12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 1Numerical Description Of Size And Load Configuration Effects In Glulam Structures Matthias Frese Senior Research Scientist, Dept. of Timber Structures and Building Construction, Karlsruhe Institute of Technology, Germany Hans J. Blaß Professor, Dept. of Timber Structures and Building Construction, Karlsruhe Institute of Technology, Germany ABSTRACT: The limit states design requires realistic strength values for the verification of reliable glulam structures. Here, the problem arises that the probabilistic system effect strongly affects load-car-rying capacities. Considering this effect, modification factors for the glulam bending and tensile strength values are determined, based on strength simulations on various member sizes and structural systems. Using the proposed factors in the practical design of the treated problems leads to a structural reliability, which is more consistent, since the proposed factors cover the inherent system effects correspondingly. 1. INTRODUCTION The design rules for timber structures are based on the limit states method rather than on a direct ap-plication of the reliability analysis. With respect to safety, the 5th percentiles of the resistance and the strength (fk), respectively, are among others employed to describe the limit state. In a practical design situation, the 5th percentile, originally re-lated to a member with reference size, is adjusted with modification and calibrated safety factors to meet the target reliability, see EN 1990. Thus, the limit states design is a compromise for a reliability analysis. The magnitude of the 5th percentile of the strength has, therefore, a crucial influence on the reliability of timber members or structures made thereof. Related to this compromise, the paper aims at showing variations in the 5th percentile of glulam strength values which are obviously caused by changes in member sizes, in the structural system or in the load configuration but actually related to the probabilistic system effect as concisely de-scribed e.g. by Thelandersson (2003). Since the examined changes basically repre-sent variations of the stressed volume, the prob-lem treated here concerns the theory of Weibull applied to timber by e.g. Colling (1986a/b) and Isaksson (2003). The direct and practical benefit of the work is the calculation of more balanced strength values for members or structures, which partly deviate from standardised reference sizes. Thus, the pre-sented strengths have the meaning of effective strength values (fk,ef) coupled with a special mem-ber size or system. According to the concept in EN 1990 for calculating design values, the paper pro-poses a row of modification factors (ki, i = 1-9) to adapt the 5th percentile of the strength to changed conditions often occurring in practice. The corre-sponding format of this adaption is given in Equa-tion (1). k,ef i kf k f (1) The size- and system-dependent strength var-iation is treated by means of the 5th percentile based on the counting method. Thus, restrictions occur regarding the interpretation of the true structural reliability. The method used for the strength variation employs Monte Carlo Simulations. The members and structures were modelled and computed with an individually configured stochastic finite ele-ment program for glulam. Performing a parameter 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2variation, the load-carrying capacity and strength, respectively, of bending and tension members as well as continuous beams and frames were deter-mined. Similar works, which gave impulses for these examinations, were performed by Foschi et al. (1996) as well as Hansson and Thelandersson (2003). A further benefit of the work concerns the ap-plication of finite element modelling on structures which are more complex like continuous beams or frames. Thus, the experiences gained in this nu-merical examination should also help to identify deficiencies in modelling and purposeful amend-ments. The paper is organised as follows: Section 2 describes the approach including the paper’s gen-eral concept, the members and systems with ref-erence sizes, the computer model and the per-formed variations. In section 3, the simulated strength values are stated in normalised form and described in three separate subsections. Based on these results, the modification factors are finally proposed. 