Load carrying capacity of cracked beams

Transcription

1 oad carrying capacity of cracked eams Steffen Franke 1, Bettina Franke 2, Noëlie Magnière 3 ABSTRACT: The increasing numer of wood structure amongst large and potentially pulic uildings gave a new impulse to the assessment of timer structures. For assessing the state of timer elements, cracks are a key indicator. Therefore, experimental and numerical investigations on not cracked and partly cracked timer memers were carried out and analysed. The results show no influence on the stiffness and modulus of elasticity for partly cracked eams. The experimental results were used for the development of analytical and validation the numerical solutions for the assessment of the residual load carrying capacity of cracked timer memers. Several models predicting the residual load carrying capacity depending on the crack situation are presented. KEYWORDS: Timer structures, Assessment, Numerical investigations, Cracks, Stiffness, oad carrying capacity 1 INTRODUCTION 1 The growing numer of timer structures especially high performing and large span timer memers gave a new impulse to the field of assessment of timer structures. When assessing a timer structure, one of the first and most ovious sign of damage is the presence of cracks in the elements. One type of cracks develops y implication ecause of climate variation and another type of cracks develops ecause of over stressed situations. These cracks propagate in the grain direction as a consequence of the low resistance in tension perpendicular to the grain. They are significant as they represent 7% of the types of failure in timer elements [1-2]. Although easy to detect in dry conditions, no reliale method has yet een designed to estimate the impact of cracks on timer elements. Current international standards mostly provide rules of thum or guidelines developed over the years through practical experience (see for instance [3-4]). Considering their origins, these rules are not consistent from one country to the other. Therefore, the need for assessment methods of cracked eam resistance has ecome prominent. Hence, a twostep study was conducted to evaluate the impact of cracks on timer elements. First experimental test series and a numerical model ased on information from structure assessments was setup and secondly analytical solutions were derived for practical use. 1 Steffen Franke, Bern University of Applied Sciences, Institute for Timer Construction, Structures and Architecture, Solothurnstrasse 102, 204 Biel, Switzerland. steffen.franke@fh.ch 2 Bettina Franke, Bern University of Applied Sciences 3 Noëlie Magnière, Bern University of Applied Sciences 2 CASSIFICATION OF CRACKED BEAMS For the assessment of cracks, the crack kind, the measured sizes as well as the numer has to e judged individually for each case. Particular the type of structure, the structural system as well as the function of the memer within the complete structure have to e considered, e.g. main or second girder. Further, the current as well as the future condition of use e.g. the climate conditions are essential for the assessment. For the limit state design, existing cracks reduce the effective cross section of the memer mainly for tension and shear stress ut also may influence the capacity under ending, compression and torsion stress or of notches and holes. The weakest point in a timer structure is often the low tensile strength perpendicular to grain which leads to cracks in the cross-section and along the span of the memer. For the assessment of such cracks, the standard SIA 269/ gives some guidelines how to consider existing cracks in the evaluation of timer structures. Furthermore, the international standard DIN for strength grading of solid softwood provides some specifications on maximum crack depths as well. In this standard, the maximum crack length is restricted to 1 m. In research papers y Frech (1986) and Radovic & Wiegand (200), further specifications are given for the consideration of cracks. However, there is no consistency in the restrictions in the crack ratio. The pulished criteria in the standards and the research reports vary in a wide range from R = 0.12 to 0. and neither the actual stress situation nor the stress comination or the position of the crack along the span which is an important criterion, nor the crack length are always considered. For the analyses of the influence of a cracks, six main different cases of occurred cracks were defined and evaluated. The six main cases, as shown in Tale 1, are

