Hysteretic damping of wood framed buildings

Engineering Structures 25 (2003) 461–471 www.elsevier.com/locate/engstruct

Hysteretic damping of wood framed buildings A. Filiatrault a,∗, H. Isoda b, B. Folz c a

c

Department of Structural Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0085, USA b Building Research Institute, 1 Tachihara, Tsukuba, Ibaraki 305-0802, Japan Department of Civil and Structural Engineering Technology, British Columbia Institute of Technology, 3700 Willingdon Avenue, Burnaby, B.C., Canada V5G-3H2 Received 29 January 2002; accepted 17 October 2002

Abstract The direct-displacement seismic design of wood buildings requires knowledge of the global load–displacement behavior of the building and the variation of the global equivalent viscous damping with displacement amplitude. This paper proposes a simple numerical model for the determination of these parameters. The proposed model is validated with results of shake table tests of a full-scale two-story wood framed house. The model is then used for the monotonic and cyclic pushover analyses of full-scale woodframe buildings, representing various construction qualities. From these analyses, a simple equation is proposed for the variation of the global equivalent viscous damping with building drift.  2003 Elsevier Science Ltd. All rights reserved. Keywords: Damping; Earthquakes; Design; Pushover analysis; Seismic; Wood; Wood structures

1. Introduction A recent important development in earthquake engineering has been the elaboration of performance-based concepts for the seismic design of structures. This approach, based on the coupling of multiple performance limit states and seismic hazard levels, overcomes several of the shortcomings of the traditional force-based seismic design procedure, which has been the cornerstone of building code requirements to date. Although the performance-based seismic design approach has advanced for some types of structures, such as reinforced concrete buildings and bridges, to the point where it may be ready for incorporation into future generations of building and bridge codes, its application to wood framed buildings has only been recently formulated [1]. Since inter-story drift is a key parameter for the control of damage in wood framed buildings, it is rational to examine a performance-based seismic design procedure wherein displacements are at the core of the seismic

Corresponding author. Tel.: +1-858-822-2161; fax: +1-858-8222260. E-mail address: [email protected] (A. Filiatrault). ∗

design process. In this regard, the direct-displacement approach, proposed by Priestley [2,3] for reinforced concrete structures, is an attractive seismic design procedure for wood framed buildings. Two necessary requirements of the direct-displacement design strategy are the detailed knowledge of the global non-linear monotonic load–displacement behavior (pushover) of the building, as well as the variation of the global equivalent viscous damping with displacement amplitude. These requirements represent a difficulty for wood framed buildings since knowledge of their behavior at the system level is not well established. The objective of this paper is to propose a simple numerical model for the seismic analyses of complete wood framed buildings that can be used for their displacement-based seismic design.

2. Overview of displacement-based seismic design procedure The basic elements of the displacement-based seismic design procedure for wood framed buildings are briefly summarized in this section. More details can be found elsewhere [1].

0141-0296/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0141-0296(02)00187-6

462

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

The central concept of the direct-displacement method, as originally proposed by Priestley [2,3], is that the seismic design of a structure is based on a specified target displacement for a given seismic hazard level. For this purpose, the structure is modeled as a single-degreeof-freedom (SDOF) system with equivalent elastic lateral stiffness and viscous damping properties representative of the global behavior of the actual structure at the target displacement. The first step in the design procedure is the definition of the target displacement ⌬t that the building should not exceed under a given seismic hazard level. The seismic hazard associated with the target displacement must then be defined in terms of a design relative displacement response spectrum corresponding to the equivalent viscous damping exhibited by the structure at the target displacement. Once the design performance level and associated seismic hazard have been defined, a potential structural lateral load-resisting system must be specified. Most light framed wood buildings in North America use wood shear wall assemblies as the primary lateral loadresisting structural system. Furthermore, wall finish materials such as gypsum wallboard and stucco can also contribute significantly to the lateral stiffness and strength of wood framed buildings [4]. In order to capture the energy dissipation characteristics of the structure at the target displacement, an equivalent viscous damping ratio must be determined. For this purpose, a damping database must be established for the selected structural system from the global hysteretic behavior of the structure. Note that a nominal damping ratio needs to be added to this hysteretic damping to account for the energy dissipation characteristic of other structural and non-structural elements in the structure [5]. Knowing the target displacement and the equivalent viscous damping of the building at that target displacement, the equivalent elastic period of the building Teq can be obtained directly from the design displacement response spectrum. Representing the building as an equivalent linear SDOF system, the required equivalent lateral stiffness kreq can be obtained kreq ⫽

