ASSESSMENT OF DYNAMIC CHARACTERISTICS OF MULTI- STOREY TIMBER BUILDINGS

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ASSESSMENT OF DYNAMIC CHARACTERISTICS OF MULTI- STOREY TIMBER BUILDINGS Johannes Hummel 1, Werner Seim 2 ABSTRACT: This paper discusses the impact of the natural frequency of multi-storey timber

How stiffness parameters for simple two-dimensional analysis models can be derived from the different contributions of deformation are demonstrated.

1 ASSESSMENT OF DYNAMIC CHARACTERISTICS OF MULTI- STOREY TIMBER BUILDINGS Johannes Hummel 1, Werner Seim 2 ABSTRACT: This paper discusses the impact of the natural frequency of multi-storey timber structures, focusing on force-based seismic design. Simplified approaches to determine the frequency of light-frame and cross-laminated timber structures are investigated. How stiffness parameters for simple two-dimensional analysis models can be derived from the different contributions of deformation are demonstrated. Based on findings from a parametric study, recommendations are given for the determination of the natural frequency of timber buildings up to eight storeys. Effects of higher modes will also be discussed with reference to the response spectrum method. KEYWORDS: force-based seismic design, natural frequency, fundamental period, mass, stiffness, light-frame, CLT 1 INTRODUCTION 12 Fundamental dynamic characteristics must be determined for the seismic design of multi-storey timber structures by means of the response spectrum method [1]. In this context, the most relevant parameter is the Eigen frequency f, respectively, the Eigen period T in addition to the behavior factor q. This paper focuses on the Eigen frequencies of multi-storey timber structures. We refer to relevant literature regarding the behavior factor q, e.g. [2],[3]. Figure 1: Response spectrum and range of period, respectively, frequency for different building heights 1 Johannes Hummel, University of Kassel, Department of Structural Engineering, Building Rehabilitation and Timber Engineering, Kurt-Wolters-Straße 3, Kassel, Germany. jhummel@uni-kassel.de 2 Werner Seim, University of Kassel, Department of Structural Engineering, Building Rehabilitation and Timber Engineering, Kurt-Wolters-Straße 3, Kassel, Germany. wseim@uni-kassel.de Timber structures with one to three storeys exhibit in most cases a natural frequency, which lies mainly on the plateau of the response spectrum (domain 2, see Fig. 1). The frequency was found by analytical and experimental investigations [4]-[8]. This indicates that the seismic action of low-rise buildings is less sensitive to the structural period. On the other hand, the determination of the Eigen frequency is decisive for timber structures with four or more storeys. Here, the determination of the period basically dominates the seismic forces. Different methods to determine the period of multistorey timber buildings were discussed by SEIM ET AL. [5] and others [11]. In addition to a full modal analysis, the RAYLEIGH method [12] seems to be applicable for timber structures up to a certain height (see section 2.2). Comparatively simple two-dimensional (2D) analysis models will be related to more realistic threedimensional (3D) analysis models in this paper. Full 3D modal analyses will be used to check the performance of simplified 2D models and to illustrate the effect of higher modes and the 3D interaction between walls and slabs. In both cases, it is necessary to define stiffness parameters which reproduce different contributions to the overall deformation of the lateral load resisting system. Which contributions are decisive to predict the period in sufficient accuracy will be demonstrated. Stiffness parameters will be determined systematically for the light-frame and cross-laminated timber (CLT) construction types. A reference structure was defined as a basis for this study and to develop the analysis model (see section 4).

