associated constants. A dimension of the foundation could be determined flexibly according to size of dowel, expectancy of deformation and purpose of analysis. In this study, the foundation having the diameter of 4.5 times nail diameter was selected and after the trial-and-error runs, the conversion factors were found as A=0.95, B=0.65 and C=1.9. The corresponding independent material constants were determined as //FE =1093 MPa with yε =0.019

and, ⊥FE = 275 MPa with yε =0.027.

3. Experimental studies

3.1. Materials

Three pieces of 105 (width) ×105 (depth) × 4000 (length) (mm) Douglas-Fir conditioned at moisture content 15% were selected as the mother wood for test specimen. Then, for matching sampling purpose, each piece of the lumber was cut into the respective test specimens for wood compression test, nail embedment test, nail withdrawal test, nail joint test and ASTM D 143 compression perpendicular to grain test. A galvanized common wire having the shank diameter of 3.35mm and the length of 65mm was used. According to ASTM F 1575 the nail yield stress of 756 MPa was determined by the bending test. For a single nail test 2.15mm-thick T-shape steel plate was used.

3.2. Compression test of wood

Wood compression tests were conducted with 25×25×102 (mm) (1×1×4 in.) small prismatic specimen in longitudinal, radial and tangential directions according to ASTM D 143 (Fig 2). The specimen should be subjected to full contact area loading so that uniform compressive stress could be induced. In order to eliminate effect of unfavorable failures at the region of loading contact ends, a pivot gauge equipment was installed to the test specimen. The equipment had the gauge length of 50mm and played a role of transferring the deformation to the measurement of a direct current differential transformer (DCDT) by pivot action.

3.3. Nail embedment test

Nail embedment wood block was cut to the dimension of 51×38×51(mm) (2×1.5×2 in.) in three directions. Additionally, having the different size of 51×38×38 (mm) thirty 45-degree to grain wood blocks were prepared for the purpose of the model verification. A half-hole was drill-grooved on the loaded surface of the block. Perfectly fit nail groove was used to remove any influences of pre-stresses which might be produced due to the smaller wood groove than the nail shank. For each principal direction 15 tests were performed with loading rate of 0.1mm/sec.

Fig 2 Compression test setup

4. Results

4.1. Compressive modulus of elasticity of Douglas-Fir

The stress-strain curve from each compression test was fitted by a bilinear elastic-plastic curve. Then, the average compressive MOE and the average yield strain should be determined. The average data were taken as the representative material constants. Compared with values from literatures the MOE’s did not show much difference (Table 2). It means that the compression tests might not be necessary for the constant determination if the tolerance allowed.

4.2. Simulation on wood compression

Compared with the average bilinear curve fits on the experimental curves the load-deformation curves of 3D FE simulation showed good results (Fig 3). As shown in the perpendicular to grain curve it was hard to define a post-yield modulus. This is why in the wood material model the tangent modulus was assumed as 0.01×elastic MOE.

Since the compression test was devised to induce the uniform stress over the wood specimen it can be thought that the simulation involved only simple constitutive material behaviour. Thus, ASTM D143 standard compression perpendicular to grain test was performed then, the strength behavior was compared with the simulated results for the validation of the wood material model. Note that the wood specimen is subjected to partial contact area loading which generates non-uniform stresses over the wood and wood shear failure at the edges of a loading steel block. Although the plasticity-based material model is not capable of prediction for shear brittle failure the 3D FE simulation showed acceptable results (Fig 4).

Table 2 Compressive MOE and yield stress in three directions

Repetition Yield stress (MPa)

MOE (MPa)

Yield stress1

[Jessome] (MPa)

MOE2 [Wood Handbook]

(MPa)

Longitudinal 45 44.5 16961 34.1 13600 Radial 45 4.6 923 - 925 Tangential 45 5.3 740 - 680 Transverse 3 - 5.0 832 6.0 803 1 Values from A.P. Jessome’s report (Jessome 1977)

2 Values from Wood Handbook. 3 Average values of the radial and the tangential (perpendicular to grain direction)

Fig 3 Compression of simulated results in wood compression: parallel to grain (left) and perpendicular to grain (right)

4.3. Nail embedment behavior

Comparisons of the load-embedment curves between the experiment and the simulation were shown in Fig 5, Fig 5 and Fig 7Fig 8. Good matching results in parallel/ perpendicular to grain direction should be always obvious because the process of determination of the conversion factors forced the simulated curves to match the experimental curves. However, the simulated results of the 45 degree to grain test could be taken as an estimate of the model validation. The simulated load-embedment curve of 45-degree to grain test revealed a good match to the experiment. It means that complex internal stresses in wood were represented successfully by the constitutive model developed.

0

5000

10000

15000

0.00 0.50 1.00 1.50 2.00Embedment (mm)

Loa

d (N

) .

ExperimentNo foundation model3D FEA

0

2000

4000

6000

8000

0.00 0.50 1.00 1.50 2.00Embedment (mm)

Loa

d (N

) .

Experiment3D FEA

0

1000

2000

3000

4000

0.00 0.50 1.00 1.50 2.00Embedment (mm)

Loa

d (N

) .