4π2Weff , gT2eq

(1)

where Weff is the effective seismic weight acting on the building and g is the acceleration of gravity. The actual equivalent lateral stiffness kaeq of the building at the target displacement ⌬t can be determined from the results of a static pushover analysis. The actual equivalent lateral stiffness of the building must be compared to the required equivalent lateral stiffness. If these two stiffness values differ substantially, the lateral loadresisting system of the building must be modified. If the

actual lateral stiffness of the building is nearly equal to the required lateral stiffness, the design process is completed by computing the required base shear capacity, Vb, of the building Vb ⫽ kaeq⌬t.

(2)

This base shear can then be used to design the other elements of the structure. The direct-displacement design strategy requires detailed knowledge of the global non-linear monotonic load–displacement behavior (pushover) of the building as well as the variation of the global equivalent viscous damping with displacement amplitude. These requirements can be considered as a limitation of the directdisplacement design procedure since knowledge of the behavior of wood framed construction at the system level is not well established. To obtain this information, system level testing is required in parallel with the development of specialized numerical models for wood framed construction.

3. Review of numerical modeling of wood framed buildings In low-rise wood framed structures subjected to earthquake loading, shear walls are commonly used in North America as the primary component of the lateral loadresisting system. A large number of numerical models, of varying complexity, have been formulated to predict the static racking response of wood shear walls. In the simpler models, the non-linear global wall response is fully attributable to the non-linear load–deformation behavior of the sheathing-to-framing connectors [6–9]. The sheathing is generally assumed to develop only elastic in-plane shear forces. It was also found that bending of the framing members contributed little to the global wall response [10]. Consequently, in a number of these studies, the framing members are assumed to be rigid. These models generally provide good agreement with the load–displacement response obtained from tests. However, because of their simplicity, they are not able to capture the detailed interaction and load sharing between the components of the shear wall under the imposed lateral loading. More sophisticated finite element models have also been proposed [11–14]. In these models, the framing members comprise beam elements and the sheathing is represented by plane stress elements or plate bending elements. The sheathing-to-framing connectors are modeled using springs with non-linear load–deformation characteristics. Also, gap and bearing elements have been included along the interface between the sheathing panels. Obviously, these models are able to capture more fully the inter-component response within the wall. However, with this increased model complexity, a

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

greater computational effort must be expended. Interestingly, the overall global load–displacement predictions of these models produce essentially the same level of correlation with experimental data as the simpler models discussed previously. Several non-linear dynamic analysis models have also been developed for predicting the seismic response of wood shear walls. The simplest of these are SDOF lumped parameter models [5,15]. These, however, have limited application because the model must be calibrated, in each case, to full-scale test data. Others have extended existing static shear wall models to perform non-linear dynamic time history analysis under seismic input [9,16]. Between the bookends of static ultimate load analysis and the full non-linear dynamic analysis lies the cyclic analysis of shear walls. Recently, a formulation for the structural analysis of wood framed shear walls under general cyclic loading has been elaborated [16]. The numerical model, integrated in a computer program named cashew (Cyclic Analysis of wood SHEar Walls), predicts for sheathed shear walls with or without opening the load–displacement response and energy dissipation characteristics under arbitrary quasi-static cyclic loading. In formulating this structural analysis tool, a balance has been sought between model complexity and computational overhead. The proposed model was validated against full-scale tests of wood shear walls subjected to monotonic and cyclic loading. This model is used in this study to calibrate the parameters of an equivalent SDOF hysteretic shear elements that provide the global cyclic response of wood shear walls that are part of complete three-dimensional wood framed buildings.

