A PROPOSAL FOR A STANDARD PROCEDURE TO ESTABLISH THE SEISMIC BEHAVIOUR FACTOR q OF TIMBER BUILDINGS
Ario Ceccotti1, Carmen Sandhaas2
ABSTRACT: A simplified method for the determination of the seismic behaviour factor q of timber buildings is presented. The proposed approach is a hybrid approach with element testing such as cyclic testing of wall elements combined with numerical modelling using the test results as input parameters for complete building models. The method combines non-linear-in-the-time-domain dynamic modelling of 2D or, better, 3D building models. The models are spring - lumped mass models. The mechanical behaviour of the buildings is determined by the springs; all other components are rigid. The springs are calibrated on reversed cyclic testing data of large-scale elements such as shear walls. With this method, computationally efficient and stable models can be developed which can cover many different geometrical setups or mass distributions. Subjecting these building models to different earthquakes and increasing the seismic intensity until near-collapse, behaviour factors q for the simulated construction typologies can be derived.
KEYWORDS: seismic design, behaviour factor q, modelling, timber buildings
1 INTRODUCTION 123 With the last earthquakes in the world and the awareness of the increased importance of highly performing timber construction in seismic regions, the urge of developing proposals for an effective seismic design of timber construction became more and more important. A well known simple method (when meeting certain construction rules) to design structures under earthquake loading is the method used in seismic codes as EC8 [1], for example, where a simple global linear-elastic analysis is required for timber structures. Then a so-called behaviour factor q is introduced reducing “the forces obtained from a linear-elastic analysis, in order to account for the non-linear response of a structure, associated with the material, the structural system and the design procedures”. Once the elastic seismic actions are reduced by q, designers are allowed to verify stresses on structural elements and connections using the same design values as of the relevant static code (i.e. Eurocode 5, in Europe). Actually q represents the ability of the structure to dissipate energy and to withstand large deformations 1
Ario Ceccotti, CNR-IVALSA – Trees and Timber Institute, Italian National Research Council, Via Biasi 75, 38010 San Michele all’Adige (TN), Italy. Email:
[email protected] 2 Carmen Sandhaas, Timber Structures and Wood Technology, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands. Email:
[email protected]
without ruin. At the same time, however, it is evident as per se, that chosen q values are dependent from the design code itself. In fact q is just a modification factor in a design procedure and it simply must be such that the objectives of the design procedure are satisfied. This method leads to the need of establishing behaviour factors q for all different structural systems which one may want to design with this simplified approach. Fullscale testing may be used, but is very expensive and it is nearly impossible to cover all the different structural systems, the variations between the different systems and different earthquake loadings to cover the response spectra for different regions. Therefore, small-scale tests including quasi-static reversed cyclic testing on components or substructures are done whose outcomes are then used for non-linear dynamic modelling of buildings. This paper proposes a combined testing-modelling approach to determine q-factors for timber building systems. The necessary testing includes monotonic and cyclic tests whose results are used to develop a mathematical model. These models are then subjected to different seismic actions and the peak ground acceleration (PGA) values of the earthquakes are increased until reaching a near-collapse failure criterion. With this PGAnear-collapse, a behaviour factor can be derived. The developed modelling technique in the timedomain using a nonlinear dynamic FE-package is presented and the method to derive q-factors is explained.
2 RELEVANCE OF q-FACTOR IN SEISMIC DESIGN This paper proposal to quantify q values for timber buildings, is simply that the peak ground acceleration (PGA) of different earthquakes at which near-collapse status of the building was reached is divided by the design PGA with which the building was designed elastically, according to the code in use:
However, this concept is difficult to use in timber constructions as typical load-displacement curves do not have a well-defined yield point. An example of a monotonic test on a X-Lam shear wall is given in Figure 1. No clear yield point can be determined. As the ductility concept is sensitive to the location of the yield point, the problems correlated with the uncertainty in defining a yield point are evident. 120
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Another method to determine the behaviour factor q is to approach the issue from the reaction side and not from the action side. Strictly speaking q-factor can also be defined by the ratio:
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where Relastic = seismic base shear assuming linear-elastic behaviour, Rplastic = seismic base shear real accounting for real non-linear behaviour. In some way this definition is code independent, i.e. it represents a “real” q-value, instead of a “conventional” design-code based q-value, as it is considered here for the sake of this paper. The higher the q-factor, the lower the seismic base shear. In other words, the more energy a structure is dissipating, the higher the q-factor. It is therefore of crucial importance that a proper behaviour factor q for different structural systems is evaluated in order to undertake a reliable seismic design.