2. METHODS OF THE EXAMINATION 2.1. Concept The first and main focus of the parameter varia-tions concerns the member size and the structural system. These variations are the basis for simu-lated strength values applying to conditions which deviate from the reference systems. The second focus (partly superposed with the first one) is on the variation of the material quality (MQ). The purpose is to examine, whether different material qualities and material strengths, respectively, have an influence on the resulting effective mem-ber or system strength modification. Thus, several common and theoretical strength grading pro-cesses were empirically represented with the com-puter model. They result in different material qualities (A to I). The material quality A is the re-sult of a simple strength grading process (leading to high yield and low strength) while quality I is the result of a demanding process (leading to low yield and high strength). Table 1 gives the outline of the numerical ex-amination. The 1st column contains the nine mod-ification factors to be determined. The 2nd one contains the corresponding member (bending or tension) and system type (continuous beam or frame), respectively for which the k-factors are valid. The 3rd one shows which parameters are subject to variation. Except for the frames and ten-sion members, size or structural and material as-pects are examined together. The size variation of the bending members is based on a constant h/ℓ-ratio (=1/18) while that of the tension members is performed with h- and ℓ-values which are inde-pendent of each other. The last column contains the total number of performed single simulations. Table 1: k-factors and parameter variation. Member type/ system Variation Simu-lations k1 Bending mem-ber Size and mate-rial quality 30000 (+3000) k2 Tension mem-ber Length and depth 70000 k3 Continuous beams 2 spans and material quality 9000 k4 3 spans and material quality 9000 k5 4 spans and material quality 9000 - 2-hinged frames h = ∞ 3000 k6 h = 3ℓ/2 3000 k7 h = ℓ/2 3000 k8 h = ℓ/4 3000 k9 h = ℓ/8 3000 Once a load-carrying capacity (F or M = function of F) is simulated with the computer model, the effective strength is calculated with this capacity and the corresponding cross-sec-tional values. In case of the continuous beams and the frames, the effective strength is always calcu-lated with the global maximum bending moment present in the structure in the state of ultimate fail-ure. 2.2. Reference systems The initial configuration of the member size and of the system is shown by the reference systems 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 3with a beam depth h of 600 mm and a dependent length ℓ (Figure 5). In case of the bending and ten-sion members, the parameters , and are set to 1. For the continuous beams the bending member with = 1 and n = 1, respectively, states the ref-erence system. = ∞ (infinite) describes the ref-erence system for the frames; statically seen, a simple beam with uniformly distributed load (q). Further details are: The position of the con-centrated load F is in the third points of the bend-ing member. The action line of F coincides with the centre line in the tension member. For simpli-fication reasons, frames are modelled as 3-span-beams with equal lengths of the end spans (Figure 1). Thus, normal forces are not modelled and the bending resistance of the frame corner is assumed to be the same as in the straight section. Figure 1: 3-span beams as a substitute for frames 2.3. Computer model The computational strength examinations were conducted with a validated finite element based computer model. It is based on the structural anal-ysis system ANSYS (release 14) and on its corre-sponding processors. The integrated design lan-guage was employed to control the general pro-gramme flow and to build the model with realistic mechanical properties of glulam. Figure 2 shows the basic model used to sim-ulate the bending strength. It exemplifies the bending test later shown in Figure 5 and applies to arbitrary members and structures analogously. The member size of the modelled body depends on the numbers of elements (150 mm in length and 30 mm in depth) along the total length and depth. The main features of the computer model are described in the following. For further model characteristics, see Frese (2010). Figure 2: Discretised glulam model with selected ele-ments under tensile and compressive stresses. To obtain one of the material qualities (A-I), a corresponding grading process is activated prior to the Monte Carlo analysis. That results in statis-tically distributed values of features (knots and density) which influence the mechanical proper-ties (Figure 3, 1st step). After being empirically represented by regression equations (Fig. 3, 2nd step), stochastically distributed and auto-corre-lated mechanical properties of glulam are as-signed discretely and systematically to the ele-ments of the model (Fig. 3, 3rd step). In doing so, the natural occurrence of knots and glulam