4. Description of proposed numerical modeling The three-dimensional non-linear seismic analysis model for woodframe buildings in this study is referred to as a pancake model [4]. This model simulates the three-dimensional seismic response of a woodframe construction through a degenerated two-dimensional planar analysis. The general purpose computer program ruaumoko [17] is used herein to construct the pancake model. A typical woodframe building is modeled as a planar pancake system with all floor and roof diaphragms superimposed on top of each other. The pancake model of a simple two-story woodframe building tested recently on a shake table [4] is shown in Fig. 1. The lateral load-resisting system of this simple test structure, as described later, is composed of only perimeter shear walls. The foundation of the structural model is connected to the floor diaphragm with four zero-height nonlinear shear springs representing the four first story shearwalls. Similarly, the floor diaphragm is connected to the roof diaphragm with four additional zero-height

463

non-linear shear springs representing the second story shearwalls. The roof and ceiling diaphragm, having a high in-plane stiffness in this particular case, are modeled by four plane stress quadrilateral finite elements with very high in-plane stiffness. The floor diaphragm is modeled by plane stress quadrilateral finite elements. The in-plane stiffness of the floor diaphragm can be calibrated to simulate the in-plane response of the floor diaphragm [18]. Frame elements are used along the four edges of the floor diaphragm to connect the corners of the quadrilateral elements to the shear wall elements. The bending stiffness of the frame elements is assumed very small to allow free deformations of the diaphragm. The axial stiffness of the frame elements is assumed very high in order to distribute the in-plane forces of the shear elements along the edges of the floor diaphragm. Each wall in the structure (composed of wood, gypsum and/or stucco) is modeled by a single zero-height non-linear in-plane shear spring using the Wayne Stewart hysteresis rule [14] shown in Fig. 2. This hysteresis rule incorporates stiffness and strength degradations, typical of the racking response of wood, gypsum and stucco walls, and is defined by nine independent physical parameters: Fy, equivalent lateral yield strength; k0, initial lateral stiffness; Rf, post-yield stiffness factor; Fu, ultimate lateral capacity; Fi, intercept force; PTRI, trilinear stiffness factor beyond the ultimate capacity; PUNL, unloading stiffness factor; b1, softening factor; and a1, reloading stiffness factor. These required input parameters for wood, stucco, and gypsum walls can be obtained either from the specialized computer platform cashew, described earlier, or from available experimental data [16,18]. It must be noted that this simple modeling procedure is used to capture the global seismic behavior of wood framed buildings. This simplified approach is not intended to model every connection between the building elements. Roof-to-wall connections or anchor bolts, for example, are not considered, as they are believed to be of secondary importance in the global seismic response of the structure.

5. Validation of numerical model The numerical pancake model described in Section 4 was validated by comparing its predictions with the results of shake table tests recently conducted on a fullscale two-story woodframe house [4]. This section briefly presents the test structure and its physical properties incorporated in the numerical model along with a comparison between some experimental and numerical results.

464

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

Fig. 1.

Pancake model of two-story woodframe test structure.

Fig. 2.

Wayne Stewart degrading hysteresis rule.

5.1. Description of test structure The test structure used to validate the numerical pancake model represents a two-story single family house incorporating several characteristics of modern residential construction in California. Fig. 3 shows plan views of the first and second floors of the test structure, while Fig. 4 shows exterior wall elevations. The major structural components of the test structure are identified in these two figures. A photograph of the test structure is also shown in Fig. 1.

The footprint of the test structure was 4.9 m × 6.1 m. The sill plates were founded on a 38-mm thick perimeter concrete foundation pad placed on top of a stiff steel base that was attached to the shake table. The anchor bolts were welded to the top of the steel frame and embedded in the concrete foundation pad. Shaking was unidirectional along the short dimension of the structure (north–south direction). The construction of the test structure was at full-scale. The plan dimensions, however, are smaller than would be of a typical residence due to limitations of the shake

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

Fig. 3. Plan views of the test structure.

Fig. 4.

Elevation views of the test structure.