2 ANALYSIS METHODS If specific requirements concerning the regularity of the building in plan and in elevation are fulfilled, then the simplified response spectrum method can be applied [1] for

2 2 ANALYSIS METHODS If specific requirements concerning the regularity of the building in plan and in elevation are fulfilled, then the simplified response spectrum method can be applied [1] for seismic design. While the (full) modal response spectrum method incorporates all relevant modes of vibration, the simplified response spectrum method also known as the lateral force method only considers the first mode. 2.1 MULTIMODAL ANALYSIS The response spectrum method which considers the contribution of multiple modes of vibration to determine the component forces for seismic design follows the subsequent basic steps: mass and stiffness for 2D or 3D discretization of the building structure with realistic assumptions, computing periods T i for modes i in global x- and y- direction by means of simplified calculation methods or modal analyses, reading the spectral acceleration from the response spectrum for each mode i, determining component forces for each mode, and combining component forces. Eurocode (EC) 8 [1] provides information on how many modes have to be considered. The number of relevant modes depends on the contribution of masses. Most common finite element software is currently able to perform full modal analyses. Such programs support the combination of component forces directly. Torsional modes as a reason for eccentricities can be considered if a 3D analysis model is used (see section 5.2). In this context, the challenge does not lay in the calculation procedure, but in the realistic discretization of the 3D structure. 2.2 SIMPLIFIED ANALYSIS FOR MDOF SYSTEMS If torsional and higher mode effects can be excluded, then the lateral force method can be applied in combination with a simple 2D analysis model (see Fig. 2). As the distribution of mass and stiffness dominates the structural period, the simplified model must reflect the basic characteristics of the real structure. To simplify the structure, a fundamental requirement is that the center of stiffness and masses are nearly on the same vertical line over the building height. Eurocode 8 [1] contains specific criteria for structural regularities. The reference building (see section 4) fulfils the criteria. If the distribution of stiffness and mass is known, the fundamental period T 1 and the natural frequency f 1 of timber structures can be determined by means of the RAYLEIGH method: n n 2 T1 2 mi ui RFi u i i 1 i 1, f1 1 T (1) The parameters m i, u i and RF i stand for mass, horizontal deformation and equivalent horizontal force, 1 respectively, at each floor level i. Structural masses are applied as lumped masses, as illustrated in Figure 2. The deflections u i are calculated by a simple linear static analysis for equivalent lateral forces RF i. The equivalent lateral forces are defined as the product of mass m i and the gravity constant g (RF i = m i g). Figure 2: Generalized model of a multi-mass oscillator for application of the RAYLEIGH method The determination of stiffness properties, which are required to calculate the deflections u i, is explained in section 3. 3 MASS, STIFFNESS PROPERTIES AND FLEXIBILITIES Floor and roof slabs are significantly heavier than wall assemblies due to the requirements of the building construction and building physics of common multistorey timber buildings. Therefore, it is a reasonable simplification for timber structures to concentrate structural masses at the floor level. Masses of the walls are assigned to each slab, as shown in Figure 2. The full dead load and a share of the live loads quasi-permanent gravity loads are taken into account to determine structural masses. Eurocode 8 [1] gives basic guidance for the determination of stiffness with reference to seismic design, such as the slip in the joints of the structure shall be taken into account. The definition of an appropriate stiffness aiming at the calculation of the natural frequency was discussed by SEIM ET AL. [5],[10]. Four deformation shapes related to EI, GA, K φ and K s can generally be distinguished for a beam element; see Figure 3 and Equations (Eqs.) (2) to (5). (a) (b) (c) (d) Figure 3: Relation between stiffness and deformation characteristics

The different contributions of deformation of timber shear walls (see section 3.1 and 3.2) must be traced back to these four basic cases.

3 F h EI (2) 3 u EI F h GA (3) u K K s GA 2 F h (4) u,inst F u (5) s The parameters F and h are the lateral load and the height of the structural member considered, respectively.

1 LIGHT-FRAME WALL As light-frame wall elements are composed of different parts (see Fig. 7a), the deformation of each part contributes to the overall deformation of the wall element.

Several codes and standards (e.g. [13],[14],[15]) provide formulae to determine the different deformation contributions for light-frame constructions. 3.

GA, rotation (rocking) due to contact and tensile anchoring u φ, and slip of shear anchoring u s.

However, the deformation of the CLT element is very low in comparison to the contributions rotation and slip, see e.g. [16].