spe-cific characteristics as finger joints are taken into account with regard to the local mechanical prop-erties. That enables a computation of realistic re-sulting tensile and compressive stresses (t and c) for the loaded member. An orthotropic mate-rial model is used. In the compression zone of simple beams under bending ideal elastoplasticity and in the tension zone, in principle, linear elas-ticity until tensile failure is assumed. A uniaxial failure criterion is used for both compression and tension. With the continuous beams and frames, the compression and tension zone alters its posi-tion. For simplification reasons, linear elasticity in 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 4the spatially limited compression zones of contin-uous beams and frames is modelled instead of ideal elastoplasticity. Figure 3: Computation of mechanical properties The load (F or q) is applied through a step-wise displacement. That results in a particular number of load steps for a simulated test. During these load steps, element failure in the tension zone outside the outermost laminations is allowed (s. Figure 4, a); corresponding elements are iden-tified after each load step and are deactivated by multiplying their stiffness by a severe reduction factor. That leads to an apparently non-linear be-haviour in the tension zone. Figure 4: Locally limited failure accepted during sim-ulated loading (a) and ultimate member failure termi-nating loading (b). Figure 4, b exemplifies the ultimate failure criterion used for the simulated tests. This crite-rion is based on the assumption that no further loading is mechanically possible if the locally cal-culated tensile stress (confer t, Figure 2) of any element in the outermost laminations equals the individual element tensile strength. In case of ten-sion members, failure in either of the two outer-most laminations constitutes ultimate failure. 2.4. Variations of the reference systems and sim-ulation of the load-carrying capacities Figure 5 shows the parameter variations for the two basic members and the two systems. The var-iation range of , , , , n and the material quality is stated by the corresponding lower and upper value. Using the computer model, each defined single parameter combination is examined with 500 up to 3000 single simulations resulting in the total number of simulations given in Table 1. Each single simulation finally leads to an individual load-carrying capacity used to calculate a corre-sponding effective strength value. 3. RESULTS 3.1. Effective and normalised strengths The effective bending (fm) and tensile strength (ft), respectively, is computed according to Eq. (2) with the simulated load-carrying capacity Fmax or qmax. max maxm2max maxmt max / F Mf W Wq Mf W Wf F A (2) Here, is a common tabular value to com-pute the global maximum bending moment ac-cording to elementary beam theory. With the con-tinuous beams, depends on the number of spans (n) and the load configuration (e.g. n = 2 → = -1/3). For the frames, the -values are independ-ent of the stiffness ratio between beam and col-umn, which amounts to 1; in the present cases, -values are calibrated to the frame length ℓ (e.g. = 3/2 → = 1/12 or = 1/8 → = -1/13). W is the section modulus and A the cross-sectional area. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 5 Figure 5: Reference configurations as well as size and material variations with the basic members and systems Table 2: Reference strengths. 1 k Material fk,ref N/mm² 2 k1 A - 24.2 3 C - 27.5 4 D - 28.6 5 E - 31.4 6 G - 32.3 (32.8*) 7 I - 36.7 8 k2 G 1/5-1 28.5 9 k3-k5 A - 22.9 10 B-H - 24.8-35.2 11 I - 38.6 12 k6-k9 G - 33.8 *Assuming linear elasticity in the compression zone for comparable purpose, based on 3000 simulations For the graphic representation, the strength values f (either 5th percentiles of simulated popu-lations or individual simulated strength values) are normalised according to Eq. (3) where fk,ref is the reference 5th percentile stated in Table 2. norm k,ref/f f f (3) With the normalisation, two cases are distin-guished: first, normalising of 5th percentiles of simulated populations and second, of single sim-ulated values. With the first case, which applies to the bending and tension members (later on treated in Figure 6), fnorm amounts to approximately 1 for the corresponding reference member ( = = 1). With the second case, which applies to the contin-uous beams and frames (treated in Figure 7 and 8), the normalised individual strength amounts to 1 for the cumulative frequency of 0.05 and the ref-erence system (n = 1 or = ∞). 