SIMPLIFIED SEISMIC ANALYSIS OF WOODFRAME STRUCTURES

Transcription

1 3 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 24 Paer No. 245 SIMPLIFIED SEISMIC ANALYSIS OF WOODFRAME STRUCTURES Bryan FOLZ and Andre FILIATRAULT 2 SUMMARY A simle numerical model to redict the dynamic characteristics, quasi-static ushover and seismic resonse of woodframe buildings is resented. In this model, the building structure is comosed of two rimary comonents: rigid horizontal diahragms and nonlinear lateral load resisting shear wall elements. The actual three-dimensional building is degenerated into a two-dimensional lanar model using zeroheight shear sring elements connected between adjacent diahragms or the foundation. The hysteretic behavior of each wood shear wall in the building can be characterized using an associated numerical model that redicts the walls load-dislacement resonse under general quasi-static cyclic loading. In turn, in this model, the hysteretic behavior of each shear wall is reresented by an equivalent nonlinear shear sring element. With this simle structural model, the resonse of the building is defined in terms of only three-degrees-of-freedom er floor. This numerical model has been incororated into the comuter rogram SAWS - Seismic Analysis of Woodframe Structures. The redictive caabilities of the SAWS model are comared with shake table tests erformed on a full-scale two-storey woodframe house as art of the recently comleted CUREE-Caltech Woodframe Project. It is shown in this study that the SAWS comuter rogram rovides reasonably accurate estimates of the dynamic characteristics, quasi-static ushover and seismic resonse of this test structure. Furthermore, the SAWS rogram requires a minimum amount of data inut and rovides a fast comutational turn-around time to analyze a given structure. As a result, this simle numerical model may be a useful structural analysis tool for racticing engineers and researchers. INTRODUCTION Light-frame residential wood buildings are by far the most common structures constructed in North America. These building systems range in size from small, single-story dwellings to large, multi-level, multile-occuancy condominiums and aartments. The rimary structural comonents of these building are tyically horizontal floor diahragms, horizontal or sloed roof diahragms and vertical shear walls. Each of these structural elements is generally comosed of sawn lumber framing members attached to sheathing anels using dowel connectors, such as nails, and/or adhesives. Tyically, these elements are interconnected through nailed connections to make u the overall structural system for the building. With Faculty, British Columbia Institute of Technology, Burnaby, Canada. bfolz@bcit.ca 2 Deuty Director of the Multidiscilinary Center for Earthquake Engineering Research and Professor of Structural Engineering, State University of New York, Buffalo, NY, USA. filiatrault@mceermail.buffalo.edu