(a) (b) (a) (b) (c) (d) Figure 5: Deformation contributions of CLT walls (c) (d) Figure 4: Deformation contributions of light-frame walls The following five contributions can be distinguished: slip

3 The different contributions of deformation of timber shear walls (see section 3.1 and 3.2) must be traced back to these four basic cases. Which assignment might be convenient to predict the natural frequency is studied in section 5. 3 F h EI (2) 3 u EI F h GA (3) u K K s GA 2 F h (4) u,inst F u (5) s The parameters F and h are the lateral load and the height of the structural member considered, respectively. The displacement parameters u EI, u GA, u φ and u s correspond to the four types of deflection which are bending, shear, rotation and slip, respectively. 3.1 LIGHT-FRAME WALL As light-frame wall elements are composed of different parts (see Fig. 7a), the deformation of each part contributes to the overall deformation of the wall element. Four out of these five deformations are illustrated in Figure 4. The sliding of the wall element due to the slip of shear anchoring is comparable to the sliding of CLT elements (see section 3.2). Several codes and standards (e.g. [13],[14],[15]) provide formulae to determine the different deformation contributions for light-frame constructions. 3.2 CLT WALL Four main contributions to deflection for loading inplane can be distinguished for CLT wall elements: bending deformation of the CLT element u EI, shear deformation of the CLT element u GA, rotation (rocking) due to contact and tensile anchoring u φ, and slip of shear anchoring u s. The CLT can be considered as a solid material with effective stiffness properties for bending and shear. However, the deformation of the CLT element is very low in comparison to the contributions rotation and slip, see e.g. [16]. This indicates that the flexibility of the anchoring of the CLT wall element dominates overall deformation. (a) (b) (a) (b) (c) (d) Figure 5: Deformation contributions of CLT walls (c) (d) Figure 4: Deformation contributions of light-frame walls The following five contributions can be distinguished: slip of fasteners joining sheathing and frame u K, shear deformation of sheathing u G, axial deformation of studs u E, compression in the contact zone between stud and plate and slip of the tensile anchoring u φ, and slip of shear anchoring u s. Analytical models based on mechanical principles are capable of predicting the load displacement characteristics of anchored CLT wall elements, see e.g. [9],[10]. 4 REFERENCE STRUCTURE In order to carry out a parametric study on multistorey timber structures (see section 5), reference structures were defined. Aspects of regularity were considered in the formation of the building. The reference structures were developed in accordance with FEMA 695 [17] and requirements of the building s physics, fire safety and usage with respect to gravity loads, respectively, structural masses were fulfilled. The dimensions are oriented towards common modular dimensions. The

The cross-section of the CLT elements was designed for lateral and vertical loads. Basic requirements of EC 8 [1] to exclude P-Δ-effects have been considered in the design.