2 1) full section (as reference), 2) horizontal split, 3) through crack, 4) non through crack, ) opposite cracks, and 6) numerous cracks. For these different cases, analytic or numerical solutions as well as experimental test series are availale. The analytical solution concentrates on the cases 1 to 4. The results will e compared with the numerical model and the experimental test series. 3 EXPERIMENTA TEST SERIES 3.1 MATERIA For the analysis of the prolem, spruce glulam eams with a cross section of 10/600 mm and a length of 244 mm were used. The memer was produced with lamellas with a thickness of 40 mm. For each lamella, the density and modulus of elasticity were determined Tale 1: Definition of different crack cases Failure case Cross section Side View 1: No Crack; Full Section (Reference Case) 2: Horizintal Split 3: Through Crack h (1- α )h αh (1- α )h αh 4: Non through crack l c crack efore assemling the glulam memer. According to the results, the lamellas were graded and arranged due to their strength and stiffness-aility in T11, T18 and T26 according to EN 14080:2013. The inner part of the cross section of the eam consists of lamellas of class T11 or T18 to provide a homogeneous distriution of E- modulus. The outer three layers always consists of higher strength class lamellas of T18 or T26 according to EN 14080:2013, shown in Figure 1. Therefore, test series 1 and 2 are different in the layered cross section respectively the stiffness of the eam. The moisture content of the test specimen was around 13.7 M% and the mean density 476 kg/m 3. The characteristics of each lamella of the glulam memers are summarized in [] and [6]. 3.2 TEST PROGRAM AND SETUP The tests were performed as a 3-point ending test, as shown in Figure 2. In order to investigate the effect of cracks within the eam structure, samples with and without cracks and different crack depths are tested. The crack length was over the total span of the memer in the middle of the cross section. The deepness varies etween 3, and 6 of the eam width. In total, 3 tests with 10 eams were performed, compare Tale 2. Each eam of test series 1 was tested without cracks up to of the estimated load capacity. Artificial cracks of 3 and depth were then created and tested in each case again up to of the estimated load capacity. Finally, the crack depth was increased to 6 and tested until failure. The four different test configurations are shown in Figure 1. Therefore, a direct comparison could e made for the different cracked situations. Test series 2 was done in similar manner ut with 3 crack depth only. To prevent failure under compression perpendicular to grain, reinforcement at the loading and support points were prepared. Glued-in steel rods using the GSA technology was used. The steel rods were shorter than ½ of eam depth and, therefore, do not influence the ehaviour of the eam under shear stress. T18 (1- α )h SERIE 1 T11 4 mm 7 mm 90 mm αh d c l c T18 : Two opposite cracks Type 0 Type 1a Type 1 Type 1c ack side T26 h 6: Numerous cracks front side SERIE 2 T18 4 mm lamella open crack possile crack plane h T26 Type 0 Type 1a Figure 1: Arrangement of the lamellas

3 The displacement at midspan and earing and the load applied are measured continuously during the test for the determination of the modulus of elasticity, modulus of rigidity, stiffness. For the displacement measuring for the determination of modulus rigidity an optical measuring device was used. 4 RESUTS 4.1 EXPERIMENTA TEST SERIES The failure ehaviour of test series 1, type 1c and test series 2, type 1a was similar showing a typical shear failure close to the support, as shown in Figure 4. The load displacement curves oserved were evaluated according to Figure 6 - in terms of load carrying capacity y the crack propagation/initiation load F crit and. In theory, depending on the crack depth, the quotient of the tested load-carrying capacity and the load-carrying capacity of an uncracked memer are in linear relation. Regarding the stiffness of the glulam eam, the modulus of elasticity and the shear modulus are not influenced y the cracks investigated. The determined values do not show a change with the appearance of cracks or changing crack depth, as summarized in Tale 3. The quotient of cracked to uncracked tested stiffness is approximately, see Figure 7. Only samples with crack depths of 6 show a lower quotient, means, the actual stiffness is smaller than the stiffness of the uncracked eam. - the corresponding displacement u crit - in terms of stiffness y the slope K in the elastic range (10-4 of F crit ). The appearance of cracks and their dimension influence the load-carrying capacity. By enlarging the crack depth, the load-carrying capacity decreases, as shown in Figure F oading/support plates 300 mm 600 Figure 4: Cracked eam after testing, Type 1a, Series 2 10 A B [mm] Figure 2: Experimental test setup for test series 1 and 2 Failure oad F crit [kn] Serie 2: Type 1a Crack depth 3 Serie 1: Type 1c Crack depth Crack depth d c / [%] Figure : Failure loads for experimental test series 1 and 2, ox plot with mean value and range of Minimum and Maximum 00 Figure 3: Test setup of timer memer uncracked Tale 2: Experimental test setup for test series 1 and 2 Test series 1 Test series 2 Failure case No cracks 3 deep crack deep crack 6 deep crack No cracks 3 deep crack Type 0 Type 1a Type 1 Type 1c Type 0 Type 1a No. of specimen oading procedure linear to linear to linear to linear to failure linear to linear to failure oad F [kn] 400 F crit F crit 100 K F crit u crit Displacement u [mm] Figure 6: Example of evaluation of a load-displacement curve