2 this tye of construction the resulting building tyically has a high degree of structural redundancy. In addition, the inelastic resonse of the building under severe lateral loading conditions may dislay significant degradation in strength and stiffness. These comlicating factors have contributed to aucity in dedicated three-dimensional structural analysis models for woodframe building []. This is in contrast to the abundance of static and dynamic structural analysis tools available to evaluate the resonse of building frames comosed of other construction materials such as structural steel or reinforced concrete. In the modeling of woodframe structures, it has been noted that if all of the structural details within a woodframe building is catured within a finite element model, the required comutational overhead can overwhelm ractical comuting caability [2]. Therefore, model reduction techniques must be imlemented to roduce usable numerical simulations. Another maxim at work is: the more detailed the model, the greater the deendency on test data to calibrate the model. Unfortunately, the requisite test data is not always readily available. With these considerations in mind, this aer resents a simle and versatile numerical model that redicts the dynamic characteristics, quasi-static ushover and seismic resonse of light-frame wood buildings. The basic modeling aroach is to degenerate the actual three-dimensional building into a twodimensional lanar model comosed of zero-height shear wall sring elements connecting the floor and roof diahragms together or to the foundation. All diahragms in the building model are assumed to have infinite in-lane stiffness. Using this simle modeling aroach, the resonse of the building is defined in terms of only three-degrees-of-freedom (3-DOF) er floor. Each shear wall sring element is calibrated to reflect the strength and stiffness degrading hysteretic characteristics of the associated shear wall it is modeling in the structure. Requisite test data is limited to that necessary to characterize the behavior of the shear walls. This three-dimensional numerical model has been incororated into the comuter rogram SAWS - Seismic Analysis of Woodframe Structures [3]. This aer rovides an overview of the model formulation. As well, the redictive caabilities of this model are comared with exerimental results from recent shake table tests erformed on a full-scale two-story wood frame house. It is shown that the model redictions are in good agreement with the test results with resect to both the dynamic characteristics and seismic resonse of the building structure. MODEL FORMULATION Structural Configuration of the Building Model For illustrative uroses, the structural configuration of a tyical woodframe building is discussed in terms of the simlified single-story building layout shown in Fig. a. The main structural elements that comose the building are: the exterior and/or interior shear walls, the interior artition walls and the floor and/or roof diahragms. The building structure is assumed to be attached to a rigid foundation. In the modeling of the structure, it is assumed that both the floor and roof elements have sufficiently high inlane stiffness to be considered as rigid elements. This is exected to be a reasonable assumtion for tyically constructed diahragms with a lanar asect ratio of the order of 2:, as suorted by exerimental results from full-scale diahragms tests [4]. Further to this, it is assumed that an equivalent horizontal diahragm can adequately model a sloing roof. As a final modeling ste, the threedimensional building structure is degenerated to a lanar model by assigning zero-height to all of the shear wall elements that connect the horizontal diahragms to the foundation. Each shear wall in the building structure is, in turn, reresented by an equivalent single-degree-of-freedom (SDOF) shear element via a calibration rocedure discussed in a subsequent section. Alying this overall modeling aroach to the building structure illustrated in Fig. a roduces the simlified building model shown in Fig. b.

3 2 3 A B C Diahragm (Rigid) y A B C 3 Generic Point on the Rigid Diahragm: (x, y ) 2 y f x Foundation Shear Wall, ty. Origin: (,) Θ V U x f y Global Degrees-of-Freedom for Rigid Diahragm Shear Element-to-Diahragm Attachment Point x SDOF Shear Element (Zero-Height), ty. Foundation, ty. (a) Figure : (a) Comonents of a single-storey woodframe structure. (b) Model of the single-storey woodframe structure. It is noted that this degenerated lanar model does not cature the overturning and flexural resonse of a building. However, this is not viewed as a significant limitation of this model for its intended alication. Most woodframe buildings are low-rise structures so that overturning is not tyically significant and the deformation mode is rimarily one of shear. Kinematic Assumtions for the Building Model To analyze the simlified building model resented in Fig. b, a global rectangular coordinate system is rescribed as shown so that the entire building structure is contained within the first quadrant. The global degrees-of-freedom of each diahragm are then assigned at the origin of the coordinate system. Under the assumtion that each diahragm is rigid, only three global degrees-of-freedom are required to comletely describe its rigid body motion: two translations U and V and one rotation Θ. Under a general dislaced state of the diahragm secified by the values of U, V and Θ, the resulting linearized dislacement of any oint on the diahragm can be exressed in matrix form as: (b) u v θ = y x U V = ΛD Θ where Λ is a transformation matrix and [ ] T D = U,V, Θ is the global dislacement vector. () Load Vector and Stiffness Matrix Formulation Consider the alication of external forces f x and f y located at any oint on the diahragm, with the line of action of f x arallel to the x-axis and the line of action of f y arallel to the y-axis, as shown in Fig. b. These forces are statically equivalent to the following system of forces alied at the origin: F F M U = y f x f y = V Θ x f x T Λ f y (2)

