NUMERICAL INVESTIGATION OF THE LOAD CARRYING CAPACITY OF LAMINATED VENEER LUMBER (LVL) JOISTS WITH HOLES

Transcription

1 NUMERICAL INVESTIGATION OF THE LOAD CARRYING CAPACITY OF LAMINATED VENEER LUMBER (LVL) JOISTS WITH HOLES Manoochehr Ardalany 1, Bruce L. Deam 2, Massimo Fragiacomo 3 ABSTRACT: Predicting the load carrying capacity of wood beams with holes requires a good model of the complex microscopic material properties that influence crack development and propagation. Classical stress analysis methods are not able to be used to predict beam strengths because the stress distribution around a hole varies from tension to compression. Numerous methods, including the Weibull weakest link theory and probabilistic fracture mechanics, have been used to predict the failure loads of wood beams with the large holes often required to install building services. Crack development is best characterized using the Linear Elastic Fracture Mechanics (LEFM) method. The linear elastic Finite Element Method (FEM) was used to identify highly stressed regions within LVL beams where the stress intensity is most likely to initiate cracking. Their Stress Intensity Factors (SIFs) were then calculated at the crack tip using FEM by introducing progressively increasing crack lengths. The LVL beams were modelled using ABAQUS membrane (shell-planar) elements and orthotropic material properties. Fine element meshes were used, along with element singularity at the crack tip for the stress intensity and contour calculations. The current paper uses LEFM to predict the strengths of a series of Laminated Veneer Lumber (LVL) beams that had a range of hole-diameters and -positions within the beam. A closed form elastic solution was shown to provide poor failure predictions. Mixed mode fracture is always likely in beams containing holes, so strengths were calculated using several mixed mode fracture criteria proposed by others. The LVL beam strengths were found to be well predicted by one of the mixed mode fracture criteria. KEYWORDS: LEFM, FEM, fracture mechanics, mixed mode fracture criteria, LVL, holes, stress intensity factor. 1 INTRODUCTION 123 Openings are often required to allow services to pass through timber beams. Their presence, location and size relative to the beam depth influences the beam strength [1]. In extreme loading conditions, cracks will develop around the holes and extend, causing beam failure at much lower loads. The normal beam design procedures are therefore unable to predict the load carrying capacity of cracked wood beams. Fracture mechanics methods are required to avoid the impossible infinite 1 Manoochehr Ardalany, Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand. mar149@student.canterbury.ac.nz 2 Bruce L. Deam, University of Canterbury, Christchurch, New Zealand. bruce.deam@canterbury.ac.nz 3 Massimo Fragiacomo, Department of Architecture, Design and Urban Planning, University of Sassari, Italy. fragiacomo@uniss.it stresses that are predicted at the crack tip using linear elasticity. The strength computed using fracture mechanics provides a conservative estimate (lower bound) of the ultimate beam strength [2]. Fracture mechanics uses SIFs to predict the strength from the stresses at the crack tip. However, closed form solutions are difficult to formulate for the SIFs. Some formulations (e.g. [3]) require complex calculus and are only available for limited crack lengths, loading conditions and boundary conditions. Another, popular method for stress intensity calculation uses the linear elastic finite element method (FEM). FEMs are well developed and have fewer limitations on the boundary and loading conditions. However, FEMs require additional material properties and the crack tip region needs to be modelled with special elements and fine meshing. Commercial finite element programs such as ABAQUS [4] incorporate these features for fracture modelling. The failure mode for a beam with a hole is normally a combination of fracture modes I (opening) and II