4 Tale 3: Experimental test results for test series 1 and 2 Test series 1 Test series 2 K/K ref [%] Failure case No cracks 3 deep crack deep crack 6 deep crack No cracks 3 deep crack 12 8 Modulus of Elasticity E [MPa] Modulus of Rigidity G [MPa] Stiffness K [N/mm] Serie 1 Serie 2 4 Crack depth Crack depth Crack depth 3 Crack depth 3 2 Crack depth Crack depth 3 num. Sim. Crack depth 6 Crack depth 6 num. Sim Numer of experimental tests [-] Figure 7: Stiffness for different cracked/uncracked eams of experimental test series and numerical simulation 4.2 NUMERICA SIMUATION AND VAIDATION The numerical model developed for this study simulates a three point ending test of straight glued laminated timer eams presenting initially one open crack, as shown in Figure 8. The numerical model was developed using the finite element software ANSYS. The timer memer is defined using 8-node elements (SOID 182) with a wooden orthotropic linear elastic material. The element sizes are ased on the growth ring width and descrie a 2-dimensional macro scale model. The three principal material directions of wood (R, T and ) have een approximated to the main axes of the model (X, Y and Z) and orientates on the mostly horizontally oriented growth rings of the lamellas of the glulam eam. A uniform cross section is used to model the layered cross section of the glulam material ecause, as for sawn timer, the failure as crack growth occurs etween the fires and not in the glue layers. For the loading and earing areas, steel plates were modelled and connected y using a friction law. In order to simulate the presence of an initial crack and its further crack propagation, contact elements have een used. The rittle failure ehaviour of wood under tension stress depends on the angle etween the stress and cell respectively fire orientation. Based on a micro scale level, the fracture under tension stress parallel or perpendicular to the grain is characterised y an intercellular and/or cellular fracture, whereas the cells under shear perpendicular to the grain mostly peel off. Therefor the contact elements were placed on the surfaces of the crack plane. The crack system reflects an R-crack system; crack opening in radial direction and crack propagation in longitudinal direction. These contact elements follow a ilinear Cohesive Zone Material (C.Z.M.) law. The cohesive zone material used, considers the single fracture mode I (cracking under tension stress transverse to the crack plane) and fracture mode II (cracking under shear stress in crack plane) as well as the important mixed mode using a quadratic fracture criteria. Further information according the definition of the numerical model and the validation are provided in [6], [7], [8]. The numerical model was applied to the experimental test series using the material settings of Tale 4. Figure 9 shows the comparison of the load displacement curves with the experimental test series. In oth cases, a clear load drop due to a shear failure close at the support is visile. The load capacity for series 1 shows a lower level which could e caused y the material settings chosen for the crack opening and less stiffness of the numerical model compared to the experimental test series, as shown in Figure 7 and Figure 9. d c y c A Figure 8: Definition of glued laminated timer eam and crack parameters in numerical test series Tale 4: Material parameter used in numerical simulation Wood - Glulam Modulus of elasticity E E R E T Modulus of rigidity G R G T G RT Poisson ratio v R v T v RT Cohesive zone material Tension strength f t,90 Shear strength f v Fracture energy G ci G cii Steel Modulus of elasticity E Poisson ratio v Friction c Specific* MPa MPa MPa 8.0 MPa Nmm/mm MPa 0.2 Friction coefficient 0.6 * specific values according lamella arrangement F Crack plane 12' B 600 [mm]