465

466

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

table. The lateral load-resisting system of the test structure parallel to the shaking direction (east and west wall elevations) consisted of exterior shear walls. The openings of these walls were designed in an attempt to reproduce the torsional eccentricity that would be induced by a large garage door opening on one side of a residence. The lateral load-resisting shear walls also supported the gravity loads together with a first floor interior bearing wall and a glue-laminated beam at the floor level. In the north and south wall elevations, exterior shear walls with symmetric window openings provided additional torsional restraint to the test structure. All exterior shear walls were sheathed with 9.5-mm thick oriented strand board (OSB) panels that were fastened to the framing with 8-penny box gun nails. The second floor structural walls were tied to the first floor walls by steel straps (see Fig. 4). The first floor walls were tied down to the foundation by hold-down devices at the end of the wall piers [4]. The structure was tested with interior and exterior wall finish materials. The wall finish materials consisted of interior gypsum wallboard and exterior stucco. The 12mm thick gypsum wallboard panels were installed on all interior wall and ceiling surfaces. All surfaces were taped, mudded and painted. The panels were oriented horizontally on the walls and fastened with 32-mm long drywall screws spaced at 400 mm along the vertical studs. The ceiling panels were fastened with the same screws spaced at 300 mm in the center. The exterior stucco finish was applied in three coats for a total thickness of 22 mm. Cylinder tests on the stucco indicated an average compressive strength of 9 MPa at the time of testing. The stucco was attached to the wood framing by a galvanized 17-gage steel wire lath fastened to the OSB sheathing and vertical studs by 20-mm long staples. A wood pedestrian door was also installed in the west wall of the first story and windows with aluminum frames were installed in all walls. The garage opening in the east wall of the first story did not have a door installed. 5.2. Experimental set-up The shake table tests were performed on the 3 m × 5 m uniaxial earthquake simulation facility of the Powell Laboratory at the University of California, San Diego [19]. The electronic digital control of the shake table incorporates an adaptive inverse control (AIC) scheme. This technique generates the inverse transfer function of the shake table and specimen and uses it to modify the input signal in order to improve system fidelity. The test structure was secured to the shake table by a stiff steel base frame anchored to the top plate of the shake table. This frame incorporated stiff cantilevered outriggers in the east and west directions to accommodate the plan dimensions of the test structure.

5.3. Parameters of numerical model The principle elements used to construct the pancake numerical model of the test structure have been described earlier and are shown in Fig. 1. Additional shear spring elements, not shown in Fig. 1, were also included to model the in-plane racking behavior of the interior gypsum walls and exterior stucco walls. Table 1 lists the physical parameters of the Wayne Stewart hysteresis rule for each of the shear spring elements used to model the walls of the test structure. The hysteretic parameters for each stucco and gypsum wall were estimated from available cyclic test data on wall assemblies [18]. These parameters were obtained by adjusting the experimental strength and stiffness values according to the actual length of full wall piers in each wall line. The hysteretic parameters for the OSB shear walls were computed by the computer program cashew [16,18]. The roof diaphragm was assumed rigid and modeled by plane stress quadrilateral finite elements with very high in-plane stiffness. The floor diaphragm was modeled by plane stress quadrilateral finite elements. The inplane stiffness of the floor diaphragm was calibrated using the results of preliminary quasi-static tests performed on the diaphragm of the test structure [18]. A constant in-plane stiffness (elastic modulus times thickness) of 27 kN/mm was used for the quadrilateral floor diaphragm elements. The seismic weight assigned to the model was computed using the dead load of all elements. This dead load was distributed as lumped seismic weights at the nodes according to their tributary areas. Total seismic weights of 61.4 and 48.0 kN were used for the floor and roof level, respectively. Furthermore, given the simple construction and architectural details of the test structure, only very small viscous damping was added on top of the hysteretic damping captured by the load–deformation rule of the various shear spring elements. Rayleigh-type viscous damping was assumed with ratios of 0.1% of critical in the first and second elastic modes of vibration. 5.4. Comparison between experimental and numerical results The predictions of the numerical pancake model were compared against the results of the most severe seismic test conducted. For this purpose, one horizontal component of the ground motion obtained at the Rinaldi recording station during the 1994 Northridge, California, earthquake was used as input motion on the shake table. This near-field ground motion, exhibiting a peak ground acceleration of 0.89g, was reproduced at full-scale. Fig. 5 compares the relative displacement time history measured at the center of the roof of the test structure during the test against the predictions of the numerical pancake model. The model under-predicts the peak roof

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

467

Table 1 Hysteretic parameters for shear spring elements in numerical pancake model of test structure Element