4 proposed floor plan might be a simple representation of a town house (see Fig. 6). The number of storeys varied from four to eight. All relevant parameters of the reference structures are summarized in Table 1. Sufficient strength of the sheathing is provided. The cross-section of the CLT elements was designed for lateral and vertical loads. Basic requirements of EC 8 [1] to exclude P-Δ-effects have been considered in the design. This leads to a value for the total shear in each storey i of V 10 P d / h (6) tot tot r and the total gravity load P tot of each storey was limited to d r /h = 2 %. (a) (b) Figure 7: Reference walls: (a) light-frame and (b) CLT Slabs were designed based on the service limit state. Data for the slabs are required for 3D modeling. 5 PARAMETRIC STUDY The parametric study aims for a comparison of 2D and 3D analysis models. The analysis models were implemented based on the definitions as documented in section 4. The quasi-permanent gravity loads were applied to determine the masses m according to Eq. (7). Figure 6: Floor plan of the reference structure Table 1: Data of the reference structure Dimensions length in X-direction m length in Y-direction 7.50 m storey height 2.75 m Number of storeys Number of walls wall length in X-direction m in Y-direction m Gravity loads TF* CLT dead load roof slab kn/m² dead load floor slab kn/m² live load floor slab kn/m² dead load walls kn/m² * light-frame (timber-frame) The lateral loads have been applied to design the shear walls. The wall elements depicted in Figure 7 were taken as reference for the design of shear walls. The wall length was adapted for longer walls. The wall height h is the storey height deducting the thickness of the floor slab. Shear walls were anchored with hold-downs and angle brackets. The number of connectors was determined for the lateral loads applied under consideration of the full dead load. m ( G +0.3 Q )/ g (7) k k G k and Q k stand for the dead load and live load, respectively. The gravity constant g was taken into account at 9.81 m/s². 5.1 EQUIVALENT 2D MODEL The study on 2D models focuses on the application of the RAYLEIGH method. In this context, the determination of appropriate stiffness parameters with respect to different combinations of deformations based on the definitions in section 3 is essential. The combinations as considered in this study are summarized in Tables 2 and 3. Here, the specific contributions of deflection of light-frame and CLT wall elements were assigned to the four types of deflection. Five variants were considered for each construction type, whereas variant V1 represents the reference variant. The stiffness properties of a single shear wall are then calculated by means of Eqs. (2) to (5). The flexibility of the connection for all calculations was taken into account by the slip modulus K u. The number of deformation modes was reduced one by one from variants V1 to V5. Thereby, it was presumed that the overall deformation is dominated by shear deformations. Some oversimplified variants V4 and V5 for light-frame and V5 for CLT were included to show the impact of inadequate assignment of deflections. The cross-section of the members of the timber frame (studs and rails) was designed for tensile and compression loads in the outer studs. The spacing of fasteners was adapted depending on the storey shear.

Table 2: Deformation contributions for light-frame V1 V2 Assignment of deflections Parameters u EI = u E u GA = u G + u K EI GA u φ = u φ,tf u s = u s,tf K φ K s u EI = u E u GA = u G + u K + u s,tf

contributions for CLT V1 V2 Assignment of deflections Parameters u EI = u EI,CLT u GA = u GA,CLT EI GA u φ = u φ,clt u s = u s,clt K φ K s u EI = u EI,CLT u GA = u GA,CLT + u s,clt EI GA u φ = u

5 Table 2: Deformation contributions for light-frame V1 V2 Assignment of deflections Parameters u EI = u E u GA = u G + u K EI GA u φ = u φ,tf u s = u s,tf K φ K s u EI = u E u GA = u G + u K + u s,tf EI GA u φ = u φ,tf K φ - V3 u EI = u E + u φ,tf u GA = u G + u K + u s,tf EI GA V4 u EI = u E + u G + u K + u φ,tf + u s,tf EI - V5 u GA = u E + u G + u K + u φ,tf + u s,tf - GA Table 3: Deformation contributions for CLT V1 V2 Assignment of deflections Parameters u EI = u EI,CLT u GA = u GA,CLT EI GA u φ = u φ,clt u s = u s,clt K φ K s u EI = u EI,CLT u GA = u GA,CLT + u s,clt EI GA u φ = u φ,clt K φ - V3 u EI = u EI,CLT + u φ,clt u GA = u GA,CLT + u s,clt EI GA V4 u φ = u φ,clt u GA = u GA,CLT + u s,clt K φ GA V5 u GA = u E + u G + u K + u φ,tf + u s,tf - GA The reference structures with four, six and eight storeys are represented as multi-mass oscillator, as shown in Figure 2. The corresponding lumped masses and lateral loads are presented in Table 4. Table 4: Lumped masses m i and lateral loads RF i for the 2D MDOF system Lumped masses (to) Lateral loads* (kn) Level of TF CLT TF CLT roof slab floor slab * RF i = m i g (equivalent loads for the RAYLEIGH method) The reference structure must be transferred to a plane MDOF system, as depicted in Figure 8, to be able to perform the parametric study on the simple multi-mass oscillator. Here, it is assumed that the shear walls are cantilever beams which are rigidly coupled by the floor slab. The stiffness of the cantilever beam was determined storey-wise by means of Eqs. (8) to (11) for the X- and Y-direction (see Fig. 8). EI EI wall, i (8) GA GA wall, i (9) K K s K (10), wall, i K (11) s, wall, i It should be noted that such a simplified analysis model does not allow one to consider the interaction between perpendicular walls and walls and slabs. Effects of 3D interaction will be discussed in section 5.2. Figure 8: Transfer of the reference structure towards the simplified analysis model, here a four-storey building The first Eigen frequency is determined according to RAYLEIGH with Eq. (1). The lateral displacements u i can be computed with a linear static analysis. However, the analysis software must be able to consider shear deformations. In this study, the commercial computer software RFEM [18] has been used to determine the displacements u i for each variant of combination, V1 to V5, separately for X-and Y-direction. Masses m i have been transferred to equivalent lateral loads RF i as defined in section 2.2 (see Table 4). Exemplarily, the MDOF system for the six-storey building is depicted in Figure 13a. The frequencies calculated are summarized in Table 5 for the X-direction and in Table 6 for the Y-direction. Table 5: RAYLEIGH frequencies (in Hz) for X-direction Number Variants of storeys V1 V2 V3 V4 V5 Light-frame CLT Table 6: RAYLEIGH frequencies (in Hz) for Y-direction Number Variants of storeys V1 V2 V3 V4 V5 Light-frame CLT The RAYLEIGH frequencies were compared with the frequencies obtained by full modal analysis. The deviation of the first frequency (f 1 ) of the two methods is illustrated in Figures 9 to 12. It shows that the RAYLEIGH method yields excellent results for different types for multistorey timber structures. The deviation over all 60 values is between 0.5 and 11 % and is, on average, 2 %.