5 4.3 NUMERICA RESUTS OF VIRTUA EXTENDED TEST SERIES For the investigation of the influence of one main crack, a parameter study with more than 400 numerical simulations were carried out and analyzed. Therefore, the glued laminated timer eam used was of constant dimensions (width w =140 mm, depth h = 600 mm, and span = 12 m), compare Figure 8. The dimensions were taken as a alance from analyses of existing timer structures regarding dimension, shape and loading situation, [6]. The material settings were defined according to a moisture content of 12 M% and the corresponding values oserved y Neuhaus (1981). The moisture content of 12 M% is the equilirium moisture content according to the normal climate 20 C temperature and 6% relative humidity. For the fracture mechanic parameters, approved values y FP (1999), arsen & Gustafsson (1990), and Franke (2008) were used. Tale summarizes the average material parameters. The characteristic of a considered initially open crack is entirely defined using four parameters (, y c, c and d c ) as illustrated in Figure 8. These parameters can identically e expressed using four ratios: - the crack origin ratio ( / ), - the crack height ratio (y c / h), - the crack length ratio (l c / ) and - the crack depth ratio (d c / w). oad F [kn] oad F [kn] Displacement u [mm] Series 1 Type 1c - 90 mm Series 2 Type 1a - 4 mm Experimental test series Numerical Simulation Experimental test series Numerical test series Displacement u [mm] Figure 9: oad displacement curve of experimental test series and numerical simulation These parameters enale to simulate the performances of the eam for various crack locations and dimensions. The complete programme of the numerical test series and their parameters considered are summarized in Tale 6. The numerical tests are carried out as a displacement controlled three-point ending test. The eam is considered to e failed and the simulation is stopped shortly after when either the initially open crack propagates or a new crack is initiated. The results of the experimental and numerical simulations were analyzed in order to answer the question: Which parameters of one main crack influence the stiffness and residual load-carrying capacity? The results reveal that the distinction etween through cracks (cracks over the complete width of the cross-section) and not through cracks (cracks partially over the width of the cross-section) should e made. Indeed, in terms of stiffness and load-carrying capacity, these two cases have a different ehavior. First, the simulations reveal that not through cracks could e neglected regarding the gloal stiffness of the eam. Moreover, a calculation model was proposed to estimate the reduction of stiffness for a eam presenting a partial-length or fulllength through crack, [6]. This simple calculation model shows a good correlation with the results otained from the numerical simulations, [6]. Tale : Material parameter used in virtual numerical test series Wood - Glulam Modulus of Elasticity E E R E T MPa Modulus of Rigidity G R G T G RT MPa Poisson ratio v R v T v RT Cohesive zone material Tension strength f t, MPa Shear strength f v 4.0 MPa Fracture energy G ci G cii Nmm/mm 2 Steel Modulus of Elasticity E MPa Poisson ratio v 0.2 Friction Friction coefficient 0.6 Tale 6: Crack parameter used in numerical test series Crack origin Crack height y c y c Crack length l c Crack depth [mm] / [mm] y c / h [mm] l c / [mm] d c / d m* m* m* m* m* corresponds to a crack centred at mid span l c d c d c

6 Trough / Not Through cracks Trough / Not Through cracks 8 8 K crit /K ref [%] 6 4 F crit /F ref [%] Not trough cracks Through cracks 2 Not trough cracks Trough cracks Simulation [-] Simulation [-] Figure 10: Ratio of the stiffness for the complete numerical test programme Figure 11: Ratio of load capacity for the complete numerical test programme Through cracks, l c = 600 mm 3 30 y = 1E-0x x R² = Through cracks, lc = 6000 mm y = 3E-06x x R² = F crit /F ref [%] Cracks's height y c [mm] F crit /F ref [%] Crack's height y c [mm] Figure 12: Dependency of the residual load carrying capacity and the position of the crack over the depth of the memer for a crack length of 600 mm (left) and 6000 mm (right) However, regarding the load carrying capacity, no clear dependency is distinguishale etween through and not through cracks. Several trends have een highlighted to correlate the residual load-carrying capacity of a cracked eam with the crack characteristics [6]. In the case of through cracks, the residual load carrying capacity correlates with the crack position over the height of the memer if the crack length keeps constant, as shown in Figure 12. The critical load ratio follows a paraolic trend, whereas the three coefficients of this paraolic trend depend on the crack length. The coefficient dependency on the crack length can e expressed with a power function of the crack length l c. Therefore, the ratio of the residual load carrying capacity F crit to the load capacity of a not-initially-cracked timer eam F ref can e expressed as followed: With: The application of the empirically derived equation