4 where F = [ F ] T U, FV, M Θ is the global force vector that is conjugate to the global dislacement vector D. The force transformation relationshi given by Eq. (2) can be used to reresent all tyes of loads alied to the diahragm including inertia forces as well as the restoring forces develoed in the SDOF shear elements. For a given building structure let N x and N y denote the number of SDOF shear elements arallel to the x- axis and y-axis, resectively. For the examle building model shown in Fig. b N x = N y = 3. For the i-th SDOF shear element aligned arallel to the x-axis, let k ix denote its in-lane stiffness. Similarly, let k jy denote the in-lane stiffness of the j-th SDOF shear element aligned arallel to the y-axis. Out-of-lane stiffness in all shear elements is assumed to be zero. This assumtion is exected to be reasonable for an isolated shear wall, however it also imlies that intersecting shear walls behave indeendently of each other, which tyically is not the case. The force develoed in a SDOF shear element resulting from a dislacement of the diahragm can be related to the global forces in the structure through Eq. (2). Also, the dislacement of the SDOF shear element can be related to the global dislacement of the diahragm through Eq. (). Alying and combining these two stes over all SDOF shear elements in the building model results in the generation of the global stiffness matrix K: Nx Ny T T ( Λ K Λ ) ( Λ K Λ ) K = + (3) ix= ix ix ix jy= where K ix and K jy are given, resectively, by: jy jy jy k ix K ix = and K jy = k jy (4a,b) As resented, K ix and K jy aly to a 3-DOF building model such as shown in Fig. b. In Eq. (3), the coordinates in the transformation matrices Λ ix and Λ jy identify the oint of attachment of the SDOF shear element to the diahragm. Extension of the formulation of the global force vector and stiffness matrix to multi-story building structures is straightforward. For the global stiffness matrix K resented above, it is imortant to note that the contributing stiffness of each SDOF shear element is, for the roblem at hand, load (or dislacement) history deendent. A general characterization of the hysteretic resonse of the SDOF shear elements is resented in the subsequent section. SDOF Hysteretic Model of Wood Shear Walls The force-deformation resonse of a wood shear wall with dowel-tye sheathing-to-framing connectors is nonlinear under monotonic loading and additionally exhibits inched hysteretic behavior with strength and stiffness degradation under general cyclic loading. It is well recognized that the global forcedeformation resonse of a wood shear wall is qualitatively very similar to that of the individual sheathingto-framing connectors used in the construction of the wall [5]. Consequently, when roerly calibrated, the same hysteretic model used for sheathing-to-framing connectors can be alied to model the global cyclic resonse of a shear wall. The authors have reviously develoed a general hysteretic model for sheathing-to-framing connectors, which is defined in terms of a number of ath following rules to reroduce the resonse of a connector under arbitrary cyclic

5 loading [5]. This same hysteretic model is adoted herein to model the global cyclic resonse of a wood shear wall. Force, F (kn) F 5 K δ F r K r 2 K (δ u, F u ) Dislacement, δ (mm) δ F Force, F (kn) r 3 K D E F Dislacement, δ (mm) A F F I B O r 4 K C 2 H Dislacement (mm) K K O G (δ u,f u ) r 2 K Cyclic Loading Protocol I A D G I H (a) Figure 2: Force-dislacement resonse of a wood shear wall under: (a) monotonic loading and (b) cyclic loading. The hysteretic model is fitted to test data for a 2.4 m by 2.4 m shear wall with 9.5 mm thick OSB anels [8]. Figure 2a shows the assumed force-deformation behavior of a wood shear wall under monotonic loading to failure. The monotonic racking resonse, in terms of to-of-wall force F and dislacement δ (see the insert in Fig. 2a) is modeled by the following nonlinear relationshi: ( F + r K δ ) [ ex( K δ / F )] δ) F + r K [ δ sgn( δ) δ ], sgn( δ), δ δu F = sgn( u 2 u δu < δ δf (5), δ > δf This force-deformation model is characterized by six hysically identifiable arameters: F, K, r, r 2, δ u and δ F. Phenomenologically, Eq. (5) catures the crushing of the framing members and sheathing along with yielding of the connectors. Beyond the dislacement δ u, which is associated with the ultimate load F u, the load-carrying caacity is reduced. Failure of the wall under monotonic loading occurs at the dislacement δ F. Next, consider the force-deformation resonse of a shear wall under the cyclic loading shown as an insert in Fig. 2b. The basic ath following rules which define the hysteretic model are identified and briefly discussed. In Fig. 2b, force-deformation aths OA and CD follow the monotonic enveloe curve as exressed by Eq. (5). All other aths are assumed to exhibit a linear relationshi between force and deformation. Unloading off the enveloe curve follows a ath such as AB with stiffness r 3 K. Here the wall unloads elastically. Under continued unloading, the resonse moves onto ath BC that is characterized by a reduced stiffness r 4 K. The very low stiffness along this ath exemlifies the inched hysteretic resonse dislayed by wood shear walls under cyclic loading. This behavior occurs because of (b)