3 DETERMINATION PROCEDURES FOR BEHAVIOUR FACTOR q Basically, there are three methods to derive a behaviour factor q for building systems. First attempts to define the behaviour factor q were related to the concept of static ductility ratio as the ratio of ultimate displacement over yield displacement. In EC8 [1], construction typologies are assigned to ductility classes. Three ductility classes exist: low ductility class with a correspondent upper limit value of q=1.5; medium ductility class with a correspondent upper limit value of q=2.5; high ductility class with a correspondent upper limit value of q=5. The three different classes must each fulfill certain requirements of static ductility ratio in order to ensure that the given q-factors may be used. For instance, in medium ductility class, “the dissipative zones shall be able to deform plastically for at least three fully reversed cycles at a static ductility ratio of 4” [1]. The strength impairment between cycles should not exceed 20%.
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Figure 1: Monotonic test on X-Lam shear wall
For instance, in Figure 1 the yield point could be defined at a displacement of 5mm but also at a displacement of 10mm which leads to a twice as high ductility and therefore to a different behaviour factor q. A clear definition of the onset of plasticity is needed in order to derive a correct yield displacement. The second method is purely experimental. A large experimental programme can be set up with full-scale earthquake tests. This approach includes the following steps: design the building with q=1 (elastic) and a chosen PGAdesign value; undertake full-scale shaking table tests on the building increasing the seismic intensity until a previously defined near-collapse criterion is reached; note the PGAnear-collapse value during which the near-collapse state was reached during the test; solve for qtest as the ratio PGAnear-collapse over PGAdesign; qtest is the experimentally established behaviour factor q. The thusly established behaviour factor q is only valid for the tested building and the chosen earthquakes. In order to generalise the q-factor, more tests on different buildings (same construction technology, different geometry and masses) using different earthquakes must be done. This of course is very costly and timeconsuming and rather a theoretical approach as it is not practicable. The last method is a hybrid approach which is nearly exclusively used in research. Instead of undertaking fullscale testing, buildings are modelled and subjected to earthquake loading. The modelling can take place at many different modelling levels; each level requiring however test results as input parameters. At what construction scale these tests must be undertaken depends on the scale of the model itself. For instance, if
the modelling starts at material level, then material tests must be undertaken to establish the mechanical properties necessary for mathematical modelling. This may include material tests on timber and tension tests on fasteners. An example would be explicit modelling of nails fixing the sheathing to a timber frame for woodframe houses. The next hierarchical level are models starting at the scale of structural elements. If considering again woodframe houses, this next hierarchical modelling could start at wall-level using complete shear-wall tests as input parameters neglecting the behaviour of the single nails. However, due to computational limitations, whole buildings can hardly be modelled starting at material level. More simplified models such as models that start at shear-wall-level, calibrated on connection and element tests are more promising. The differences within this method lie hence in the different model scales and assumed simplifications. The procedure with mathematical models to establish the behaviour factor q is similar to the experimental method, the difference lies in less effort to do testing: model design building with q=1 using test results as input parameters; apply earthquakes on the numerical building model and increase PGA until near-collapse state of building is reached; solve for q. Therefore, the most promising approach seems to be a hybrid approach with higher-level element testing such as cyclic testing (for instance according to EN12512 [2]) of wall elements combined with numerical modelling using the test results as input parameters for complete building models. Testing is necessary to establish system properties under fully-reversed cyclic loading. The complex loading conditions appearing during an earthquake are thus simplified using cyclic loading protocols. Like this, the models will be calibrated on larger-scale testing than pure material testing. For an efficient determination of the behaviour factor q, nonlinear dynamic computer models of whole, simplified buildings are better suited than models simulating every single building component. The simplified and fast models have another advantage when considering the number of simulations that must be carried out. A large amount of necessary simulations is needed as the building’s eigenfrequency must be changed and many different earthquakes must be applied. Such an approach for a reliable and effective seismic modelling will be presented in the following paragraph.
construction typologies are simulated at the “biggest logical scale” at which testing is still possible in terms of costs, time and feasibility. The biggest logical scale mostly means complete shear walls. Simplified models calibrated on these, more realistic, test results will be efficient and reliable. As for the testing, this means reversed cyclic testing of shear wall assemblies. The shear walls may be single wall panels (with fasteners) or walls with and without openings assembled with single wall panels. Which ones are chosen, depends strongly on the complexity of the model, for instance if 2D or 3D and on the dimensions of the building to be modelled. A part from tests on shear walls, other test results may be needed in order to properly model a complete structure. For instance, the behaviour of interstorey connections or vertical joints between wall elements may be necessary together with information on the rigidity of the diaphragms (floors and roofs). 4.1.1 Traditional timber-frame and woodframe Due to the structural similarity, traditional timber-frame buildings and modern woodframe buildings can be modelled analogously. Figure 2 shows a typical woodframe wall with a window opening. A single wall panel is marked by the dashed circle. Figure 3 was taken from [3] where seismic behaviour of traditional timber-frame buildings was assessed. Above, a wall panel with an infill of hazelnut branches and mortar (slaked lime and sand) is shown. Below, a stiffer panel with stone infill is shown. Figure 4 shows a typical cyclic force-displacement graph of the panel with stone infill.