Experiment3D FEA

Fig 4 ASTM D 143 perpendicular to grain compression test and the 3D FE simulation

Fig 5 Perpendicular to grain load-nail embedment curves

Fig 6 Parallel to grain load-nail embedment curves

Fig 7 Load-nail embedment curves of 45 degree to grain test

Fig 8 Stress ratio in the 3D FE 45-degree to grain model

In Fig 5, the dotted-curve showed the simulated curve from a model using only the wood material model without the foundation portion. The curve is far away from the experimental results. It confirmed that wood behavior under a nail does not follow the standard strength properties, which justified the wood foundation model approach.

4.4. Feasibility study on statistical approach to the three-dimensional material models

Since the material model developed has only two experimental independent material constants including elastic MOE and yield point, applying a technique of correlated random variable generation to the independent constants a random 3D wood material model can be constructed. However, with this method a 3D FE simulation requires much execution time to obtain statistically enough number of results on a common personal computer basis. Thus, simply using linear relationships amongst the independent constants approximations of 95th percentile limit and 5th percentile limit in the material model were used to study feasibility of statistical approach.

Starting with picking Nth percentile parallel to grain MOE from the 2-P Weibull distribution fit, the required independent material constants should be determined by the linear relationships with the picked MOE. Then, applying the recipe for the 3D material model in Table 1 Nth percentile wood material model should be established. Note that all the test specimens were obtained by the matching sampling method along the length of the mother lumber. Thus, the linear relationship amongst the independents was obtainable in this study (Fig 9).

)385.0R(764.2E002.0

)405.0R(52.16E0016.0

)467.0R(9.1114E1148.0E

2y

2////y

2//

=+×=

=+×=

=−×=

⊥⊥

⊥

σ

σ

����������

⊥//, ��� � �� � � ������� � �� � � � ��� � � ������������ � ��� � � !� �" � �� �#!�$���&%��'�������'#� ()+*� � ��� #!� ��� #�#

Fig 9 Two parametric Weibull distribution of parallel to grain MOE and the linear relations with other independent constants

Fig 10 Estimates of 95th percentile limit and 5th percentile limit in nail embedment curve-

parallel to grain (left) and perpendicular to grain (right)

Desirable estimations of the percentile limit curves were produced from the model analysis (Fig 10). It was thought that better prediction of variations in post-yield could be achieved if the correlated random variable generation method was applied.

4.5. Three dimensional model of a nail joint

With the wood foundation model incorporated, three-dimensional FE model of nail joints were developed. The model provided successful outputs (Fig 11). The details of the analysis can be found in accompanying paper.

5. Conclusions

Three-dimensional wood-related material model developed showed good capability of three-dimensional finite element analysis of wood behaviour. The feasibility study on the statistical analysis revealed that the material model has the good flexibility for the material constant input. The wood foundation approach worked well for the modeling of wood under a nail. It enabled 3D FE analysis of a nail joint.

6. References

[1] Bodig, J. and B. A. Jayne. 1982. Mechanics of wood and wood composite. Van Nostrand Reinhold Company Inc.

[2] Moses, D. M. 2000 Constitutive and analytical models for structural composite lumber with applications to bolted connections. Ph.D dissertation. Dept. of Civil engineering, Univ. of British Columbia.

[3] Chen, C. J. T. L. Lee and D. S. Jeng. 2003. Finite element modeling for the mechanical behavior of dowel-type timber joints. Computers and structures. 81:2731-2738.

[4] Kharouf, N. G. McClure and I. Smith. 2005. Postelastic behavior of single- and double-bolt timber connections. J. of structural engineering. 131(1):188-196.

[5] Patton-Mallory, M. 1996. The three-dimensional mechanics and failure of single bolt wood connections. Ph.D dissertation. Civil engineering department, Colorado state university.

[6] Guan, Z. and P. Rodd. 2001. DVW-local reinforcement for timber joints. J. of structural engineering. 127(8):894-900.

[7] Foschi, R. O and T. Bonac. 1977. Load-slip Characteristics of connections with common nails. Wood Science. 9(3):118-123.

[8] Ni. C. 1997. Behaviour of nailed timber joints under reversed cyclic load. Ph.D dissertation. Faculty of Forestry and environmental management, The univ. of New Brunswick.

[9] Forest Products Laboratory. 1999. Wood handbook-Wood as an engineering material. Gen. Tech. Rep. FPL-GTR-113.

[10] Saliklis, E. P., T. J. Urbanik and B. Tokyay. 2003. Bilinear modelling of cellulosic orthotropic nonlinear materials. J. of pulp and paper sci. 29(12):407-411.

[11] Foschi, R. O. 1974. Load-slip Characteristics of Nails. Wood Science. 7(1):69-76. [12] Jessome, A.P. 1977. Strength and related properties of woods grown in Canada. Forestry

Tech. Rep. 21. Eastern forestry products laboratory, Ottawa, Ontario.

Fig 11 Three-dimensional FE analysis of a single nail joint with steel plate