STRUCTURAL SAFETY
Structural Safety 29 (2007) 308–321 www.elsevier.com/locate/strusafe
Random imperfection fields to model the size effect in laboratory wood specimens Sara Casciati *, Marco Domaneschi Department of Structural Mechanics, University of Pavia, via Ferrata 1, I 27100 Pavia, Italy Available online 26 September 2006
Abstract The composite nature of a wood continuum prevents one from extrapolating the results of laboratory tests on standard wood specimens to structural elements of significant size. Therefore, these elements are usually tested under standardized loading conditions in order to detect a sort of average material behaviour. In this paper, the initial step consists, instead, of testing the material specimens. The extension of the results to structural elements is then pursued by introducing a random field, or, in a discretized model, a random array of imperfections. The calibration of the suitable spatial distribution of the imperfections is then investigated by a mixed experimental– numerical approach, for a reference beam. The analyses on the relative finite elements model are iterated to match the response of the full scale laboratory tests. 2006 Elsevier Ltd. All rights reserved. Keywords: Biaxial tests; Finite element model; Imperfections; Laboratory tests; Random field; Wood specimens
1. Introduction ‘‘Black locust’’ specimens of square section were carefully prepared and visually graded by Prof. Uzielli of the University of Florence [1]. The specimens were cut along the likely principal directions of an orthotropic model consistent with the fibres orientation [2]. The wood presents its fibres organized in parallel planes, so that the three reference directions will be: (i) along the fibres, (ii) orthogonal to the fibres and parallel to their planes, and (iii) orthogonal to the fibres and to their planes. The identification of the parameters of an orthotropic material was achieved by performing several static tests on different specimens in each of the above directions. This material characterization is the initial step for a further study. A beam of square section, made of the same material of the specimens, is used to undergo a classical bending test. Its finite element model is also generated within the general purpose software described in Ref. [3]. Following the idea originally formulated in [4], a random array of imperfections is added to the model to make the numerical response match the experimental results. The correlation structure of such an array is also discussed. *
Corresponding author. Tel.: +39 0382 985 787. E-mail address:
[email protected] (S. Casciati).
0167-4730/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.strusafe.2006.07.014
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The approach proposed in this paper provides a short path following which one can account, into the modelling, for the spatial variability of the material properties, as well as for the size and shape effects. The idea also offers a timely basis for extending structural identification schemes to the monitoring of existing (old and ancient) timber buildings. 2. Theoretical framework Probabilistic risk assessment of structural systems is today widely accepted and provides the basis for the so-called ‘‘Performance Based Design’’ [5]. In this context, probabilistic models for loadings and material properties are needed in order to prevent the designer from the extensive statistical effort of data gathering. Some areas of structural mechanics still offer criticism to the method, due to the fact that they are mainly characterized by experimental bases whose results are difficult to extrapolate to different situations. This is mainly related to both the spatial variability of the material properties, and to the size and shape effects. Timber structures are the example investigated in this paper. The variety of the wood species and the inhomogeneity of the material properties imply that an EU probabilistic code for this type of structures is still under development, because it requires the a priori availability of a large number of databases and of a robust mechanical model, whose related uncertainties are to be identified and probabilistically characterized. The strength distribution within a structural member is, indeed, dependent on several factors, among which one recalls the element size and the knot-size frequencies within each grade. Many studies which address the issue can be found in literature [6–8], and some of their main results are summarized in the following. One could refer to [9] for a general overview on the subject, and, in particular, for the problem of modelling the variability of bending strength within and between members of structural timber, under different length and loading configuration effects. However, a different approach, which can be used for a wide range of applications and considers wood just as an example of strong spatial variability of the material properties, is here formulated in view of progress in structural monitoring and diagnostics. In general, one assumes that a global fragility analysis [10] needs to be performed from scratch. The following steps can be envisaged as: (1) (2) (3) (4)
material selection; extensive testing of specimens in all the structural situations involved in the system; running numerical structural analyses on mathematical models; accounting for the model error and imperfections which are included in the following general expression: zðx; yÞ ¼ z0 ðx; yÞ þ e
ð1Þ