Fy (kN)

k0 (kN/mm) Rf

Fu (kN) Fi (kN) PTRI

OSB, level 1, west OSB, level 1, east OSB, level 1, north and south OSB, level 2, east and west OSB, level 2, north and south Stucco, level 1, west Stucco, level 1, east Stucco, level 1, north and south Stucco, level 2, east and west Stucco, level 2, north and south Gypsum, level 1, west Gypsum, level 1, east Gypsum, level 1, north and south Gypsum, level 2, east and west Gypsum, level 2, north and south Gypsum, level 1, bearing wall Gypsum, level 2, interior partition east and west

32.3 36.6 24.0 22.1 39.5 14.1 6.49 12.9 6.49 12.9 9.16 4.23 8.45 4.23 8.45 8.45 9.9

3.48 2.56 2.51 2.17 3.52 6.35 2.93 5.86 2.93 5.86 5.41 2.50 4.99 2.50 4.99 4.99 5.55

46.7 58.2 37.5 35.4 49.5 21.0 9.74 19.4 9.74 19.4 12.4 5.74 11.4 5.74 11.4 11.4 12.8

0.071 0.083 0.069 0.071 0.049 0.082 0.082 0.082 0.082 0.082 0.064 0.064 0.064 0.064 0.064 0.064 0.064

7.52 8.36 5.83 5.29 8.39 3.51 1.65 3.25 1.65 3.25 2.31 1.07 2.14 1.07 2.14 2.14 2.38

⫺0.07 ⫺0.09 ⫺0.05 ⫺0.06 ⫺0.05 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.06 ⫺0.04 ⫺0.04 ⫺0.04 ⫺0.04 ⫺0.04 ⫺0.04 ⫺0.04

PUNL

b1

a1

1.14 1.19 1.19 1.12 1.20 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45 1.45

1.09 1.08 1.12 1.09 1.10 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09

0.79 0.79 0.78 0.77 0.91 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38

Fig. 5. Experimental and numerical roof relative displacements of test structure under Rinaldi ground motion.

displacement by only 9% and captures well the phase of the experimental response. Similarly, Fig. 6 compares the experimental and numerical absolute global hysteretic behaviors (base shear—roof central relative displacement) of the test structure. The stiffness and energy dissipation characteristics of the test structure are reasonably well predicted by the numerical pancake model. Based on these comparative results, it was concluded that the numerical pancake model could predict with sufficient accuracy the seismic response of wood framed buildings and could be used as an analysis tool in displacement-based seismic design procedures. 6. Evaluation of hysteretic damping in wood framed buildings 6.1. Description of full-scale building structures In order to investigate the hysteretic damping exhibited by wood construction, four different wood framed

Fig. 6. Global hysteretic response of test structure under Rinaldi ground motion: (a) experimental, (b) numerical.

building configurations were modeled by the pancake approach discussed above. These buildings are prototypical buildings developed by the CUREE-Caltech Woodframe Project in California for use in loss estimation and benefit-to-cost ratio analysis [20]. Only a

468

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

brief description of these building configurations is presented herein. More detail information can be found elsewhere [18]. The four buildings considered are: 1. Small house: A one-story, two bedrooms, one bath house built ca. 1950 with a simple 110 m2 floor plan on a level lot. Prescriptive construction is assumed. Wood framed floor cripple walls are included in the poor- and typical-quality construction variants. These construction variants are described in the following section. 2. Large house: An engineered two-story single family dwelling of approximately 225 m2 on a level lot with a slab on grade and spread footings. This building is assumed to have been built as a housing development ‘production house’ in either the 1980s or 1990s. 3. Small townhouse: A two-story 140–170 m2 unit with a common wall. Part of the dwelling is over a twocar garage. It is on a level lot with a slab on grade and spread foundations. It is recently built and the seismic design is engineered. 4. Apartment building: A three-story, rectangular building with 10 units, each with 75 m2 and space for mechanical and common areas. All walls and elevated floors are woodframe. It has parking on the ground floor. Each unit has two bedrooms; one bath and one parking stall. It would have been constructed prior to 1970 and ‘engineered’ to a minimal extent. Pancake models were constructed for three different construction variants (representing superior-, typicaland poor-quality construction) of each of the four basic building types. A detailed description of these construction variants can be found elsewhere [18]. These 12 basic models were then used in this study in order to perform monotonic and cyclic pushover analyses in each principal direction of each building in order to characterize their hysteretic damping. Therefore, a total of 24 different cases were available to develop an equivalent viscous damping model to be used for the displacement-based seismic design of wood framed buildings.