2 (shearing) [5], producing mixed mode fracture. Therefore, it is important to use an accurate mixed mode fracture criterion to provide useful strength predictions. The work outlined in this paper is a part of a larger study developing a design method for LVL beams containing holes. This paper outlines the prediction and verification of the strength of LVL beams with holes that were tested previously [6]. 2 EXPERIMENTAL DATA The experimental data were from a series of experiments performed at the University of Canterbury, New Zealand on LVL beams with different hole sizes and locations [6]. The research report contains full details of the experimental programme and loads which caused the cracking and failure of beam. Beams were simply-supported with dimensions of mm (see Figure 1). Figure 1: General geometry of the beam with holes (dimensions in mm) The beams were tested in the universal loading machine under three-point bending. The holes were located mainly in the high shear regions because this is the most likely location of penetration. The diameters varied from 32 mm to 255 mm. The load-deflection data at the mid-span were produced during the tests. The two parameters L and D which signify end distance and edge distances respectively (see Figure 1) for the beams in the experiments are defined in Table 1. Table 1: Details of holes in the LVL beams Beam # Hole Diameter (mm) End distance (L ) in mm Edge distance (D ) in mm FINITE ELEMENT MODELLING Coupled FEM and LEFM were used for the prediction of the load carrying capacity of the beams with the holes. The FEM was used to obtain the stress intensities at the crack tip for mode I and II for different crack lengths while LEFM was used for the prediction of failure load. Wood may be described as an orthotropic material for its elastic phase which requires twelve constants, nine of which are independent, including three moduli of elasticity, three shear moduli and six Poison s ratios in elastic phase [7]. However, the nine constants decrease to four independent values in the 2D modelling. The Finite Element Modelling was implemented in the ABAQUS package [8]. The procedure is summarized in the following: Orthotropic elastic modelling and analysis of the LVL beam with the hole in order to obtain the most probable location of crack initiation, which is assumed as the point with the highest stress concentration. The direction of the crack will be perpendicular to the maximum principal stress. Cracks with different lengths are introduced into the model. The seam option of ABAQUS is used for the crack modelling, where seam is a crack which can open if loaded. The model is analysed under unit loading. After re-meshing and re-analysis the SIFs at the crack tip in mode I and II are obtained. Very fine meshing is used in the crack tip along with special elements because the software uses rings of elements for the calculation of the SIFs in opening and shearing modes. The SIFs at the crack tip are then obtained for different crack lengths. Assuming a crack length, the failure load is calculated. Using the Wu s mixed mode fracture criterion [9, 1], the failure load of the LVL beam with hole is calculated. Moreover, the failure load using other mixed mode fracture criteria [11-13] are calculated as well for comparison purposes. The predicted failure load will be checked again by inserting into the main finite element model and obtaining the stress intensities and checking Wu s criterion. The used finite element model is basically a 2D orthotropic model with shell elements loaded at the mid-span in stress plane regime. Unfortunately at the time of writing this paper the exact material properties of the Radiata pine LVL were not available. The following data are based on few experiments performed by the authors.

3 K I k I F Table 2: Material Properties Used in FE Modelling E (MPa) 5 G (MPa)) 1 K II k II F ν.3 4 MESHING OF THE CRACK TIP The meshing of the model was done with special care. The crack tip was meshed using six node triangular thin shell elements with five degrees of freedom namely (STRRI65).. A very fine (about 1 mm) mesh was used around the crack tip. In this case, eeight node doubly curved thick shells namely (S8R) were used. The crack introduced into the model used the seam option in ABAQUS. In order to improve the accuracy of the SIF calculations, the mid-side side nodes on the sides connected to the crack tip was moved to quarter point closest to the crack tip [4]. The mesh used for the calculation of the SIFs is presented in Figure 2. The use of fine meshing around the crack tip leads to reliable values of the SIFs. (3) Where K I stress intensity factor of LVL in mode I (opening), K II stress intensity factor of LVL in mode II (shearing), K IC fracture toughness of LVL in mode I, K IIC fracture toughness of LVL in mode II, ki, k II SIFs in mode I and II respectively obtained in the case of beam under unit loading, and finally F = failure load. No Mixed mode 1 KII/KIIc E (MPa) 12 (2).8.6 (KI/KIc)+(KII/KIIc)^2= KI/KIc 1 Figure 3: Wu's Mixed Mode Fracture racture Criterion 5 MIXED MODE CRITERIA FRACTURE When a beam with a notch or a hole is subjected to bending, two fracture modes usually exist at the crack tip, namely opening and shearing. In other words, a combination of opening and shearing of the crack tip will simultaneously occur and govern the fracture mode of the whole system. Wu [9] suggested a mixed mode fracture criterion for failure load prediction. The Wu s mixed mode fracture is a general fracture criterion for anisotropic composite materials [14]. Figure 3 provides graphical representation of the mixed mode fracture criterion.. The plot represents a fracture envelope for an arbitrary fracture mode mode. If the specimen goes into the pure modee I and mode II, there will be no interaction and the fracture envelop envelope will be a straight line. By inserting equation (2) and (3) in the equation (1) (1), the following equation can be obtained obtained: I F" ( KI$ II F" ) KII$ ) ( 1 By solving the second order equation above the F will be calculated. For LVL the fracture toughness of the material was derived based on extensive experiment experimental work carried out at the University of Canterbury Canterbury. The primary results of that study refer to the fracture toughness of LVL in mode I and II in the grain direction TL, where T stands for Tangential and L for Longitudinal direction. The first character in (TL) represents the direction perpendicular to crack plane and the second indicates the crack direction. K I (MPa m).38~.57 K II (MPa m) 2.78~ ~3.95 Figure 2: Meshing the crack tip The proposed fracture criterion is defined as below: KI K II ( ) ( ) 1 K IC K II (1) (4) (5) (6) The above values are based on a series of experiments on specimens of mm dimensions while for the mm specimens experiments show about 2% increase for the fracture toughness in mode I, and 4% increase in mode II. All of the specimens had moisture content in the range 8 to 1 percent. However, previous research undertaken at the University of Canterbury on New Zealand Radiata Pine