Figure 2: Typical woodframe wall, detail one panel [4]
(a)
4 MODELLING APPROACH 4.1 MODELLING OF CONSTRUCTION SYSTEMS The hybrid approach will be exemplified on two main modelling techniques for two different construction typologies. As already stated, the main background is the strong simplification of entire buildings for modelling purposes in order to obtain efficient numerical models for the estimation of the behaviour factor q. Therefore,
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Figure 3: Traditional timber-frame panels, (a) hazelnut branches infill and mortar, (b) stone infill [3] 80
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Figure 4: Cyclic test result of panel with stone infill [3]
Both panel types can be modelled by spring models with lumped masses where rigid members are representing the framing and rotational springs are representing the global behaviour of the panel. Such a model is presented in Figure 5. The rotational springs are calibrated on cyclic test results; only these springs represent the behaviour under lateral loading. The timber framing is modelled as being rigid in order to be sure that only the springs are working. The masses are added as lumped masses on the upper nodes of the panels. Such a model can be used to represent one wall panel, but also whole walls with more than one panel can be modelled if the tests are carried out on these whole wall assemblies and not only on panels. Due to the model approach, the shear walls are assumed to deform in shear only. Any other deformation, for instance uplift and translation due to rigid body motion, is included in the rotational springs. All four springs are identical.
effects such as friction are implicitly included in the springs. Such a model is easy to control as its behaviour is completely governed by the spring elements. With this approach, it is possible to model constructions whose seismic behaviour is defined by horizontal displacements of the wall elements with no significant uplift. 4.1.2 X-Lam buildings Cross-laminated timber (X-Lam) constructions can also be modelled on the wall element scale doing tests on assembled X-Lam walls with fasteners and with or without openings. However, X-Lam constructions cannot be modelled with the same models as timber-frame buildings as these structures are made with very rigid panels in comparison to the connections. The assumption of pure shear deformations does not hold any more. The wall behaviour is governed by the connections; there are no significant contributions from the rigid cross-laminated timber panels. A possible model of a X-Lam shear wall is shown in Figure 6 together with an example of a typical X-Lam wall element. A connection detail is also shown in Figure 6.
Figure 5: Model of platform frame or timber-frame panel
Wall models according to Figure 5 are then assembled to represent whole buildings. If the models are in 2D, no decision must be taken on the rigidity of the diaphragms. If instead the models are in 3D, then of course the diaphragms must be modelled as well. Either they are considered being rigid or further spring elements are modelled to simulate their elastic of plastic flexibility. Another parameter apart from the springs that influences the mechanical behaviour of such wall models in a dynamic analysis is the equivalent viscous damping. Values must be attributed for damping in order to undertake a non-linear dynamic analysis. All other
Figure 6: Example with detail (above) and model (below) of X-Lam wall
With the current connection typology for X-Lam buildings, the panels themselves remain rigid with no change of angle. The wall is hence modelled with infinitely rigid members and, if wished, with additional
4.2 CALIBRATION OF SPRINGS Plastic deformation capability and hence energy dissipation are very important concepts for earthquake design. It is exactly these concepts that are taken into account by the behaviour factor q. Models must hence be able to reflect the ductility and energy dissipation capacity of the modelled structure. As the mechanical behaviour of the structures is governed by the springs due to the chosen modelling approach, a suitable spring model must be used. Ceccotti and Vignoli [5] developed such a spring model to represent the behaviour of semirigid joints under reversed-cyclic loading. In their piecewise linear model, springs can reproduce the nonlinear pinching hysteresis loops of typical semi-rigid joints in timber constructions as can be seen in Figure 7. With the insertion of stiffness k4, the spring model is able to model the pinching behaviour but not the strength impairment; the covered area of the loops gives a measurement of the dissipated energy. The shown models are symmetric; also asymmetric models exist. The different shapes (with four or six inclinations) of the models were developed for different shapes of cyclic tests represented by the envelope curves.
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Figure 8: Cyclic tests on X-lam panel
A modelling approach for X-Lam panels was already shown in Figure 6 where vertical springs are modelling the uplift. This uplift movement results in asymmetric hysteretic cycles where asymmetric spring models must be used. Figure 9 shows a typical example of a calibrated spring with the piecewise linear asymmetric spring model. The also shown test results are from Figure 8 in terms of force – vertical displacement of the wall ends. 150 100 50 load [kN]
rigid diagonal members to further ensure the overall stiffness. Translational springs represent the global wall behaviour. It is assumed that the uplift is represented by the vertical springs. In the construction, this function is guaranteed by the hold-down anchors at the wall-ends as shown in Figure 6. The horizontal displacement is represented by the horizontal spring. Again, a damping value must be assumed in order to carry out a non-linear dynamic analysis. These wall models can then again be assembled in 2D or 3D to build up complete buildings. The following modelling step is the calibration of the springs defining the mechanical behaviour of the structural elements.