Fig. 7. Results of monotonic pushover analyses of apartment building, Y-direction.

variants of the apartment building. It is evident that the construction quality has a significant influence on the initial stiffness and ultimate strength of the building. From the pushover analysis results, the roof displacement, ⌬, corresponding to the maximum base shear was identified for each building configuration. A cyclic pushover analysis was then performed for the following roof displacement amplitudes: 0.05⌬, 0.075⌬, 0.1⌬, 0.2⌬, 0.3⌬, 0.4⌬, 0.7⌬, and 1.0⌬. These amplitudes correspond to the primary cycles of the testing protocol developed recently within the CUREE-Caltech Woodframe Project [21]. As a typical example, Fig. 8 presents results of the cyclic pushover analysis in the Y-direction for the superior-quality variant of the apartment building. The stiffness degradation and pinching response of the structure are evident. In order to capture the energy dissipation characteristics of each building configuration at a given displacement amplitude, ⌬a, an equivalent viscous damping ratio, zeqh, representative of the hysteretic damping in the structure, was computed from the global hysteretic behavior of each building configuration [22] zeqh ⫽

ED , 2πkeq⌬2a

(3)

6.2. Analysis procedure First, monotonic pushover analyses were performed in order to determine the envelope of the base shear-peak central roof displacement relationship of each building configuration. Lateral loads were applied at each floor level and followed an inverse triangular distribution, typical of first mode response. The pushover analysis was carried out as a slow dynamic analysis using a slow ramp loading function for each applied lateral load [17]. The loads were applied slowly to ensure that inertia and viscous damping forces were minimized. Fig. 7 presents typical results of the monotonic pushover analyses in one of the principal (Y) direction of the three construction

Fig. 8. Results of cyclic pushover analyses of superior-quality variant of apartment building in Y-direction.

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

where ED is the energy dissipated per cycle at the displacement amplitude, ⌬a, and keq is the overall equivalent (secant) lateral stiffness of the building at the same displacement amplitude. From Eq. (3), the variation of zeqh with ⌬a can be obtained. 6.3. Results Fig. 9 presents the variation of the equivalent viscous damping ratio with building drift (defined as the roof displacement divided by the building height) obtained from the pushover analyses of all 24 building configurations. The behavior is similar for all building configurations and can be characterized by two distinct regimes: (1) for building drifts less than about 0.4%, the damping ratio increases linearly with building drift as inelastic deformations of stucco and gypsum walls are initiated early, and (2) for building drifts larger than 0.4%, the equivalent viscous damping ratio remains fairly constant between 15 and 20% of critical. A simple equation that describes well this behavior is given by zeqh ⫽

再

0.5⌬b 0ⱕ⌬b ⬍ 0.36% , 0.18 0.36%ⱕ⌬b

(4)

where zeqh is the global equivalent viscous damping ratio corresponding to the hysteretic damping and ⌬b is the building drift in percent. A nominal damping ratio, z0, must also be considered to account for the energy dissipation characteristic of other structural and non-structural elements in the structure. From previous work, z0 ⫽ 2% of critical appears reasonable for this purpose [5]. Therefore, the following equation is proposed for the global equivalent viscous damping ratio, zeq, in wood buildings:

Fig. 9.

zeq ⫽

再

0.5⌬b ⫹ 0.02 0ⱕ⌬b ⬍ 0.36% . 0.20 0.36%ⱕ⌬b

469

(5)