on the numerical test program of through cracks shows a relatively good correlation etween numerical simulation and model proposed, as shown in Figure 13. Simulation F crit /F ref [%] Through cracks - Paraolic model for load capacity R 2 = Model F crit /F ref [%]] Figure 13: Comparison of model proposed and numerical results oserved for the residual load carrying capacity of through cracks In the case of not through cracks, no real trend could e identified; here further research would e needed to effectively correlate the crack characteristics to the eam residual load-carrying capacity. Indeed, only a trend appears: the lower the crack in the cross-section, 1 the deeper it can e without influencing the eam strength. 2 Thus, depending on the height ratio of the crack 3 y c /h (counting from the eam lower face) the allowale depth ratio of the crack varies. However, no model comining the four parameters shows a real correlation with the simulation results so far.

7 4.4 ASSESSMENT METHODS For the assessment of cracks, the crack kind, the measured sizes as well as the numer has to e judged individually for each case. Particular, the type of structure, the structural system and the function of the memer within the complete structure have to e considered, e.g. main or second girder. Further the current and the future condition of use e.g. the climate conditions are essential for the assessment. For the limit state design, existing cracks reduce the effective crosssection of the memer and, therefore, the load carrying capacity mainly under shear stress, tension stress perpendicular to grain and ending ut maye also for compression stress and torsional stress. The weakest point in a timer structure is often the low tensile strength perpendicular to grain which leads to cracks in the cross-section and along the span of the memer. For the assessment of such cracks, the standard SIA 269/ [9] gives some guidelines how to consider existing cracks in the evaluation of timer structures. Furthermore, the international standard DIN [10] for strength grading of solid softwood provides some specifications on maximum crack depths as well. The research work of Frech [11] summarizes the reason for cracks and characterizes cracks in solid wood and glued laminated timer (glulam) regarding ending or shear stress. For glulam, a crack depth of 1/6 of the eam width per side, thus, in total 1/3 of the eam width is tolerale. However in case of comined tension stress perpendicular to grain and shear stress, the limit can e exceeded already. Radovic & Wiegand [12] indicate for glulam the same limit under ending and shear stress. Whereas, they give an experienced limit of 1/8 of the eam width for transverse tension stressed areas. However, there is no consistency in the restrictions in the crack ratio. The pulished criteria in the standards and the research reports vary in a wide range from R = 0.12 to 0. and neither the actual stress situation is always considered nor the stress cominations or the position of the crack along the span which is an important criterion, nor the crack length. It can e summarized that the current practice in the assessment of existing timer structures including the influence of different crack situations on the load-carrying capacity may not e considered suitale whether to facilitate confident decisions aout the reliaility of the structure nor to evaluate the residual load-carrying capacity of structural elements, [13], [14]. Since the theoretical capacity of a eam ased on the cracked part (e. g. with small cracks near the support or edges) can e higher than the capacity within uncracked parts, the residual load-carrying capacity of the eam is the minimum of:,,,, min (2),,,, Hereafter, first analytical solutions for the assessment of the residual load-carrying capacity are provided for the four main cases. The analytical solutions are simplified to a point loaded eam at midspan. Tale 20 and 21 summarize the models evaluated for the estimation of the residual stiffness and load-carrying capacity. The residual load-carrying capacity has to e calculated with the original strength for the uncracked part and the reduced strength provided for the cracked part. The minimum of the capacities governs the validation. For the further two cases ( and 6 of Tale 1) no clear characteristics could e found. Therefore, further research with numerical and experimental tests need to e done. Tale 7: Summary of the results otained y means of the models Case 1 Full section (Reference case) Stiffness K 4 2 Horizontal split 1 3 Through crack 1 4 Not through crack No influence K = K ref Capacities respectively strength factors for calculation of residual load-carrying capacity within the cracked part :, 2, 3 :, :,, 1 :, 1 :,, 1 :, :,, 1 :, 140.