Furthermore, the results show that the simplifications V4 and V5 do not represent the system behavior for the light-frame construction sufficiently.

The assignment of u φ towards a bending deformation u EI (combination V3) leads to a slight underestimation of the frequency compared to V1.

6 Furthermore, the results show that the simplifications V4 and V5 do not represent the system behavior for the light-frame construction sufficiently. By contrast, the frequencies calculated for V2 and V3 show that the Eigen frequency is hardly affected if the displacement u s is assigned to a shear deformation u GA. The assignment of u φ towards a bending deformation u EI (combination V3) leads to a slight underestimation of the frequency compared to V1. Figure 11: Comparison of the natural frequency according to RAYLEIGH and modal analysis for CLT construction in the X-direction Figure 9: Comparison of the natural frequency according to RAYLEIGH and modal analysis for light-frame construction in the X-direction Figure 12: Comparison of the natural frequency according to RAYLEIGH and modal analysis for CLT construction in the Y-direction All in all, the variants V2 and V3 seems to be promising simplifications for the light-frame construction. It appears that all contributions of deformation should be considered separately for the CLT construction. Figure 10: Comparison of the natural frequency according to RAYLEIGH and modal analysis for light-frame construction in the Y-direction While for the CLT construction the definitions for V2 and V4 leads to similar results compared to V1, V3 and V5 do not reproduce the system behavior with sufficient accuracy. The natural frequencies are underestimated by V3 and overestimated by V5 significantly in most cases. V2 underpredicts and V4 overpredicts the natural frequencies in some cases, but the differences are comparatively small. Only the combinations V2 to V4 do not really match the frequency of V1 for the fourstorey building in Y-direction. The different behavior in the X- and Y-direction is caused by the different wall length (see section 4). (a) (b) Figure 13: Eight-storey 2D analysis model (V1) for CLT in the X-direction; (a) lateral loads (RAYLEIGH) and deformation shape and (b) mode shapes of mode 1 to 3

The frequencies of the relevant Eigen modes for the combinations V1 to V3 for the light-frame construction and the combinations V1, V2 and V4 for the CLT construction for evaluation of higher mode