6.4. Appraisal of proposed damping model The proposed equivalent damping model for wood framed buildings, given by Eq. (4), was appraised by reanalyzing a linearized version of the test structure with equivalent viscous damping representing its hysteretic energy characteristics at the maximum displacement reached under the Rinaldi record. From Fig. 6, the secant global stiffness obtained numerically at the maximum displacement of 20.2 mm (building drift of 0.38%) is 6.44 kN/mm. This value represents 33% of the initial lateral stiffness predicted in the pancake numerical model of the test structure. Therefore, all wall elements in the pancake numerical model of the test structure were assigned linear elastic properties along with a stiffness equal to 33% of their initial stiffness. Furthermore, a Rayleigh viscous damping model based on viscous damping ratios of 18% of critical in the first and second elastic modes of vibration was introduced according to Eq. (4) for a building drift of 0.38%. Fig. 10 compares the relative displacement time-history at the center of the roof of the test structure predicted by the linearized numerical pancake model with the measured values (Fig. 10(a)) and with the predictions of the non-linear numerical pancake model (Fig. 10(b)). The linearized model under-predicts the experimental peak roof displacement by 15% and under-predicts the predictions of the non-linear model by 5%. These results are quite acceptable in the context of a displacementbased seismic design procedure. The same appraisal procedure described above was

Variation of equivalent viscous damping ratio with building drift.

470

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

the small house, for which the deformations are concentrated at the top of the wood cripple walls, the overpredictions of the equivalent linear models are within 15% of the building drifts predicted by the non-linear models. This level of accuracy can be considered adequate for displacement-based seismic design purposes. 7. Conclusions

Fig. 10. Roof relative displacement of test structure under Rinaldi ground motion: (a) experiment and linearized numerical model, (b) non-linear and linearized numerical model.

applied to the typical-quality variants of the full-scale woodframe buildings described in the previous section. The Rinaldi record was used once again for this appraisal exercise. Table 2 presents a summary of the results obtained and also compares the peak building drifts predicted by the non-linear models with the ones predicted by the equivalent linear models. For all cases, the equivalent linear models over-predict the peak building drifts, which is conservative for design purposes. Except for

This paper has presented a simple numerical model for the seismic analysis of wood framed buildings. This model, referred to as the pancake model, simulates the three-dimensional seismic response of a woodframe construction through a degenerated two-dimensional planar analysis in which all floor and roof diaphragms are superimposed on top of each other. Each wall in the structure (wood, gypsum and/or stucco) is modeled by a single zero-height non-linear in-plane shear spring using a hysteretic rule that simulates the degrading stiffness and pinching, typical of the racking response of these walls. By comparing the predictions of the pancake numerical model with the results of shake table tests conducted on a full-scale two-story woodframe house, it was concluded that the numerical pancake model could predict with sufficient accuracy the seismic response of wood framed buildings and could be used for displacementbased seismic design. The numerical pancake model was then utilized to develop a simple equation for the equivalent viscous damping ratio of wood framed buildings, based on the results of monotonic and cyclic pushover analyses on 24 full-scale wood framed building configurations representing four different building types and various construction qualities. The results indicate that the equivalent viscous damping increases linearly up to a building drift of about 0.4%, as the stucco and gypsum walls deform early in the inelastic range, and thereafter stabil-

Table 2 Building drifts predicted by non-linear and equivalent linear models of full-scale building structures, typical-quality variant, Rinaldi ground motion, Y-direction Building

Small house Large house Townhouse Apartment

Initial lateral stiffness (kN/mm)

17.8 40.5 114.0 88.8

Equivalent lateral stiffness at peak nonlinear building drift (kN/mm)

0.42 11.60 21.10 6.99

Equivalent viscous Building drift damping ratio at peak non-linear building drift

0.20 0.20 0.20 0.20

Non-linear model (%)

Equivalent linear model (%)

Error (%)

4.56 0.44 0.57 2.51

5.93 0.45 0.65 2.76

+30 +2 +14 +10

A. Filiatrault et al. / Engineering Structures 25 (2003) 461–471

izes around 18% of critical, when the wood shear walls undergo inelastic deformations. By comparing the predictions of a linearized pancake model of the test structure, utilizing damping values according to the proposed equation, it is shown that the equation could be used for direct-displacement based seismic design of wood framed buildings.

