A Finite Element Analysis

The case studied is that of a beam of height^h= 450 mm and length ^l= 1600 mm, cf.

Figure 6. The plane end-sections of the beam are assumed to remain plane during de-formation and the results from the simulations are presented as formal bending stresses, (^M=(^bh2=6)). To investigate the laminating eect, ve dierent lamination thicknesses

^hwere studied, namely 50, 25, 12.5, 6.25 and 3.125 mm. In each case the length of

8

l

Initial crack (zero width) M M

Δh

h

Crack path

Figure 6: A laminated beam with an initial crack of a length equal to the lamination thickness. The dashed line represents the crack path at crack propagation.

the initial crack was assumed to be equal to the lamination thickness, as indicated in Figure 6.

The elements representing the bondline and located along the crack path are 0.8 mm long. The bondline data needed to dene its behaviour include the strengths in pure mode I and II and the corresponding fracture energies. The values of these quantities were chosen in accordance with those reported by Wernersson (1994) i.e. 6.5 and 10 MPa strength in modes I and II, respectively, the corresponding fracture energies being 360 and 980 J/m².

The wood was modelled as a linear elastic orthotropic material with the engineer-ing constants of ^Ex=16800 MPa, ^Ey=560 MPa, ^Gxy=1050 MPa, and ^xy=0.45. The engineering constants used here were chosen from Serrano (1997) and are therefore not the same as those used in the previous section. The elements representing the wood are 4-node isoparametric plane stress elements or triangular constant strain elements for mesh rening. The deformed beam at maximum load, corresponding to a bending stress of 37.9 MPa in the outermost lamination, is shown in Figure 7.

The results of the ve dierent lamination thickness simulations are shown in Fig-ure 8. The ve simulations are represented by circles, whereas the dashed lines represent results based on Equation (2) with^Gc =^GIc = 360 J/m² and ^Gc =^GIIc =980 J/m². A major outcome of the simulations is that, as the lamination thickness decreases, the crack propagation is increasingly governed by mode II.

Another way of presenting the results of the nite element analyses is shown in Figure 9, displaying the strong nonlinearity. This gure presents the formal bending stress in the outer lamination as a function of the position of the tip of the fracture process zone (as measured from the symmetry line). For all the analyses, the load reached a plateau-value. Since this corresponds to the propagation of a fully developed fracture process zone, constant in shape, LEFM can be expected to provide an accurate estimate of the peak load, provided the proper mixed-mode value of ^Gcis employed.

Figure 10 shows the stress distribution along the bond line of the outermost lam-ination at peak load for the cases of ^h being 50, 12.5 and 3.125 mm, respectively.

In the area were no damage has occurred the stress distribution is more uniform for thicker laminations. More important however is that the sizes of the fracture process

800 mm x

Figure 7: The deformed beam at maximum load. The crack has extended 50{60 mm.

The displacements are magnied by a factor of 30.

10

0 10 20 30 40 50

Figure 8: Formal bending strength, 6^M=(^bh2), versus lamination thickness for a lam-inated beam 450 mm in height. The circles represent results of FE-simulations. The dashed lines represent results based on Equation (2) for^Gc=^GIc(dashed) and^Gc=^GIIc

(dashed-dotted).

Figure 9: Formal bending stress, 6^M=(^bh2), versus crack tip position for various lami-nation thicknesses.

550 600 650 700 750 800

Figure 10: Stress distribution along the bond line of the outermost lamination at peak load. The solid lines correspond to shear stress and the dashed to normal stress.

zones dier for dierent lamination thicknesses. For a 3.125 mm lamination the fracture process zone is approximately 48 mm long, while for all the thicker laminations, it is approximately 25 mm long. This is due to the fact that for thicker laminations, the normal stress is much higher, which in turn leads to a mixed mode fracture which is as-sociated with a lower fracture energy than the nearly pure mode II fracture taking place for thin laminations. For the 12.5 and 50 mm lamination the maximum normal stress is 2.3 and 2.8 MPa respectively, while it is only 0.5 MPa for the 3.125 mm lamination (cf.

Figure 10). Since the size of the fracture process zone is associated with the fracture energy this results in a smaller fracture process zone for thicker laminations.