4 led to the formulas in the following form in the grain direction (TL) [15]: K I (MPa m),5f (7) K II (MPa m).16f (8) where F timber shear stress in grain direction MPa. With the assumption of the crack oriented along the wood fibres another mixed mode fracture criterion based on the strain energy release rate could be obtained [11, 13]: K I β K II K I (9) In the above equation β is defined as below: β ( K I K II ) (1) If equation (9) is divided by K I, remembering the meaning of β the following equation can be written: ( K I K IC ) ( K II K II ) 1 (11) This equation is very similar to Wu s fracture criterion, which can be generalized as in the following: ( K I K IC ) ( K II K II ) 1 (12) where m, n= calibration factors, to be determined via experimental testing. Eq. (12) is equivalent to Eq. (11) when m n 2. The ultimate load F can then be derived by substituting Eqs. (2) and (3) in Eq. (9): K I F (k I k II β ) (13) Fracture criteria expressed in different analytical forms can be obtained if the crack growth is assumed to be governed by the magnitude of near tip stresses rather than the strain energy [11]. Another stress-based fracture criterion can be derived by assuming that fracture occurs when the maximum principal stresses at a distance (r) in front of the crack tip attains a critical value [11]: 1 β β β K I β K I K II K I (14) In the above equation β and β are coefficients which are dependent solely on the elastic properties of the material. Such coefficients could also be obtained using analytical formulas [13], which led for the case under study to the values reported in Table 3. Table 3: Results of and calculations F K I β β β k I β k I k II (15) 6 STRESS INTENSITY FACTOR DETERMINATIONS: The values of the SIFs are fundamental for the calculation of the failure load of the beams with the holes. Two methods have been followed in this paper for the calculation of the SIFs: Closed form solutions FEMs for stress intensity calculation based on the displacement extrapolation methods. There are also other FEM based methods which exist in literature such as the virtual crack closure technique, but they have not been used in the current paper. 7 STRESS INTENSITY FACTOR DETERMINATION BASED ON CLOSED FORM SOLUTION Obtaining the SIFs with easily available finite element software is not always simple due to the small meshing required around the crack tip. Moreover, sometimes special elements are used around the crack tip which could make the analysis time consuming. Therefore, closed form solutions can be very useful to calculate the stress intensity factor. Unfortunately, there are few literature references for beams with holes due to the complexity of the problem. A series of formulas were derived by Riipola [3]. The formulation is quite complex and requires an algorithm to draw the solution as there are many variables involved. Furthermore, the formulas are limited to the beams with the holes in the shear dominant regions, and are not extended to the case of uniformly distributed loads or moment dominant regions of the beam. Figure 4 shows the beam analyzed with the different variables. β 3.25 β 2.25 Again, using equation (2) and (3), the equation (14) can be easily re-written as below: Figure 4: Riipola beam configuration for the stress intensity factor evaluation [3]