7 The frequencies of the relevant Eigen modes for the combinations V1 to V3 for the light-frame construction and the combinations V1, V2 and V4 for the CLT construction for evaluation of higher mode effects and comparison with results from three-dimensional analyses are documented in Tables 7 and 8, together with modal mass factor α. The mode shapes of the first three modes which were obtained by modal analysis of the 2D model of the eight-storey CLT building in the X-direction are depicted in Figure 13b, exemplarily. Here, the number of modes can be limited to three according to EC 8 [1], because 90 % of the total mass is taken into account by three modes. The contribution of mass of each mode can be represented by the modal mass factor α. Table 7: Eigen frequencies and modal mass factors from 2D modal analysis light-frame, variants V1 to V3 Mode V1 V2 V3 No. f (Hz) α (-) f (Hz) α (-) f (Hz) α (-) 4 storeys, X-direction storeys, Y-direction storeys, X-direction storeys, Y-direction models were implemented with shell elements for walls and slabs and discrete springs to represent connections. In order to represent the orthotropic load-bearing behavior of walls and slabs, the stiffness properties were defined directly by setting the components of the stiffness matrix. In a previous step, the stiffness values of the stiffness matrix were computed for in-plane and out-of-plane action for light-frame and CLT elements. The inherent flexibilities of light-frame element (see section 3.1) were considered in the determination of the entries of the stiffness matrix. The effective stiffness properties in each direction for the CLT element were taken into account (see e.g. [4]). Shell and spring elements are elastic. The same flexibilities of the connectors which were used to set up the 2D models were taken as input values for the discrete springs. As presumed for the 2D analysis models, no flexibilities between slab and lower wall was included. Threedimensional analysis models with four and eight storeys have been developed for light-frame and CLT (see Fig. 14). Table 8: Eigen frequencies and modal mass factors from 2D modal analysis CLT, variants V1, V2 and V4 Mode V1 V2 V4 No. f (Hz) α (-) f (Hz) α (-) f (Hz) α (-) 4 storeys, X-direction storeys, Y-direction storeys, X-direction storeys, Y-direction THREE-DIMENSIONAL MODEL Discritization and Implementation Analyses on 3D models were performed in order to validate the results of the 2D analyses. The 3D analysis Figure 14: 3D analysis model, for a four-storey building In contrast to the 2D model, structural masses are uniformly distributed Eigen frequencies and higher mode effects Full modal analyses were carried out to obtain frequencies and modal mass factors. The results of the modal analyses are summarized in Table 9. The 90 % total mass rule was again applied to determine the required number of modes again. A maximum of three modes were required. In addition to the modal analysis, the RAYLEIGH method was also applied within the 3D analysis. The RAYLEIGH frequencies of the 3D analysis are taken as references for comparison with the 2D analysis. The natural frequencies f 1 obtained by the full modal analysis and the RAYLEIGH method are compared in Figure 15.

Table 9: Results from the modal analyses of the 3D analysis models Mode No. TF CLT f (Hz) α (-) f (Hz) α (-) 4 storeys, X-direction 1 0.83 0.73 2.00 0.72 2 1.90 0.15 6.35 0.16 3 2.94 0.07 9.17 0.