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Figure 7: Hysteresis cycle of spring model, left: 4 inclinations, right: 6 inclinations
Typical outcomes of cyclic tests on X-Lam wall panels are presented in Figure 8. The results in terms of load – upper horizontal displacement of the panel are shown on the right; the tested panel is shown in Figure 6. The two horizontal lines indicate the maximum force and 80% of maximum force. The vertical line indicates the displacement v80% at 80% of maximum force.
However, there are no standardised methods to calibrate the springs. The calibration procedure is iterative and the calibration parameters are maximum force, maximum displacement and amount of dissipated energy. The correct representation of the envelope curve is prerequisite to proper modelling; a calibration only in terms of dissipated energy could lead to totally different results. As a starting point, the rules given in EN12512 [2] can be used as shown in Figure 10.
Figure 10: Curve fitting according to EN12512 [2]
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Figure 11: Cyclic test on traditional timber-frame panel with stone infill with envelope curve
The first step is the proper choice of one of the spring models shown in Figure 7. When looking at the envelope curve, the model with six inclinations is chosen although the loop is not symmetric. The model with six inclinations needs six stiffness values (from k1 to k6), two displacement values (u1 and u2) and a residual force F0. The stiffness k1 and k2 can be determined according to EN12512 [2], see Figure 10. These two stiffnesses determine the displacement value u1. All other values are then determined by “engineering judgement”. The values for k3, k4 and k5 are established visually. The stiffness k4 modelling the pinching behaviour has a large variability; a first decision on the stiffness must be taken which may be changed during the calibration process. The same is valuable for the residual force F0 which will be changed during the iteration process and which is a good calibration parameter to adjust the amount of dissipated energy.
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In the following, the calibration procedure will be explained thoroughly by means of an example. The traditional timber-frame panel with stone infill as shown in Figure 3b is taken as example. The cyclic test results with envelope curve are shown in Figure 11. The test results are already transferred from load-displacement (Figure 4) to moment-rotation as the chosen numerical model has rotational springs as shown in Figure 5. As there are four rotational springs, the global mechanical behaviour is determined by all four springs simultaneously. This must be taken into account by dividing the loading, the y-axis of the load-displacement graphs, by four. Another possibility is to multiply the simulation result of one spring by four in order to compare test and model.
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Figure 12: Determination of spring model values
With a first estimation of the stiffnesses, the displacements and the residual force, the cyclic test is practically repeated by applying the time-displacement history of the tests on the numerical model. The calculation results are then shown in terms of momentrotation of a spring. The resulting graph is superposed with the test results and analysed. Especially a proper modelling of the envelope curve is required with a good prediction of maximum force and maximum displacement. The energy dissipation is calculated by integrating the force over the displacement. The dissipated energy of the test is compared with the dissipated energy of the model. Both values should agree with a difference of ±10%. If the difference is bigger than 10% or if the envelope is not modelled properly, then the values determining the hysteresis loops must be changed. The calculation is repeated and the results are again compared. Finally, the mechanical behaviour of the tested wall is completely described by the rotational springs; pure shear behaviour is assumed and all other effects such as friction and rigid body movements are implicitly included in the springs. 4.3 DISCUSSION ON MODEL APPROACH Previously, the concept of static ductility ratio for the estimation of the behaviour factor q was discussed. Apart from the problems associated with the definition of the yield displacement, the amount of dissipated energy is also not considered in this concept. This discussion leads directly to an advantage of the proposed method. If the calibration of the hysteresis loops is carried out on the parameters maximum force, maximum displacement and amount of dissipated energy, the problems connected with an unclear yield point are avoided. Due to the possibility of modelling hysteretic loops, the real reversed-cyclic loading under a seismic load is at least partly considered. Energy dissipation can be properly taken into account. Further parameters connected with dynamic loading and energy dissipation are for instance the already mentioned damping and friction. Damping is difficult to evaluate on a global scale; even for building components such as wall elements, the damping can hardly be established. Rules of thumb exist to estimate damping. Usually, a viscous damping of 2% to 5% is estimated [6]. However,