5 A numerical algorithm was written to use the analytical formulas. The rectangular hole in the theory was replaced by an equivalent circular one with the height equals to the hole diameter (see Figure 5). The interesting point about this method is that it assumes the crack starts to propagate at an angle of 45 degrees. More details about the method can be found in [3, 16]. Figure 5: Approximation of a circular hole with an equivalent rectangle [3] 8 STRESS INTENSITY FACTOR DETERMINATION BASED ON THE FINITE ELEMENT METHOD As discussed before, the SIF can be evaluated by implementing a model made by shell elements with a very fine meshing around the holes and the crack tip in the finite element program ABAQUS.. The crack was introduced using the seam option (see Figure 6). The beam was loaded at the mid-span under unit loading and the SIFs were obtained at the crack tip. The SIFs may have positive or negative values depending on the case. The negative sign of the stress intensity factor in mode (I) means that the crack is closing while in mode (II) indicates a change in the movement direction. ABAQUS makes use of two different methods for SIF calculations: The displacement extrapolation technique: the values of k I and k II are computed starting from displacements (horizontal or vertical) of the finite elements nearest to the crack tip by means of the extrapolation technique proposed by [17, 18]. The virtual crack closure method: the method makes use of the nodal displacements and forces for the calculation of the energy release rates in the opening and shearing modes. The method calculates the work needed for the crack closure, and then the calculated strain energy release rate is transformed into the SIFs in mode I and II. Figure 6: Crack around the hole As it is clear in Figure 6 and Figure 7, the finite element method can predict the location of the crack initiation very well. The crack is most likely to initiate and propagate in places with high stress intensities namely perpendicular to the plane of maximum principal stresses. To work effectively, a very fine meshing with rings is needed in ABAQUS around the crack tip (see Figure 8). The meshing was performed with automatic meshing with emphasis on minimization of mesh distortion. In the way of SIF calculations, first the software automatically detects the primary rings surrounding the crack tip and then it adds other layers to the first layers. Figure 7: Beam with hole deformation under loading

6 9 RESULTS BASED ON FINITE ELEMENT METHOD Table 4 summarizes the results of the SIF calculations in different contour s for one of the tested beams. The SIFs based on the finite element calculations (k I and k II estimates) are presented in the Table 4 together with the predictions of the failure loads in different mixed mode fracture criteria. If the concepts of fracture mechanics are used for the prediction of the failure load, results dependent on the crack length will be obtained. The crack length prediction is dependent of the structure size and Aicher [19] reports a crack length between 2 to 7 millimetres for the small structure and large holes. The 2 mm crack length does not provide reliable values of the failure load F u because if we insert the predicted load in the model and do re-analysis, the predicted stress intensities will not satisfy Wu s criterion, so that crack length should be ignored. However, in this study a crack length between 3 to 4 millimetres was assumed as the crack length. Figure 8: Successive contour s for calculation of the SIFs [8]. Table 4: Comparison of different mixed mode fracture criteria for beam 2 Crack Length (mm) k I Estimate MPa mm k II Estimate MPa mm F (N) based on Wu criterion F (N) based on equation (13) F (N) based on equation (15) Predicted Loads (N) Based on Eq. 15 Based on Eq. 13 Based on Wu F (experimental) 42.5 kn Crack Length (mm) Figure 9: Comparison of different mixed mode crack criteria for beam 2