where the underlined symbols denote vectors of random variables. In particular, the vector x collects the structural design variables, the y’s represent the parameters of the probabilistic models. The scalar z can either take the meaning of a response variable or of the performance function itself, z0 denotes its estimation from the numerical analyses, and e is the error term. Some material properties, such as the resistance of brittle materials, show the so-called size effect, which is classically approached by linking small volumes together into a series model with the purpose of characterizing the global volume [10]. In several cases, this yields to the assumption of an extreme probability distribution model of type III (i.e., a 2-parameter Weibull) for the material strength [6]. In order to calibrate the model parameters, a fair amount of data on the strength distribution is needed, and the data availability depends on the particular type of material under consideration. In particular, data on both clear wood and laminating stock are collected in [5], being the first ones needed in order to successively apply the adjustment factors which account for the influence of knots. Numerical simulations are then carried out to obtain estimates of the quantities variability which fit the ones observed from the data. Tensile tests of individual laminations are pursued within this framework; the laminations effect is successively introduced by a strength amplification factor. In this paper, instead, clear wood specimens of solid structural timber are tested in order to characterize an orthotropic material model, so that the test results can be extended to a global material behaviour through a shortened path. In this case, the same numerical model can be used both for the data fitting and
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for the calculations of the loading effects. However, it must be emphasized that the goal of the present work is not to closely represent the spatial variability of the material properties values, but to simply reproduce its effects by matching the results of bending tests on a beam model. For this purpose, the material properties estimated from the tests on the material specimen represent an adequate initial guess. At the end of the process, the strength distribution which produces the maximum deflections observed from the beam tests is found. Furthermore, the maximum beam deflection obtained from the final calibration of the finite element model can be used to calculate an apparent modulus of elasticity for the entire beam, using the maximum deflection equations of elementary beam theory. In [11], instead, the lengthwise variability of the modulus of elasticity (MOE) of lumber is investigated by using a second-order Markov model to generate serially correlated MOEs along each equally long segment of a lumber piece. The final MOE value associated to the segment is obtained by multiplying an index, computed as the ratio between each correlated MOE and the average MOE of the lumber piece, by a single random observation from a distribution of MOE. It appears evident that, when the attention is focused on wood, the size effect is not only related with the actual dimensions of the structural members, but that it also depends on the no-homogeneity of the material, which often occurs due to the presence of knots. This situation suggested the authors to replace Eq. (1) with zðx; y; gÞ ¼ z0 ðx; y; gÞ þ e
ð2Þ
where g is a random field of imperfections across the body. The correlation structure of this random field needs to be investigated and it depends on both the size of the knots and on its regularity. The model in Eq. (2) could be translated into practice by regarding the imperfections as volumes of minimal resistance and by estimating both the per-cent of volume which ensures the best accuracy and the minimum size of these imperfection volumes. The next sections will discuss the details of this approach by referring to a case of study. It must be, however, emphasized that the simplified result loses any probabilistic nature and the approach should be thought in terms of structural robustness [12], rather than in terms of structural reliability. Indeed, the proposed procedure can be seen as a mapping of Eq. (2) into a space where the random field g is replaced by a finite number of information, which is represented by the percentage of volume and by the minimum size of the imperfections. Obviously, this process implies a loss of information by preserving the average behaviour, but not the associated scatter. 3. Wood specimens characterization A wide series of ‘‘black locust’’ wood specimens was made available by the University of Florence [1]. Their square section size is 15 mm, and they have a length of 100 mm. The specimens show different orientations of the fibre planes according to the three principal directions along which they were cut: two horizontal directions in the plane of the growth rings of the tree, oriented radially and tangentially to the rings, respectively; and one vertical out of plane direction. Fig. 1 shows the specimens oriented as their cutting directions. It can be noted that the vertical and the tangential specimens present fibres parallel to their axes, while the radial one has fibres oriented orthogonally to its axis. For each direction, a large number of specimens is tested [13]. Before starting the experiments a humidity– temperature conditioning program is applied to the specimens: 48 h under 22 C of temperature and 50% humidity. It results in about 10% of humidity in the wood specimens [2]. The wood specimens characterization is conducted by the universal bi-axial material testing machine, which is available at the Teaching Laboratory of Structural Mechanics of the University of Pavia. The tests are carried out in control of the axial displacement. The imposed displacement and the load are measured variables. The stress is then computed as the ratio between the load and the section area. A mono-axial extensometer (Fig. 2) is used to measure the local deformation. The presence of the extensometer is necessary to clean the measurements from the deformation of the testing machine and from the wedge grip inaccuracies, which affect the displacement values read by the global LVDT (linear variable displacement transducer). Each specimen is tested three times by placing the extensometer with different
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Fig. 1. ‘‘Black locust’’ wood specimens oriented as their cutting directions: Radial and tangential to the tree rings, and vertical, out of the rings plane.