4.4 FROM SPRINGS TO BUILDINGS Once the springs are calibrated by “repeating” the cyclic tests, the next step is the assembling of a whole building with the calibrated wall elements. Important parameters such as vertical loads or geometry will differ between cyclic test specimen and a wall in a building. A strategy must be developed in order to transfer the hysteretic springs calibrated on cyclic test specimens into springs determining the behaviour of a 2D or 3D building under earthquake loading. For timber-frame buildings, this is rather straightforward. The lateral stiffness is assumed to be linearly proportional; the stiffness of a wall with a length of 2m is assumed to be half the stiffness of the wall with a length of 4m. Therefore, to transfer the springs into the buildings, the stiffness values must be adjusted as well as the residual force at zero displacement (F0 in Figure 7); the displacements u1 (and u2 for the model with six inclinations) must not be touched. This approach is not possible for other construction typologies such as X-Lam. For these typologies, no linearly proportional relationship between wall length and lateral stiffness can be assumed. For X-Lam, various solutions to deal with this problem are possible. Firstly, cyclic tests on hold-down anchors and other connections can be done as the X-Lam panels are not contributing significantly to the racking stiffness of the wall assembly. The springs can then be calibrated with these results. A second possibility is to undertake tests on complete wall elements varying the wall length. Like this, the relationship between wall length and stiffness of vertical and horizontal springs can be assessed. Furthermore, a simplification of X-Lam buildings is the (linear) summing-up of hold-down anchors and shear angles. For instance, if the cyclic tests are carried out with one hold-down anchor per wall end, but in the building 3 hold-down anchors are used in corners, then the stiffness of the single spring modelling all three corner hold-downs can be assumed to be three times as high as during the cyclic test. Only one spring is then representing the action of three hold-downs. The influence of vertical loads on the stiffness and on the maximum load-carrying capacity of the specimens is considerable. In Figure 13, the superposition of cyclic test results on X-lam panels (Figure 8) with and without vertical load is shown. The decrease in initial stiffness and maximum load can be clearly seen.
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dynamic models are sensitive to the assumed damping rate [6]. Friction, on the other hand, is nearly impossible to determine as well. Friction is a powerful contribution to energy dissipation. No information is available on how much energy is dissipated in the joints and how much is dissipated due to friction. However, as friction is implicitly included in the modelling, his information does not seem necessary when modelling at a larger structural scale as proposed here. The hysteretic spring models were implemented in the non-linear dynamic software DRAIN developed at the University of Berkeley [7]. With DRAIN, 2D and 3D models can be simulated.
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Figure 13: Overlap cyclic test on X-lam panel with and without vertical load
The procedure to solve this is first of all the execution of cyclic tests with well-defined vertical loading derived from a real building situation. Tests with different vertical loading to determine the rate of change of stiffnesses can also be undertaken. Basically, few tests are sufficient to estimate the rate of change of stiffnesses of structures under different vertical loading. Within the scope of the development of a straightforward and simple method to evaluate the behaviour factor q with few tests, a conservative approach is the carryingout of cyclic tests with a lower vertical load than in real buildings. The springs calibrated on those results are then transferred unchanged into building models which are subsequently underestimating the load-carrying capacity and lateral stiffness of the modelled buildings in comparison to the real structure. Now, all necessary parameters for a non-linear dynamic analysis are defined except for the masses. However, the determination of masses and their distribution to lumped nodes will be the least problem if the construction type and the used materials are known. The modelling procedure can thus be summarised as follows: Monotonic and cyclic tests on shear walls, interstorey connections, diaphragms, etc.; Calibration of rotational or translational hysteretic springs modelling the mechanical reversed-cyclic behaviour of construction typologies; Generation of building model (2D or 3D) with properly determined and distributed lumped masses and an estimation on the damping rate; Transfer of calibrated springs into building models adjusting stiffness values in order to consider different wall lengths, numbers of fasteners or vertical loads. 4.5 DETERMINATION OF q The last steps to evaluate the behaviour factor q are very straightforward. As already stated, the behaviour factor can be evaluated on the reaction side by means of Equation (1) or on the action side by means of Equation (). Here, Equation () is chosen.