8 Table 9: Results from the modal analyses of the 3D analysis models Mode No. TF CLT f (Hz) α (-) f (Hz) α (-) 4 storeys, X-direction storeys, Y-direction storeys, X-direction storeys, Y-direction Figure 15 shows that the RAYLEIGH method yields lower natural frequencies than the modal analysis. It must be noted that lumped masses are taken into account within the RAYLEIGH method, whereas a uniform distribution is applied for 3D modal analysis. The different distribution of masses leads to this deviation in the frequency. 5.3 COMPARISON AND DISCUSSION The results from the 3D analyses are compared with the preselected variants from the 2D analyses. Figure 15 presents the first frequencies which were obtained for 3D and 2D models by means of modal analysis and the RAYLEIGH method. This comparative illustration generally shows that the 2D analysis provides reasonable natural frequencies. It has already been stated that difference in the frequency between the 3D modal analysis and the RAYLEIGH method is produced by the different mass distribution. However, Figure 15 also reveals some weaknesses of the 2D analysis. There are deviations up to 0.5 Hz in comparison with the 3D modal analysis. The 2D variant V1 does not match the natural frequency from 3D analysis best in any case. The simplifications V2 and V3 for light-frame and V2 and V4 for CLT sometimes meet the frequencies even better. A good correlation of the frequencies between the 2D variant V4 and RAYLEIGH 3D can be observed for CLT. One main reason for the differences is generally the bending stiffness of the slab, which is not taken into account in the 2D analyses but is in the 3D analysis. Higher order vibration effects have an effect on the response of the structure in both the 2D and 3D analyses. Based on the modal mass factor, it is shown that up to three modes have a relevant contribution. This applies to all types and is independent of the number of storeys. light-frame CLT Figure 15: Comparison of the natural frequency from 3D and 2D analyses The lateral force method uses only the first period to determine the base shear F b according to Eq. (12). F S T m (12) b a ( 1 ) The factor λ is comparable with the modal mass factor α. Depending on the period, this factor can be 0.85 or 1.0, according to EC 8 [1]. If the modal response spectrum method is applied, the base shear for each mode i can be calculated by means of Eq. (13). Fb, i Sa ( Ti ) m i (13) The values for the acceleration are read from the response spectrum (see e.g. Fig. 16). The parameter m stands for the total structural mass. Figure 16: Elastic response spectra for different soil and a g = 1.0 m/s² according to EC 8 [1] and frequency range for modes 1 and 2 It becomes obvious from Tables 7 to 9 and Figure 16 that the frequencies of the second and third mode are always at or close to the plateau of the spectrum, while the first frequency lies right on the plateau in most cases. Although the modal mass factors of the second and third mode are comparatively low (0.04 to 0.22), the contribution of these modes is not negligible due to higher spectral accelerations compared to the first mode.

9 6 CONCLUSIONS It was confirmed that the structural period of buildings with four storeys or more lies right on the plateau of the response spectrum in most cases. However, there are still further influences which affect the frequency of the structure and which were not discussed in this paper. In this study, it was presumed that openings are over the complete storey height, but there usually remain sections around openings. The contribution of these sections leads to an increase of the frequency. However, it is assumed that this influence is more pronounced for the CLT construction than for the light-frame construction. It was also shown that a distribution with lumped mass (RAYLEIGH method) leads to lower frequencies in comparison to the uniform mass distribution (3D modal analysis). Based on the parametric study on reference structures, some recommendations for the determination of dynamic characteristics of timber structures can be derived. The domain 2 and the sloping part of the response spectrum (see Fig. 1) should be used for the determination of seismic action for timber structures up to eight storeys. The study shows that a deviation of about 0.5 Hz in the frequency is possible, depending on the analysis model chosen. That can lead rapidly to a significant underestimation of the seismic forces if the frequency shifts into the first domain of the response spectrum. If 2D analysis models are used, the spectral acceleration at a period of 2.0 s should be taken into account as a minimum acceleration in the sloping part of the response spectrum. This has to do with assumptions in the development of the analysis model. This limitation provides only a lower bound of spectral acceleration to reduce the possible underprediction of the seismic action. It is recommended that the full mass m is applied to determine the base shear to account for higher mode effects within the lateral force method. The base shear should not be reduced by the correction factor λ. The components of deformation cannot be summarized to a surrogate bending stiffness. Otherwise, the structure represented by the cantilever beam is presumed to be soft. This procedure is not sufficient for the determination of the natural frequency and yields non-conservative spectral accelerations. Under consideration of the aforementioned, it is an acceptable simplification to predict the frequency, respectively, the period of the structure by means of 2D analysis models. The definition with a bending stiffness EI and a shear stiffness GA (V3) is recommended for the light-frame construction. The deformation components rotation u φ,tf and slip of the shear anchoring u s,tf of the lightframe element can be neglected as part of a conservative estimation of the frequency. Neglecting these deformation components compensates approximately for the fact that variant V3 led to lower frequencies than the reference model. Using a non-flexible cantilever beam which considers shear deformations in combination with a rotational spring K φ at the floor level (variant V4) for the CLT construction is recommended. The recommendation takes into account that the contribution of the bending stiffness is really small, while the overall deformation is dominated by shear loading. The shear stiffness GA results from the combination of the deformation components u GA,CLT and u s,clt. The bending deformation is negligible. Support conditions which cannot be rigid might be considered for the determination of the rotational spring K φ. The recommended simplification includes the anchoring of shear walls by means of hold-downs and angle brackets. The connection on the top of the wall with slab was not regarded. This connection is commonly executed by inclined self-tapping screws, which represent a very stiff connection. Thus, the slip between slab and lower wall is expected to be very small. In this case, this deformation contribution can be neglected. REFERENCES [1] EN : Eurocode 8: Design of structures for earthquake resistance Part 1: General rules, seismic actions and rules for buildings; German version EN : AC:2009. European Committee for Standardization, Brussels. December [2] Ceccotti, A., Sandhaas, C.: A proposal for a standard procedure to establish the seismic behaviour factor q of timber buildings. Proceedings of the 11th World Conference on Timber Engineering, Trentino, Italy [3] Hummel, J., Seim, W.: Performance-based design as a tool to determine behaviour factors for multistorey timber buildings. Proceedings of the 14th World Conference on Timber Engineering, Vienna, Austria. [4] Hummel, J., Seim, W.: Wall-slab interaction of multi-storey timber buildings under earthquake impact. Proceedings of the 12th World Conference on Timber Engineering, Auckland, New Zealand. [5] Seim, W., Hummel, J., Vogt, T.: Earthquake design of timber structures Remarks on force-based design procedures for different wall systems. Engineering Structures, 76: , [6] Ceccotti, A., Sandhaas, C., Okabe, M., Yasumura, M., Minowa, C., Kawai, N.: SOFIE project 3D shaking table test on a seven-storey full-scale crosslaminated timber building. Earthquake Engineering & Structural Dynamics 42: , [7] SERIES Seismic Engineering Research Infrastructures For European Synergies.