orientations: (1) along the longitudinal direction; (2) along the transversal direction, on the face parallel to the fibres plane and denoted as ‘‘a’’ in Fig. 1; and (3) along the transversal direction, on the face ‘‘b’’ orthogonal to the fibres plane. First, the tests with the extensometer in the transversal direction are performed by applying to
Fig. 2. Wood specimen equipped with a mono-axial extensometer, mounted on the universal material testing machine.
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the specimen two cycles of loading–unloading in traction, up to 0.03 and 0.06 mm, respectively. Then, the specimen with the extensometer in the longitudinal direction undergoes four cycles of loading–unloading in traction, up to 0.03, 0.06, 0.09, and 0.125 mm, respectively, and two successive cycles of loading–unloading in compression, down to 0.06 and 0.125 mm, respectively. The extensometer is finally removed, and the specimen is brought to rupture. Fig. 3a provides, as an example, the experimental stress–strain diagram obtained from the vertical specimen labelled as V3, with the extensometer placed along its axis. Instead, Fig. 3b and c correspond to the tests on the same specimen, but with the extensometer in tangential direction on faces ‘‘a’’ and ‘‘b’’, respectively. Since the measured load is always longitudinal, the transversal and longitudinal deformations are plotted together vs. time for comparison. The results in the longitudinal and transversal directions of the wood specimens are then elaborated in order to evaluate the longitudinal (E) and transversal (m) elasticity modules. Table 1 reports the values obtained for the three sets of specimens in Fig. 1. As an example, three different specimens are considered for each class. As expected, the stiffness of the wood specimen is maximum when the specimen is loaded along the fibres direction; the Poisson modulus is minimum when the load is applied orthogonally to the fibre planes. Since for an orthotropic medium, three normal and three transversal elasticity modules, together with three shear modules, for a total of nine material parameters, are needed, torsion tests are also conducted. The results of the tests are drawn in Fig. 4a and b. In the range of strain up to 0.01%, the average G values are 150 and 800 MPa on the bases and on the lateral surface, respectively.
Stress [MPa]
15 10 5 -0.001
0 -5 0
-0.0005
0.0005
0.001
-10 -15
a
Strain
Strain
Top Line = Longitudinal 0.0006 0.0004 0.0002 0 -0.0002 0 -0.0004
5
b
Strain
10
15
20
Time [s] Top Line = Longitudinal
c
Bottom Line = Transversal, Face a
0.0006 0.0004 0.0002 0 -0.0002 0 -0.0004
5
Bottom Line = Transversal, Face b
10
15
20
Time [s]
Fig. 3. Experimental diagrams obtained from the vertical specimen V3, with fibres along its axis: (a) longitudinal stress–strain relationship; (b) and (c) are the strain time histories in both the longitudinal (top line) and transversal (bottom line) directions, with the transversal extensometer on the face parallel and orthogonal, respectively, to the fibres plane.
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Table 1 Characterization of the wood specimens Vertical specimens
E (MPa) ma mb
V1
V2
V3
18425 0.3816 0.4018
17079 0.4527 0.4494
15876 0.4022 0.3614
Radial horizontal specimens
E (MPa) m
HR1
HR2
HR3
2293 0.0644
2626 0.0614
1785 0.0646
HT1
HT2
HT3
3002 0.0636 0.4592
2342 0.0865 0.3886
2511 0.0619 0.4005
Tangential horizontal specimens
Tangential stress [MPa]
E (MPa) ma mb
4 3.5 3 2.5 2 1.5 1 0.5 0 0
0.02
a
0.04 Angular strain [%]
0.06
0.08
Tangential stress [MPa]
14 12 10 8 6 4 2 0 0
b
0.01
0.02
0.03 0.04 0.05 Angular strain [%]
0.06
0.07
0.08
Fig. 4. Results of the torsion tests: (a) specimen with axis normal to the fibre planes and (b) specimen with axis along the fibre direction.