This means that the building simulated with the nonlinear dynamic software DRAIN [7] must be designed according to EC8 [1] with a certain PGAdesign and with a behaviour factor q=1 assuming a completely elastic building with no overstrength. Then this model-building is subjected to a series of earthquakes increasing the PGA’s until a previously defined near-collapse state is reached. The ratio of PGAnear-collapse over PGAdesign will give an estimation of q for each applied earthquake. The necessary resistance parameters for seismic design according to EC8 [1] can again be derived from the cyclic tests (maximum allowable displacements and drifts could also be taken from standards). Lateral stiffness and maximum load-carrying capacity are direct derivatives from the tests. With this information, the necessary wall lengths in order to just survive a design earthquake can be assigned to the building. In other words, the wall lengths are adjusted such that the resulting walls have the required lateral stiffness in order to withstand the seismic design forces. These wall lengths are purely fictitious; for soft systems the required wall lengths to resist the seismic forces may be very long. Furthermore, a near-collapse criterion can be established by means of the test results. One possibility of a nearcollapse criterion could be the displacement v80% at 80% of the maximum force as shown in Figure 8 where an additional safety margin should be considered. Naturally, other near-collapse criteria can be chosen such as maximum uplift. The chosen near-collapse criterion is directly related to the construction typology. It is more natural to choose interstorey drift as a criterion for timber-frame buildings whereas maximum uplift seems to be also a good criterion for X-Lam buildings. Another approach to design the building is not by introducing fictitious wall lengths in order to create a building that just withstands the seismic forces, but to design first the building (cross sections, connections) and then carry out cyclic tests on the thusly established wall set-up. Many ways exist in approaching the problem. The fundamental requirements must be respected though. The design of a certain building must be carried out elastically and with a certain PGAdesign, but the model of the building must be based on real non-linear hysteretic data. 4.6 CHOICE OF EARTHQUAKES A remark is given on the choice of earthquakes. In order to represent one specific seismic region, geologically possible earthquakes for this seismic region should be chosen. This may include the use of artificially generated earthquakes. Furthermore, in order to generalise the behaviour factor q, a large variety of earthquakes must be selected. If indeed a very general q-factor for a construction typology shall be derived then the earthquakes must be selected carefully. The frequency content of the earthquakes has to cover a broad range.
4.7 SUMMARY The proposed procedure is a simple and straightforward approach to determine the behaviour factor q. In few words, its principles are as follows: Global cyclic test data on shear walls are fitted to hysteretic models which are able to reproduce pinching behaviour. The fitting parameters are envelope curve and energy dissipation. The hysteretic models are basically non-linear springs. The calibrated non-linear hysteretic models, the springs, are used to represent the behaviour of a shear wall without the need of explicit modelling of shear wall components. The cyclic test data is analysed and the lateral stiffness, maximum load-carrying capacity and a near-collapse criterion are established. A building is designed elastically with q=1 and a certain PGAdesign according to the current seismic standards and the elastic seismic shear forces are identified. The shear wall models are used to model 2D or 3D buildings whose behaviour is hence completely governed by the hysteretic springs; all other components are considered as rigid. Assumptions must be taken on the rigidity of roofs and floors (in 3D models). The wall lengths are adjusted that they just resist the elastic seismic forces. The building models are subjected to accelerograms of various earthquakes covering a wide range of frequencies. The earthquakes’ PGA values are increased until the nearcollapse state is reached. The ratio of PGAdesign over PGAnear-collapse returns the behaviour factor q.
5 EXAMPLES 5.1 INTRODUCTION The following two examples will give an overview of the proposed method. Some discussed model variances will be presented as there are: Two different models according to Figure 5 and Figure 6; Two different transfers of calibrated springs into buildings: general cyclic tests with subsequently adjusted wall lengths or execution of cyclic tests on seismically designed wall specimens. The first example will be a 2D model of a traditional timber-frame house whereas the second example is a 3D model of a X-Lam building. The 2D example is structurally simple and will hence be explained thoroughly. Other models are an extrapolation from this simple model. More and more structural complexity can be added such as more springs modelling for instance interstorey uplift or horizontal translation. If even 3D models are developed, parameters such as asymmetry or deformable diaphragms can be modelled as well.
A symmetric two-storey traditional timber frame building is investigated. It is 5mx10m with an inner wall and consists of wall panels with a stone infill. The inner wall with a length of 5m is subjected to an earthquake excitation. The masses are calculated considering a influence width of 5m and assuming a traditional timber frame construction with a timber joist floor. A viscous damping of 5% was assumed. The calibration of the shear walls was shown previously in paragraph 4.2. The model of building is shown in Figure 14.