3.1.1. Variations of the member size Figure 6 shows the relation between the normal-ised 5th percentile of the bending and tensile strength, respectively, and the corresponding size parameter. The diagram for the bending strength contains individual courses for each employed material quality (MQ) and the one for the tensile strength for each value. In the variation for and , there is a clear dependence between the 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 6stressed member size and the normalised 5th per-centile. The courses change from a steep negative gradient for small - and -values to a slight neg-ative inclination for higher values. Both the mate-rial quality and the depth of tension members show a low influence on the effective strength; however, these effects have to be judged care-fully. As in reality, an increasing material quality usually reduces the strength variation within the material, the simulation results state a contradic-tion. The reason for that could be a model simpli-fication concerning the generation of the errors e in the regression equations; this generation is in-dependent of the material quality. The apparent independence of the tensile strength with regard to the member width could partly be explained by the ultimate failure criterion. The probability of weak spots in the outermost laminations is inde-pendent of the member width, causing most likely the same strength level for narrow and wide mem-bers. Figure 6: Strength and member size; bending (left) and tensile strength (right) 3.1.2. Continuous beams The diagrams in Figure 7 show parts of the cumu-lative frequency distributions between 0 and 20 % of the normalised individual values of the bending strength. The upper diagram exemplifies the sim-ulation results for material quality A and the lower one for I. The horizontal line at 0.05 and its points of intersection with the distributions help to read the normalised 5th percentiles of the bending strength for the continuous beams with 2, 3 and 4 spans. According to the agreement on the strength of the reference system, the normalised 5th percen-tile amounts to 1 for the simple beams (n = 1). Thus, the intersection points with the distributions for n = 2, 3 and 4 indicate the change in effective strength for the corresponding continuous beam. Due to the probabilistic system effect, the 2-span beam, for instance, possesses an effective bending strength which is at least 1.2 times as high as the reference strength. The resulting strength differ-ence makes in particular 2-span beams more effi-cient compared to simple beams since both of them have the same design moment (under com-parable loading and span). Figure 7: Cumulative frequency distributions of the bending strength in continuous beams: material A (top) and I (bottom). 3.1.3. Frames In agreement with the graphic representation of the results for the continuous beams, Figure 8 Normalised 5th percentiles0.70.80.91.01.11.21.31.41.50 1 2 3 4 5MQACDEGI0 2 4 6 8 101/52/53/54/51Cumulative frequency0.000.050.100.150.20MQ=An1 (ref.)234Cumulative frequency0.000.050.100.150.20Normalised strength values0.9 1 1.1 1.2 1.3 1.4MQ=In1 (ref.)23412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 7shows the same proportion of the cumulative fre-quency distribution. The vertical reference lines (through the intersection points between the refer-ence line at 0.05 and the distributions) exemplify the order of the strength modification, which is to be expected for the examined frame variations. Figure 8: Cumulative frequency distributions of the bending strength in frames; vertical reference lines show target values for the factors k6 to k9. 3.1.4. Comparison between two different load configurations The reference strength of 32.8 N/mm² (Table 2, line 6) reflects the load configuration with two single concentrated loads (2 x F) while that of 33.8 N/mm² (line 12) reflects a configuration with equally distributed load (q). In both cases, the sys-tems are simple beams, 600 mm in depth and 10,800 mm in length; the material corresponds quality G. The ratio of both values is 33.8/32.8 = 1.03. Hence, there is a slight increase in bending strength for the configuration with equally distrib-uted load. This numerically determined ratio does not contradict an analytical calculation. Based on formulas, imparted by Isaksson (2003) for the same comparison, and under the assumption of the Weibull exponent of 1/10.8 from Eq. (5), Isaksson’s formulas result in a ratio of 1.034. Consequently, the comparison shows that the nu-merical approach is also sensitive to minor changes in the load configuration. 