Together with these two values, the averages of the data in Table 1 for the other modules in each of the three directions are assumed as material properties in the numerical simulations presented in the following sections. 4. Flexional tests of wood beams The flexional test prescriptions for the parameters evaluation of a wood continuum are detailed in Ref. [2], as well as in the codes (see [14], among the others). A test facility for static experimentation is assembled as in
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a
40mm 130mm 130mm
130mm 130mm 800mm
b Fig. 5. Flexional test scheme: (a) test setup and (b) geometric characteristics.
Fig. 5a, with the geometric characteristics specified in Fig. 5b. The beams are produced of the same ‘‘black locust’’ wood utilized for the specimens characterization in the previous section. The static scheme is symmetric in loads and constraints. Three beams [6], with square section side of 40 mm, length 800 mm, and fibres along the beam axis, are considered. The strain is measured with the extensometer applied at the midpoint of the bottom surface of the beam. The tests are repeated twice by placing downward either the face orthogonal to the fibres plane or the parallel one, as shown in Fig. 6a and b, respectively. Fig. 7 summarizes the results of the experiments on the three beams in terms of: • measured displacement vs. load; • measured strain vs. computed stress (by the isotropic bending formula). The limited laboratory capacities justify the use of beams of reduced scale. Although one acknowledges the need of coupling with experiments on specimens of structural size, the tests results are sufficient to characterized layers of beam cells or beam segments, as those considered in [6,11], respectively. The integrity of the beam can then be reconstructed by amplification factors which account for the actual lay-up of the beam segments and can be calibrated on the monitoring data.
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4 3.5 3 2.5 2 1.5 1 0.5 0
[KN]
[KN]
Fig. 6. Beam configurations: (a) face perpendicular to the fibres plane downward and (b) face parallel to the fibres plane downward.
0
2
4
4 3.5 3 2.5 2 1.5 1 0.5 0 0
6
2
4
6
[mm]
25
25
20
20
15
15
[MPa]
[MPa]
[mm]
10
10 5
5 0 0
0,0005
0,001
a
0,0015
0,002
0 -0.0002
0.0003
0.0008
0.0013
b
Fig. 7. Load vs. displacement (top), and strain vs. stress (middle) for the two beam configurations: (a) face perpendicular to the fibres plane downward and (b) face parallel to the fibres plane downward. (In the two stress–strain diagrams, one of the three curves does not report all the points, since the extensometer was sliding at the end of the test.)
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5. Numerical model of the reference beam The numerical model of the beam tested in the previous section is generated by a finite element discretization using three-dimensional, 8-nodes brick elements [3,4]. Fig. 8 shows a mesh made by elements of cubic shape of side 10 mm, together with the adopted reference system. The final model consists of 1280 brick elements, 2025 nodes, and 6075 degree of freedom. Fig. 8 also illustrates the imposed boundary conditions: the external upward arrows schematize the two vertical supports, while the loads are represented by the internal downward ones. In the middle section, the boundary conditions prevent the section from any out of plane movement. The material behaviour is idealized as elastic and orthotropic. The beam is modelled with the axis parallel to the fibres planes, but its external surfaces can either be parallel or orthogonal to the fibres planes. Table 2 summarizes the values of the elasticity modules used for the numerical simulations, under the test conditions of Fig. 6b and with the reference axes x, y and z as represented in the finite element code which produced Fig. 8. For the beam oriented as in Fig. 6a, the sets of parameters related to the y and z axes are
Fig. 8. Boundary conditions including vertical displacement constraints, loads and symmetry specifications. Table 2 Material parameters, adopted values for the case of Fig. 6b Orthotropic (wood continuum) Gxy (MPa) Gxz (MPa) Gyz (MPa) Exx (MPa) Eyy (MPa) Ezz (MPa) mxy myz mzx myx mzy mxz
800 800 150 17,126 2617 2233 0.412 0.416 0.053 0.063 0.355 0.406