The same procedure was carried out for a whole series of earthquakes. The results are shown in Figure 15 in terms of PGAnear-collapse for 8 different earthquakes and 3 different panels (M5=brick infill, M4 = stone infill, M6=infill with hazelnut branches and reinforcement of the joints with screws and EPDM pads). 1,40
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Figure 15: Collapse peak acceleration of two-storey traditional timber-frame house
The modelling procedure and the used construction typologies are thoroughly presented in [8]. 5.3 X-LAM BUILDINGS Figure 14: 2D DRAIN model of traditional timber frame
From the cyclic test results, a maximum load-carrying capacity and a near-collapse criterion can be derived. Considering Figure 4, the maximum load-carrying capacity was 74kN and the collapse displacement was 80mm. However, 60mm is chosen as near-collapse criterion being thus on the safe side. The next step is the seismic design of the traditional twostorey timber frame building with q=1 and a certain PGA, here PGAdesign = 0.25g. The thusly evaluated seismic forces must be sustained by the traditional twostorey timber-frame house. The lateral resistance of the panels is the maximum load-carrying capacity of 74kN divided by the panel length. Furthermore, the resulting resistance must be reduced by a safety factor. Now, the wall length Lwall of the two-storey house model can be determined by equalising the lateral resistance with the seismic shear and solving for the required wall length. A non-linear dynamic analysis in the time domain is carried out the fully defined building using different earthquakes. The PGA values of the earthquakes are increased until the near-collapse displacement of 60mm is reached. Then, the behaviour factor can be evaluated according to Equation (). This is shown for the earthquake of El Centro that was augmented up to a PGAtest= 0.69g at which the near-collapse displacement of 60mm was reached on the ground floor level:
q
PGAnear collapse PGAdesign
0.69 2.76 0.25
(2)
The following construction typologies are buildings with X-Lam panels. As discussed previously, the modelling approach is different; shear walls with X-Lam panels cannot be modelled with rotational springs, but need translational springs as shown in Figure 6. Furthermore, the procedure to transfer the calibrated model into a 3D building is different. Here, a previous seismic design according to EC8 [1] with q=1 was undertaken on a benchmark building defining the connection layout, number of fasteners (nails, screws) and type of connections (steel angles, hold-downs). The chosen PGAdesign was taken to 0.35g which is the highest PGA of the Italian territory. Monotonic and cyclic tests were carried out on the thusly defined wall elements – an example for the ground floor panels is shown in Figure 8. Furthermore, upper-storey panels with interstorey connections were tested. The panel width was kept constant. The calibration of the springs was done (example in Figure 9 for a hold-down) using various cyclic test results as input such as cyclic tests on ground floor walls and tests on upper storey walls with an interstorey connection and less vertical loading. A 3D model of the benchmark building was developed as shown in Figure 16. The vertical displacement on ground floor and in the interstorey connections was assigned to vertical springs. These vertical springs were calibrated on the uplift of the cyclic tests on walls. Their stiffness was adjusted according to the number of hold-downs used in the respective corners and openings. All vertical behaviour was hence assigned to the hold-downs which is a simplification. In reality, also the steel angles close to wall ends are contributing. The same assumption holds for the horizontal springs. Their displacement is attributed to the steel angles alone by adjusting the
stiffnesses to the number of steel angles used per wall length. One horizontal spring is used per complete wall length assembled of more than one wall panel. The diaphragms were assumed to be rigid and a viscous damping of 5% was assumed. Different wall lengths were no issue as they were kept constant throughout the whole building. The cyclic tests were carried out on 3m long wall elements and 3m was the length of all used walls in the test and model building.
earthquakes were selected and applied with increasing PGA’s until the near-collapse was reached. Figure 18 shows the results of the evaluation of q for the eight Fattore di struttura q per Edifici XLam different earthquakes. 5 4.5 4 3.5 3
q 2.5 2 1.5 1
Point 3NE
0.5 0 Kobe
El Centro
Nocera Umbra
Northridge
Joshua
Loma Prieta Mexico City
Kocaeli
Terremoto
Figure 18: Behaviour factors q for X-Lam building [9]
Figure 16: Numerical 3D model of X-Lam building [9]
This building was then subjected to full-scale 3D shaking table tests. The differences in model prediction and test results are small from an engineering point of view. This can be seen in Figure 17 where the results of prediction and test are shown for reference point 3NE indicated in Figure 16.
Figure 17: Shaking table test results versus model prediction [9]
This numerical model was therefore cross-checked with full-scale shaking table test results. The reliability of the chosen approach is satisfying. The estimation of a behaviour factor q is hence even more reliable than in the previous example where no full-scale shaking table tests were carried out. The near-collapse criterion was established analysing the results of the cyclic tests; as criterion the near-collapse of one or more hold-downs was chosen (uplift of 25mm). Eight historical
A behaviour factor of q=3 for X-Lam constructions is resulting from this research. X-Lam buildings can therefore be assigned to Ductility Class High (DCH) according to EC 8 [1]. In order to confirm this behaviour factor, more earthquakes and different building geometries including torsional eccentricities should be analysed. A thorough discussion about the modelling and presentation of the full-scale shaking table tests are given in [9]. An independent check - and support - of this conclusion (q=3), is provided by Pozza and others in [10]. 5.4 OTHER TYPOLOGIES Also other construction typologies were already modelled with this proposed approach in order to determine the behaviour factor q needed in seismic design. A blind prediction of a platform frame building was carried out by an international group of researchers who used different modelling approaches to predict the global seismic behaviour of a symmetric two-storey building which was subsequently subjected to a fullscale shaking table test [6]. The modelling approach using a 3D DRAIN model analogous to the presented traditional timber-frame house model with rigid diaphragms performed well. This again confirms the validity and reliability of this approach. Karacabeyli and Ceccotti also modelled woodframe buildings with 2D and 3D DRAIN models [11]. They also verified the numerical models developed based on cyclic tests on shear walls with full-scale shaking table tests. Figure 17 is taken from [11] and shows the results of a 2D DRAIN model and a shaking table test in terms of force-displacement under the El Centro earthquake with a PGA = 0.35g. The model predictions are good in terms of maximum force and displacement; no information is given on the amount of dissipated energy.