10 [8] Polastri, A., Pozza, L., Loss, C., Smith, I.: Structural characterization of multi-storey CLT buildings braced with cores and additional shear walls, paper , In: INTER International Network on Timber Engineering Research, Šibenik, Croatia. [9] Gavric, I., Fragiacomo, M., Ceccotti, A.: Cyclic behavior of cross-laminated timber (CLT) wall systems: experimental tests and analytical prediction models. Structural Engineering, 141, [10] Hummel, J., Seim, W., Otto, S.: Steifigkeit und Eigenfrequenzen im mehrgeschossigen Holzbau. Bautechnik (accepted), WILEY-VCH Verlag, [11] Brunner, R., Jung, P., Steiger R., Wenk T., Wirz N.: Erdbebengerechte mehrgeschossige Holzbauten. Lignum, [12] Rayleigh, J. W. S. B.: The Theory of Sound. Macmillan & Co., London, [13] Blass, H. J., Ehlbeck, J., Kreuzinger, H., Steck, G.: Erläuterungen zu DIN 1052: Entwurf. Berechnung und Bemessung von Holzbauwerken. Bruderverlag, Karlsruhe, [14] NZS 3603: Timber Structures Standard. Wellington, New Zealand, [15] CSA : Engineering Design in Wood. Canadian Standards Association, [16] Hummel, J., Flatscher, G., Seim, W., Schickhofer, G.: CLT wall elements under cyclic loading - details for anchorage and connection. COST Action FP1004: Focus Solid Timber Solutions European Conference on Cross Laminated Timber (CLT), Graz, Austria. [17] FEMA P695: Quantification of building seismic performance factors. Federal Emergency Management Agency, FEMA, Washington, DC, [18] RFEM: 3D finite element analysis software. Version 5.04, Dlubal Software.

THE EXTENDED N2 METHOD IN SEISMIC DESIGN OF STEEL FRAMES CONSIDERING SEMI-RIGID JOINTS

THE EXTENDED N2 METHOD IN SEISMIC DESIGN OF STEEL FRAMES CONSIDERING SEMI-RIGID JOINTS Paulina KROLO 1, Mehmed CAUSEVIC 2 and Mladen BULIC 3 ABSTRACT In this paper the nonlinear static seismic analysis