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exchanged one with the other. It must be mentioned that only three transversal (Poisson) modules are independent, being the other six values related by the three relationships which govern the constitutive law of the orthotropic elastic model [15]. In the simulations, the load is introduced step by step, with the same incremental values used for the laboratory tests. The numerical analyses are developed in the linear elastic domain, under the assumption of small displacements and deformations. The results of these numerical analyses show values of the maximum displacement which are roughly one half of those measured at the end of the laboratory tests. Some analyses were also conducted in large displacement configurations, but the results are equivalent. The idea is, therefore, to introduce a random array of imperfections in order to fit the flexional tests results. These imperfections have the effect of interrupting the development of the wood fibres along the beams. 6. Adding imperfections to the numerical model In the finite element model, the local imperfections are simulated by a random distribution of isotropic elements of a very low stiffness (E = 0.001 MPa; m = 0.25). Several different configurations are investigated. In the context of a preliminary study, the location of the weak components in each configuration is generated as a sample from an uniformly distributed random variable, whose realizations can range from 1 to the assumed number of finite elements. The software environment Matlab [16] is used for this purpose. As the analyses are repeated, the sample size is gradually increased so that it represents from nearly 5% to 20% of the beam volume. The main physical response quantities are recorded. In particular, the values of the maximum beam displacement and of the maximum strain are selected as reference, since they are the quantities measured in laboratory as function of the imposed loads. It is worth noting that the strain value is strictly connected to the local situation. Thus, if one evaluates the strain near an imperfection volume, the value can result very large but meaningless for the simulation aim. Fig. 9 shows the strain distribution for a single realization of imperfections. It emphasizes how the strain is not reliable for model comparisons. Hence, the attention will be focused only on the maximum displacement values.
Fig. 9. The local influence of imperfections: strain results obtained by a run of MARC [3] with the mesh of Fig. 8. The central part of the beam is represented. The 16% of volume is covered by imperfections and the last load step is considered.
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The numerical simulations are influenced by the volume and by the location of the imperfections. In order to prevent critical situations, the beam finite element models are all developed by avoiding the presence of imperfection elements where the forces and the displacement constraints are assigned. The geometry of the body stays the one of the beam tested in laboratory, but different models can be envisaged where: 1. the imperfections are differently located; 2. the volume of the imperfections is differently assigned; 3. the size of the imperfection volume is differently selected. The last item requires a preliminary discussion about two further aspects: (a) the extension of the imperfections depends on the wood knots average size; the wood under investigation shows rather regular knots of size 1 cm; (b) once the size of the imperfections has been selected, the question arises whether or not the finite element mesh may subdivide an imperfection volume into several elements. Consistently with the first remark, three values of the imperfections size are considered: 1 cm, i.e., the knot size, 0.5 cm and 2 cm, i.e., the halved and the doubled knot size, respectively. Fig. 10 shows the results of two simulations, with reference to the case of Fig. 6b and to a mesh of elements of size 0.5 cm. The maximum displacement is plotted vs. the percentage of imperfections volume. One determines how the maximum values of the displacement varies with the amount of imperfections introduced in the finite element model and expressed in terms of global volume percentage. The simulation results present a non linear trend of the physical quantities as the number of imperfections in the model increases. It is worth remarking that: • the lower line is obtained by considering independently distributed imperfections of the same size of the elements which form the beam mesh; • the upper line represents the same beam mesh, but with clustered imperfections; specifically, the imperfections size is doubled, i.e., it is 1 cm vs. an element size of 0.5 cm. These results bring evidence to the fact that the subdivision of the imperfection volumes into several elements produces a significant increase of the estimated displacement. This distortion should be avoided and,
Fig. 10. MARC [3] simulations of the maximum displacement vs. the volume of imperfections. Beam as in Fig. 6b. The lower line corresponds to a mesh of elements of size 0.5 cm and independently distributed imperfections. The upper line represents the same mesh with clustered imperfection.