REFERENCES
Figure 19: Comparison of 2D DRAIN model and shaking table test [10]
Furthermore, they investigated in the influence of deformable floors by developing a symmetric 3D DRAIN model. The floors were modelled as deformable by attributing the deformability of a plywood floor to the cross-bracing of the modelled floor. The deformability of the cross-bracing is elastic. The deformable floors are hence not contributing to the energy dissipation. However, the uneven load transfer to the shear walls due to the deformable floors and possible dynamic interactions between walls and floors can be captured. The same 3D building was also modelled with rigid floors. The subsequent evaluation of the q-factor resulted in a decrease between 5-30% of the building with deformable floors in comparison to the building with rigid floors. However, the authors state that more research is required to asses the influence of torsional eccentricities of buildings. Again another typology is described in [12]. The numerical models evaluating the behaviour factor q on the system described in [12] and another system called Lignotrend using massive wooden panels are presented in a paper at the WCTE2010 by Blaß and Schädle. Their approach is based on the herewith proposed procedure to establish the q-factor. One more typology is described and solved in [13], by Terzi. The versatility of the proposed method with its simple procedures and calibration techniques makes it very attractive in order to establish q-factors.
6 CONCLUSIONS The proposal for a standard procedure to establish the behaviour factor q is a simple and straightforward method. It leads to computationally efficient numerical models. The proposed standard procedure is easy to apply and reliable as shown by cross-checking with fullscale shaking table tests [6, 9, 11, 12, 13]. The problem with the concept of static ductility ratio is avoided; energy dissipation and pinching behaviour of semi-rigid joints in timber structures can be taken into account.
ACKNOWLEDGEMENTS Authors do acknowledge Provincia Autonoma di Trento support of this work and the precious help of former colleague dr Maurizio Follesa.
[1] EN 1998-1:2004, Eurocode 8. Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings. CEN, 2004. [2] EN 12512:2001. Timber structures – Test methods – Cyclic testing of joints made with mechanical fasteners. CEN, 2004. [3] C. Sandhaas. Seismic behaviour of historic timberframe buildings in the Italian Dolomites. Master Thesis, CNR-IVALSA and University of Karlsruhe, 2004. [4] EN 1995 1-1:2004, Eurocode 5. Design of timber structures - Part 1-1: General -Common rules and rules for buildings. CEN, 2004. [5] A. Ceccotti, A. Vignoli. A hysteretic behavioural model for semi-rigid joints. European Earthquake Engineering, 3:3-9, 1989. [6] B. Folz, A. Filiatrault. Blind predictions of the seismic response of a woodframe house: an international benchmark study. Earthquake spectra, 20(3):825-851, 2004. [7] V. Prakash, G. H. Powell. DRAIN-2DX, DRAIN3DX and DRAIN BUILDING: Base program design documentation. Report No. UCB/SEMM-93/16. Department of Civil Engineering, University of California, Berkeley, 1993 [8] A. Ceccotti, P. Faccio, M. Nart, C. Sandhaas, P. Simeone. Seismic behaviour of historic timberframe buildings in the Italian Dolomites. Proceedings 15th International Symposium ICOMOS International Wood Committee, Istanbul, 2006 [9] A. Ceccotti. New technologies for construction of medium-rise buildings in seismic regions: The Xlam case. Structural Engineering International, 18:156-165, 2008. [10] L. Pozza, R. Scotta, R. Vitaliani. A non linear numerical model for the assessment of the seismic behaviour and ductility factor of X-Lam timber structures. Proceeding of international Symposium on Timber Structures, Istanbul, Turkey, 25-27 June 2009, 151-162. [11] A. Ceccotti, E. Karacabeyli. Validation of seismic design parameters for wood-frame shearwall systems. Canadian Journal of Civil Engineering, 29:484-498, 2002. [12] H. J. Blaß; P. Schädle. Aussteifende Wände in Einzelelement-Bauweise. Band 13 der Reihe Karlsruher Berichte zum Ingenieurholzbau. Universitätsverlag Karlsruhe, 2009. [13] E. Terzi. Seismic behavior of a wood framed construction system. Master thesis of Master Casaclima, University of Bolzano, 2008.