It turns out that the mixed mode state varies during crack propagation. The current mixed mode state is dened by:

'= arctan (^s

n) (3)

where ^s and ^n are the relative displacements between two points on either side of the bondline. Indices ^s and ⁿ denote shear and normal deformation respectively. A failure in pure opening mode (mode I) corresponds to ^s = 0 ^{) '} = 0 and a pure shear crack propagation corresponds to ^n = 0 ^{) '} = 90 . The curves of Figure 11 are given in terms of the mixed mode angle ^', as dened by (3) versus the crack tip position. The value of ^' is calculated at the peak shear stress position. Clearly, as the lamination thickness decreases, the failure is more dominated by mode II (shearing along the lamination).

Finally, Figure 12 shows how the dierent contributions of mode I and mode II fracture depend on the lamination thickness. Again, it can be seen that a thin lamination yields almost pure mode II fracture.

12

0 10 20 30 40 50 60 20

30 40 50 60 70 80 90

Crack tip propagation (mm)

Mixed mode angle, degrees

3.125 mm 6.25 mm 25 mm 12.5 mm 50 mm

Figure 11: Mixed mode angle^'as dened in Equation (3) versus crack tip position for dierent lamination thicknesses.

0 10 20 30 40 50

0 100 200 300 400 500 600 700 800 900 1000

Lamination thickness (mm)

Crack propagation energy (N/m)

Figure 12: Energy consumption for dierent lamination thicknesses at the propagation of a fully developed fracture zone (solid line). The dashed lines represent the contributions of mode I (dashed) and mode II (dashed-dotted), respectively.

5 Conclusions

The following conclusions can be drawn from the work presented in this paper:

 For a laminated beam with a weak zone, of small dimensions in comparison to the dimensions of the beam and located in the outermost lamination, the stress redistribution around and in this weak zone is much less than predicted by con-ventional beam theory.

 The failure mode along the outermost bondline of an initially cracked laminated beam depends on the lamination thickness. Thinner laminations tend to lead to pure shear failure along the bondline while thicker laminations lead to failure taking place in mixed mode.

Appendix I. References

Falk, R. H., and Colling, F. (1995). \Laminating eects in glued-laminated timber beams" J. Struct. Engrg., ASCE, 121(12), 1857{1863.

Falk, R. H., Solli, K. H. and Aasheim, E. (1992) \The performance of glued laminated beams manufactured from machine stress graded norwegian spruce." Rep. no. 77.

Norwegian Institute of Wood Technology, Oslo, Norway.

Foschi, R. O., and Barrett, J. D. (1980). \Glued-Laminated Beam Strength: A Model."

J. Sruct. Div., ASCE, 106(8), 1735{1754.

Gustafsson, P. J. and Enquist, B. (1988) \Trabalks h allfasthet vid ratvinklig urtagning.

!strength of wood beams with a sharp notch]" Report TVSM-7042, Lund Institute of Technology, Division of Structural Mechanics, Lund, Sweden (in Swedish).

Kollmann, F. F. P and C^ote, W. A. (1968). \Principles of wood science and technology.

Vol. 1." Springer Verlag. Berlin. Germany.

Larsen, H. J. (1982). \Strength of glued laminated beams. Part 5." Report no. 8201.

Institute of Building Technology and Structural Engineering, Aalborg University, Aal-borg, Denmark.

Petersson, H. (1994). \Fracture design criteria for wood in tension and shear." Proc., Pacic Timber Engrg. Conf. Timber Research and Development Advisory Council, Queensland, Australia. Vol. 2, 232{239.

Serrano, E. (1997). \Finger-joints for Laminated Beams. Experimental and numerical studies of mechanical behaviour." Report TVSM-3021, Lund University, Division of Structural Mechanics, Lund, Sweden.

Wernersson, H. (1994). \Fracture characterization of wood adhesive joints." Report TVSM-1006, PhD thesis, Lund University, Division of Structural Mechanics, Lund, Sweden.

14

Appendix II. Notation

The following symbols are used in this paper:

b = beam width#

Ex^{ E}y = moduli of elasticity in grain and across grain direction#

fmbeam = beam bending strength#

ftlam = tensile strength of lamination#

Gc = fracture energy, critical energy release rate#

Gxy = shear modulus#

h = beam depth#

^h = lamination thickness#

I = cross-sectional moment of inertia#

klam = laminating factor#

l = beam length#

Mc = critical bending moment#

 = beam depth ratio#

n^s = relative displacements across bondline#

 = Poisson's ratio# and

' = mixed mode angle at cracktip.

Chapters 3 and 4 from

Finger-Joints for Laminated Beams