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15% imperfections
Applied load [KN]
4 3.5 3 2.5 2 1.5 1 0.5 0 0.00
0.50
1.00
1.50
a
2.00
2.50
3.00
3.50
4.00
4.50
5.00
Max displacement [mm] 20% imperfections
4
Applied load [KN]
3.5 3 2.5 2 1.5 1 0.5 0 0.00
b
1.00
2.00
3.00
4.00
5.00
6.00
Max displacement [mm]
Fig. 11. MARC [3] simulations of ten realizations of load–displacement curves: (a) 15% of volume of imperfections and (b) 20% volume of imperfections. Beam as in Fig. 6b.
hence, the analyses reported in the following are conducted with the size of the imperfections identical to the size of the mesh elements. Therefore, in all the further cases, the imperfections are distributed across the body without any correlation. Five realizations of imperfections were first generated with an elements size of 2 cm. The plot of the response vs. the imperfections volume was characterized by maximum displacements with excessive average values and dispersions as the abscissa exceeded the 4%. The size of the mesh finite elements and of the imperfections is then assumed equal to 1 cm. Fig. 11 shows, for two different values of imperfections volume percentage, the load–displacement results of ten simulations, with reference to the case of Fig. 6b. The maximum displacements are plotted vs. the imperfections volume in Fig. 12. The simulations are also repeated for the case of Fig. 6a. Due to the predominant fibre behaviour, the simulation results are equivalent to the ones obtained from the configuration of Fig. 6b. This is in agreement with the similar behaviour observed from the experiments illustrated in Fig. 7a and b. On the basis of Fig. 12, one could say that a 15% volume of imperfections leads the analysis to results which well fit the ones of the laboratory tests. Finally, Fig. 13 provides the results of the simulations which are performed by assuming, for each one, a different realization of the imperfections, and by setting at 0.5 cm the size of the mesh finite elements and of the imperfections. These realizations, in number of five, are compared with the previous ten of Figs. 11 and 12. The denser mesh, with elements size of 0.5 cm, achieves a good estimation of the experimental results for a volume of the imperfections of 18–20%, vs. the 15% estimated with the elements of size 1 cm. However, the dispersion of the results is reduced.
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Max displacement [mm]
5.5 5 4.5 4 3.5 3 2.5 2 0
5
10
15
20
% of imperfection volume in the beam Fig. 12. MARC [3] simulations of the maximum displacement vs. the volume of imperfections. Beam as in Fig. 6b.
Fig. 13. MARC [3] simulations of the maximum displacement vs. the volume of imperfections. Beam as in Fig. 6b. The thick lines correspond to a mesh of elements of size 0.5 cm. The other lines report, for sake of comparison, the ten simulations for elements of size 1 cm.
7. Conclusions The experimental estimation of the elasticity parameters for wood, regarded as an orthotropic material, represents the initial step of this study. A beam of square section, made of the same material, is then tested in the laboratory according to the standard flexional experiment. A finite element model of this beam is generated. In order to match the test results, a random field of imperfections is introduced into the finite element model. It is shown that the imperfection size dominates the achievable accuracy of the resulting model. The size of the knots should be regarded as an important reference measure in establishing the imperfection size. For the specific case investigated in this paper, knots of regular size of 1 cm were observed and this resulted to be a good assumption for the imperfection size. Of course, additional investigations on the same wood and on woods of different nature are needed. Furthermore, studies on the same beam with different loadings, and on different structural systems, should be performed before that the approach formulated in this paper can be generalized. In this preliminary phase, the main limitation to the approach can be found in the rough modelling of the knots spatial variability, which can be further refined. The advantage is that the procedure allows one to perform a limited number of experiments on large-scale models and to exploit, at the same time, the results from the tests on small material specimens. The prospect which justifies the proposal was identified with the devel-
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opment of monitoring and diagnosis techniques for existing buildings. In this context, one tests several material specimens and also collects global response data (e.g., under ambient excitation) from the sensors distributed across the structure. In order to link together the local and global information, the framework proposed in this paper may result in a convenient tool toward convergence. Acknowledgements This research was developed within the COST E24 program ‘‘Reliability Analysis of Timber Structures’’, with Professor P. Castera serving as chairman. It was funded by the Italian National Research Council (CNR) through a research program, for which Prof. A. Di Tommaso, of the University of Venice, Italy, was serving as national coordinator. The authors are also indebted with Professor Uzielli, of the University of Florence, Italy, for providing them the appropriate wood specimens, and with Ing. Mauro Mottini for his assistance in carrying out the laboratory experiments. References [1] [2] [3] [4] [5] [6] [7]
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