8 CONCUSION AND VIEW The results reached show that the assessment of the residual load-carrying capacity of large span memers in wood is a vast field. Starting from the differentiation of the failure cases, here especially for cracks, the causes (load stresses, moisture induced stresses), the geometry, position and numer of cracks have already to e characterized. The definition of typical crack causes confirms that no general rules can e valid and effective for the assessment of the residual-load-carrying capacity. On the one hand, the numerical modelling of timer memers with existing cracks can provide an estimation of the residual load-carrying capacity. The numerical model developed is, therefore, a comprehensive instrument. Not only one main crack can e defined ut also multiple crack situations can e analysed. The accuracy of the result depends on the material parameters known for the definition in the numerical model. On the other hand, first analytical solutions for the assessment of the residual load-carrying capacity are provided. In order to increase the significance of the results, the numer of experimental test series should e increased. Further, an association of the analytical, experimental and numerical results reached has to e carried out. ACKNOWEDGEMENT The research work was proudly supported y the State Secretary for Education and Research (SERI). Parts of the present results are done y Nicolas Giordano in a project paper and Noëlie Magnière in her master thesis. REFERENCES [1] M. Vogel, Üerwachung von Bauwerken, in Proceedings, Holzautag in Biel, Switzerland, Biel, [2] H. Blass and M. Frese, Schadensanalyse von Hallentragwerken aus Holz, Karlsruher Institut für Technologie (KIT), Karlsruhe, [3] APA the Engineered Wood Association, Evaluation of Check Size in Glued aminated Timer Beams - EWS R47E, [4] SN 0269/:2011 Existing Structures - Timer Structures, 2011, Swiss Society of Engineers and Architects. [] N.Girodano (201) Vergleich von der Steifigkeit und Tragfähigkeit gerissener und ungerissener BSH- Balken, Student project, Bern University of Applied Sciences, Switzerland. [6] B. Franke, S.Franke, N. Magnière, R. Steiger, R. Jockwer (2016) Assessment of the resiudual load carrying capacity of large span memers in wood, Research Report, ISBN [7] N. Magnière, S. Franke, B. Franke (2014) Investigation on elements presenting cracks in timer structures, World Conference on Timer Enginnering, WCTE, Queec, Canada. [8] N. Magnière (2013) Residual load carrying capacity of cracked timer elements - a numerical investigation, Master Thesis, Bern University of Applied Sciences, Switzerland. [9] SIA 269/: Erhaltung von Tragwerken - Holzau, SIA-Schweizerischer Ingenieur- und Architektenverein, Zürich, Schweiz. [10] DIN :2008 Standard Strength grading of wood - Part 1: Coniferous sawn timer. DIN German Institute for Standardization, Berlin, Germany. [11] Frech, P. (1986): Beurteilungskriterien für Rissildungen im konstruktiven Holzau. Research Report T188, Frauenhofer, IRB-Verlag, Germany. [12] Radovic, B., Wiegand, T. (200): Oerflächenqualität von Brettschichtholz. auen mit Holz 7, pp [13] Dietsch, P., Kreuzinger, H. (2011): Guideline on the assessment of timer structures: Summary, Engineering structures 33, issue 11, pp [14] Dietsch (2012) Einsatz und Berechnung von Schuverstärklungen für Brettschichtholzauteile, Docotral thesis, Technische Universität München, Germany.