7 Based on the Wu s mixed mode fracture criterion, using a 4mm crack length the predicted failure load would be 44.2 kn which is quite close to the experimental failure load. The predictions based on equations (13) and (15) will yield the failure load of 42.8 kn and 44.3 kn respectively. The comparison between the experimental results and the analytical predictions shows that the Wu s mixed mode fracture criterion, equation (13) and equation (15) give roughly equal approximations. Figure 9 graphically shows a comparison between different mixed mode fractures criteria for one of the beams tested (beam #2). Moreover, a sensitivity analysis in terms of the effect of the fracture toughness on the outcomes of the study was performed using the Wu s criterion. The results show that the failure load is sensitive to the change in fracture toughness in mode I (opening) while changing of LVL fracture toughness in mode II (shearing) have mild effects on the results. A number of other mixed mode fracture interaction equations have also been developed which do not include mode II namely shearing [2]. Although the sensitivity analysis shows the lesser effect of mode II in comparison with mode I, its mild effect on the results should not be ignored because it is possible that in some cases of the hole position, its role increases. Due to the low influence of mode II in current beams fracture criteria with the same value of the exponent (m) will give similar results even with different values of the exponent (n)..5 Stress intensity factor (MPamm^.5) ki kii Crack Length (mm) Figure 1: Crack length- SIFs in mode I and II for beam 2 Figure 1 gives the calculated SIFs in mode (I) and mode (II) for beam 2. In the model four contour s (see Fig. 8) were considered using the software. The first contour value was ignored because it is very close to the crack tip and may give wrong results. The second and the third contour s are plotted which are in good agreement with each other. The authors experience in modelling with regard to the meshing leads to the conclusion that the third contour yields the best values if fine meshing has been used. This is because the third ring less affected by the crack tip and is not so far from the crack tip. However, the manual of the software suggests few first contours ignored. Predicted loads (N) First contour Second contour Third contour Fourth contour Crack length (mm) Figure 11: Prediction based on the Wu's mixed mode fracture criterion using different contour s As it can be seen in Figure 11, the predictions are in good match with each others. However, first contour gives slightly higher values. As it stated before, the prediction could be affected highly if

8 the meshing around the crack tip not be small enough. Predictions based on Eq. (15) are very close to the Wu s criterion because the effect of the shearing mode is very low. However, this could not be generalized because the holes in the current study are mainly in shear dominant areas and the effect could be different in moment dominant areas. 7 6 Predicted loads (N) First contour Second contour Third contour Fourth contour Crack length (mm) Figure 12: Prediction based on equation (13) mixed mode fracture criterion using different contour s 7 6 Predicted Loads (N) First contour Second contour Third contour Fourth contour Crack length (mm) Figure 13: Predictions based on equation (15) mixed mode fracture criterion using different contour s 1 RESULTS BASED ON CLOSED FORM SOLUTION AND COMPARISONS The details of the prediction of the failure load based on the closed form solution were presented in [3, 16]. However, the closed form solution shows almost more error in predictions in comparison with the finite element Table 5: Summary of experimental-numerical-analytical comparison methods. A possible reason is perhaps the assumption in the SIF evaluation of a 45 degrees angle for the location of the maximum principal stresses exactly at the hole perimeter. Unfortunately, the method suffers from a weakness in terms of boundary conditions and applied load which limits the possibility of using the method for beams with holes. Beam # Predicted Load carrying capacity by closed form solution (kn) Experimental data (kn) Error percentage in closed form solution(%) Error percentage in Wu s mixed criterion (%) Error percentage in Eq (13) (%) Error percentage in Eq (15) (%)