3.2. Modification factors The diagram in Figure 9 combines the representa-tions in Figure 6. In place of the single courses for the different material qualities, a nonlinear regres-sion curve represents the dots, stating the normal-ised 5th percentiles for the bending strength. The same procedure was applied to the tensile strength. A nonlinear regression curve was fitted to all the depth-dependent normalised 5th percen-tiles (circles). Eq. (4) and (5) define the course of both curves and the calculation of k1 and k2, re-spectively. 1/8.2911 1 600with ( in mm)k hh (4) 1/10.821 1 5400with ( in mm)k (5) Figure 10 shows the relation between the nor-malised 5th percentiles of the bending strength for the 2-, 3- and 4-span-beams and the absolute ref-erence strength for the different material qualities. and Figure 9: Bending and tensile strength modification Figure 10: Determination of the factors k3 to k5. Cumulative frequency0.000.050.100.150.20Normalised strength values (k6-k9)0.9 1 1.1 1.2 1.3inf. (ref.)3/21/21/41/8Depth (k1) and length factor (k2)0.70.80.91.01.11.21.31.41.50 1 2 3 4 5 6 7 8 9 10Norm. 5th pctl. (bending)Eq. 4Norm. 5th pctl. (tension)Eq. 5Normalised 5th percentiles (k3-k5)0.91.01.11.21.31.4Reference strength (n = 1) in N/mm²20 25 30 35 40MQ=A MQ=In 2 3 412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 8As an example, vertical lines mark the quali-ties A and I. The regression lines were fitted to each of the three relations. They prove more or less that the strength modification is independent on the simulated material quality and strength, re-spectively; all straight lines run almost horizon-tally. Based on Figure 10, the three modification factors can be read as constant values. Eq. (6) specifies their order. 3452 1.233 1.074 1.13n kn kn k (6) The modification factors for the examined frame variations are specified in Eq. (7). As no material variation is performed in the frame ex-amination, the factors were directly taken from Figure 8. The values below agree with those of the vertical reference lines running through the inter-section points between the distributions and the 5th percentile line. Since each of the frames is exam-ined with 3000 single simulations, stable k-factors for the frames can be assumed, although no mate-rial variation was superposed. 67893 / 2 1.041/ 2 0.991/ 4 1.151/ 8 1.26kkkk (7) 4. CONCLUSIONS The numerical examination of bending and ten-sion members and various structural systems made of glulam reveals pronounced differences in strength values caused by a change in the stressed member size and the applied structural system, re-spectively. The application of the modification factors, proposed in the paper, lead to a more balanced re-liability in particular between small and large as well as short and long glulam members under bending and tension, respectively. The examina-tion of various structural systems shows that 2-span-beams and 2-hinged-frames with short and stiff columns provide a favourable probabilistic system effect so far not fully reflected if a stand-ard verification of the bending strength is per-formed. The extension of the strength simulation to more complex structural systems still requires suitable input data concerning stress interactions between normal and shear stresses, its implemen-tation in the computer model and appropriate models including the effect of normal forces in frames. The interactions concern particularly ar-eas with high discontinuity as supports or frame corners. 5. REFERENCES DIN EN 1990:2010-12 Eurocode: Grundlagen der Tragwerksplanung (Basis of structural design). Thelandersson, S. (2003). Introduction: Safety and Serviceability in Timber Engineering. In: The-landersson, S. and Larsen, H.J. (Ed.): Timber Engineering. Wiley & Sons, Chichester. Colling, F. (1986a). Influence of volume and stress distribution on the strength of a beam with rec-tangular cross section – Derivation of a general relationship with the help of a 2-parameter Weibull-distribution. Holz als Roh- und Werkstoff 44, 121-125. Colling, F. (1986b). Influence of the volume and the stress distribution on the strength of a beam with rectangular cross section – Determination of the fullness-parameters, examples. Holz als Roh- und Werkstoff 44, 179-183. Isaksson, T. (2003). Structural Timber – Variability and Statistical Modelling. In: Thelandersson, S. and Larsen, H.J. (Ed.): Timber Engineering. Wiley & Sons, Chichester. Foschi, R.O., Prion, H.G.L., Folz, B. and Timusk, P.C. (1996). Reliability based Design of Glulam Beams. International Wood Engineering Con-ference, New Orleans, USA, Vol. 1, 125-130. Hansson, M. and Thelandersson, S. (2003). Capacity of timber roof trusses considering statistical sys-tem effects. Holz als Roh- und Werkstoff 61, 161-166. Frese, M. (2010). Computer-aided simulation of glu-lam strength parallel to grain. IV European Con-ference on Computational Mechanics, Paris, France, Paper ID 382.