[4]

[5] [6] [7]

Acknowledgements The research project described in this paper was funded by the Consortium of Universities for Earthquake Engineering (CUREE) as part of the CUREE-Caltech Woodframe Project (Earthquake Hazard Mitigation of Woodframe Construction), under a grant administered by the California Office of Emergency Services and funded by the Federal Emergency Management Agency. We greatly appreciate the input and coordination provided by Professor John Hall of the California Institute of Technology and by Mr Robert Reitherman of the Consortium of Universities for Research in Earthquake Engineering. The authors also acknowledge Ms Kelly Cobeen and Mr John Coil who provided construction drawings for the woodframe buildings modeled in this paper. Opinions, findings, conclusions and recommendations expressed in this paper are those of the authors. No liability for the information included in this report is assumed by the Consortium of Universities for Research in Earthquake Engineering, the California Institute of Technology, the Federal Emergency Management Agency, or the California Office of Emergency Services.

References [1] Filiatrault A, Folz B. Performance-based seismic design of wood framed buildings. J Struct Eng ASCE 2002;128(1):39–47. [2] Priestley MJN. Myths and fallacies in earthquake engineering— conflicts between design and reality. Bull New Zeal Natl Soc Earthquake Eng 1993;26(3):329–41. [3] Priestley MJN. Displacement-based approaches to rational limit states design of new structures. In: Keynote address, Proceedings

[8] [9] [10] [11] [12] [13] [14]

[15] [16] [17]

[18]

[19]

[20]

[21]

[22]

471

of the 11th European Conference on Earthquake Engineering, Paris, France. 1998. Fischer D, Filiatrault A, Folz B, Uang C-M, Seible F. Shake table tests of a two-story woodframe house. Structural Systems Research Report No. SSRP 2000/15. Department of Structural Engineering, University of California, San Diego; 2000. Foliente GC. Hysteresis modeling of wood joints and structural systems. J Struct Eng ASCE 1995;121(6):1013–22. Tuomi RL, McCutcheon WJ. Racking strength of light-frame nailed walls. J Struct Eng ASCE 1978;104(7):1131–40. Gupta AK, Kuo GP. Behavior of wood-framed shear walls. J Struct Eng ASCE 1985;111(8):1722–33. Gupta AK, Kuo GP. Wood-framed shear walls with uplifting. J Struct Eng ASCE 1987;113(2):241–59. Filiatrault A. Static and dynamic analysis of timber shear walls. Can J Civil Eng 1990;17(4):643–51. Itani RY, Cheung CK. Nonlinear analysis of sheathed wood diaphragms. J Struct Eng ASCE 1984;110(9):2137–47. Gutkowski RM, Castillo AL. Single- and double-sheathed wood shear wall study. J Struct Eng ASCE 1988;114(2):1268–84. Dolan JD, Foschi RO. Structural analysis model for static loads on timber shear walls. J Struct Eng ASCE 1991;117(3):851–61. White MW, Dolan JD. Non-linear shear-wall analysis. J Struct Eng ASCE 1995;121(11):1629–35. Stewart WG. The seismic design of plywood sheathed shear walls. PhD thesis. University of Canterbury, Christchurch, New Zealand; 1987. Dolan JD. The dynamic response of timber shear walls. PhD thesis. University of British Columbia, Vancouver, Canada; 1989. Folz B, Filiatrault A. Cyclic analysis of wood shear walls. J Struct Eng ASCE 2001;127(4):433–41. Carr AJ. RUAUMOKO—inelastic dynamic analysis program. Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand; 2000. Isoda H, Folz B, Filiatrault A. Seismic modeling of index woodframe buildings. Structural Systems Research Project Report No. SSRP-2001/12. Department of Structural Engineering, University of California, San Diego, La Jolla, CA; 2001. Filiatrault A, Kremmidas S, Seible F, Clark AJ, Nowak R, Thoen BK. Upgrade of first generation uniaxial seismic simulation system with second generation real-time three-variable digital control system. In: Proceedings of the 12th World Conference on Earthquake Engineering, Auckland, New Zealand, Paper No. 1674, on CD-ROM. 2001. Porter KA, Beck JL, Seligson HA, Scawthorn CR, Tobin LT, Young R, et al. Improving loss estimation for woodframe buildings. Report Published by the Consortium of Universities for Research in Earthquake Engineering. Richmond, CA; 2001. Krawinkler H, Parisi F, Ibarra L, Ayoub A, Medina R. Development of a testing protocol for wood frame structures. CUREE Publication No. W-02. Richmond, CA; 2000. Clough RW, Penzien J. Dynamics of structures, 2nd ed. New York: McGraw-Hill, 1993.