9 As the hole becomes closer to the edge, the predictions becomes disturbed and less reliable. This could be attributed to the beam acting like a frame in these limit cases. Although closed form solutions for determining stress intensity factors exist, due to the assumptions their use should be considered with care. The method is fast in terms of calculation but it is restricted only to some limited cases of loading, and generalization of the method is too difficult. Table 5 shows the comparison between different mixed mode fracture criteria. In most of the cases the Wu s mixed mode fracture criterion and fracture criterion based on equation (15) gives better results than the other methods. The other important point about the predictions made using these fracture criteria is that it gives conservative results in most of the cases. However, an interesting point in Table 5 is that in the cases where the diameter of the hole is small and does not play an important role in the failure of the beam such as for beam #6, its role could be ignored and therefore the current method shall not be used for these kinds of problems. Besides, this implies that a lower limit should be defined for the effect of the holes on the strength of the beams. More experimental and numerical investigation will be required to define a lower limit for it. The Authors idea on this issue is that a percentage of the hole diameter to the depth of the beam, say about 1 to 2 percent, should be considered for the lower limit which the effect of the hole could be ignored. However, this needs experimental and numerical verification. Finally, it is important that the critical crack length in the numerical analysis is assumed from 2 to 5 millimetres. This critical length is very important because it will be the basis for the derivation of a design method. Still more numerical modelling, especially of beam with holes in moment dominant areas, is required to yield final conclusions. It is the authors opinion that locating the holes close to the neutral axis will improve the beam behaviour by reducing the shearing mode and, the opening mode effect too. 11 CONCLUSIONS Coupled finite element method and Linear elastic fracture mechanics can be used for the prediction of the failure load of LVL beams with holes. FEM can be used for the evaluation of the stress intensity factors at the crack tip, while LEFM can be used for the prediction of the failure load. Wu s mixed mode fracture criterion seems to be the most appropriate criterion. Furthermore, the critical crack length for the stress intensity factor evaluation should be assumed between 2 and 5 mm in the case of small structure. In most of the cases, Wu s mixed mode fracture criterion gives conservative estimate of the load carrying capacity. Finally, a lower limit in terms of hole diameter to beam depth ratio can be defined so that the effects of the hole on the strength of LVL beams can be ignored below this limit. Further development of this research which is currently undergoing as a PhD program at the University of Canterbury includes the extension of the method from a theoretical mixed FEM and LEFM to a design method.. 12 ACKNOWLEDGEMENT: The authors would like to extend their gratitude for the financial support for the project received from the Department of Civil and Natural Resources of the University of Canterbury, New Zealand. 13 REFERENCES [1] Guan, Z.W. and Zhu, E.C.: Finite element modelling of anisotropic elasto-plastic timber composite beams with openings. Journal of Engineering Structures, 31:394-43, 29. [2] Pattonmallory, M. and Cramer, S.M.: Fracture-mechanics - a tool for predicting wood component strength. Forest Products Journal, 37(7-8):39-47, [3] Riipola, K.: Timber beams with holes: fracture mechanics approach. Journal of Structural Engineering, 121(2): , [4] Habbitt, Karlsson, and Sorensen, ABAQUS, Theory Manual, Version : ABAQUS Inc. [5] Ranta-Mauns, A. Application of fracture mechanics to timber structures. In CIB-W18A on Timber Structures. pages: , 199.

10 [6] Handerson, P., Analysis of holes in HYSPAN beams (unpublished work). 24. p. 66. [7] Green, D.W., Wood: strength and stiffness, in Encyclopaedia of Materials: Science and Technology. 21. p [8] Habbitt, Karlsson, and Sorensen, ABAQUS, User Manual, Version , ABAQUS Inc. [9] Wu, E.M.: Application of fracture mechanics to anisotropic plates. Journal of Applied Mechanics, 34(4):967-&, [1] Wu, E.M.: Application of fracture mechanics to anisotropic plates. Journal of Mechanical Engineering, 9(4):134-&, [11] Jernkvist, L.O.: Fracture of wood under mixed mode loading I. Derivation of fracture criteria. Journal of Engineering Fracture Mechanics, 68(5): , 21. [12] Jernkvist, L.O.: Fracture of wood under mixed mode loading II. Experimental investigation of Picea abies. Journal of Engineering Fracture Mechanics, 68(5): , 21. [13] Romanowicz, M. and Seweryn, A.: Verification of a non-local stress criterion for mixed mode fracture in wood. Journal of Engineering Fracture Mechanics, 75(1): , 28. [14] Smith, I., Landis, E., and Gong, M., Fracture and fatigue in wood. 23, Chichester, West Sussex, England ; Hoboken, NJ: J. Wiley. viii, 234 p. [15] Dean, J., Gibson, J.A., and Moss, P.J. The fracture properties of timber. In Timber Engineering. pages:97-115, [16] Riipola, K. and Fonselius, M.: Determination of critical J-Integral for wood. Journal of Structural Engineering, 118(7): , [17] Kuang, J.H. and Chen, L.S.: A displacement extrapolation method for two-dimensional mixed-mode crack problems. Journal of Engineering Fracture Mechanics, 46(5): , [18] Ballerini, M. and Rizzi, M.: Numerical LEFM analyses for the prediction of the splitting strength of beams loaded perpendicular-tograin by dowel-type connections. Journal of Materials and Structures, 4(1): , 27. [19] Aicher, S., Schmidt, J., and Brunold, S. Design of timber beams with holes by means of fracture mechanics. In CIB-W18/28 on Timber Structures. pages [2] Mall, S., Murphy, J.F., and Shottafer, J.E.: Criterion for mixed mode fracture in wood. Journal of Engineering Mechanics, 19(3):68-69, 1983.