6 the reviously induced crushing of the framing members and sheathing anels around the connectors (in this case as the wall followed the ath OA). Loading in the oosite direction for the first time forces the resonse onto the enveloe curve CD. Unloading off this curve is assumed elastic along ath DE, followed by a inched resonse along ath EF, which asses through the zero-dislacement intercet F I, with sloe r 4 K. Continued re-loading follows ath FG with degrading stiffness K, as given by α δ K = K (6) δmax with δ = ( F K) and α a hysteretic model arameter which determines the degree of stiffness degradation. Note from Eq. (6) that K is a function of the revious loading history through the last unloading dislacement δ un off the enveloe curve (corresonding to oint A in Fig. 2b), so that δ max = βδ un (7) where β is another hysteretic model arameter. A consequence of this stiffness degradation is that it also roduces strength degradation in the resonse. If on another cycle the shear wall is dislaced to δ un, then the corresonding force will be less than F un which was reviously achieved. This strength degradation is shown in Fig. 2b by comaring the force levels obtained at oints A and G. Also, with this model under continued cycling to the same dislacement level, the force and energy dissiated er cycle is assumed to stabilize beyond the second loading cycle. In total, arameters are required to define this hysteretic model. To obtain these arameters for a articular wood shear wall, an analysis tool such as the CASHEW (Cyclic Analysis of SHEar Walls) comuter rogram [6] can be emloyed. The CASHEW rogram assumes that a shear wall is comosed of in-connected rigid framing members, elastic shear deformable sheathing members and nonlinear sheathing-to-framing connectors that follow the same hysteretic model described above. With the CASHEW rogram, a given wall is first subjected to the CUREE-Caltech Testing Protocol [7], after which a fitting rocedure extracts the arameters to reresent the wall resonse by an equivalent SDOF shear wall sring element. This model reduction aroach has been successfully evaluated against exerimental tests. As an examle, the force-deformation resonse shown in Fig. 2b was roduced using the arameters obtained from the CASHEW rogram for a 2.4 m by 2.4 m shear wall with 9.5 mm thick oriented strand board (OSB) sheathing anels that was also tested cyclically and under an earthquake ground motion on a shake table [8]. Comarison between the exerimental results and the redicted resonses of the reduced shear wall model showed good agreement [5]. In recent years, there has been a roliferation of cyclic tests erformed on full-scale wood framed shear walls. The resulting database of exerimental results can be used to secify equivalent SDOF shear sring elements for each of the walls that have been tested. This aroach has been utilized by the authors to calibrate shear sring elements to model shear walls sheathed with stucco and gysum wall board [3]. Dynamic Analysis The governing equations of motion for a three-dimensional structure when subjected to an earthquake ground motion are given by: ( D(t) ) = Mr& x (t) MD& ( t) + CD& (t) + Fsw g (8) where M is the global mass matrix; C is the global viscous daming matrix; F sw (t) is the global restoring force vector generated by all of shear wall elements in the structure; D & (t), D & (t) and D (t) are, resectively, the global acceleration, velocity and dislacement vectors relative to the ground; & x g (t) is the ground acceleration and r is a vector couling the ground motion inut with the global degrees-of-freedom

7 excited by that motion. The right hand side of Eq. (8) must be relaced by the more general exression M [ r U & x gu (t) + r & x V gv (t) ] for the case of bi-directional ground accelerations & x gu (t) and & x gv (t) alied arallel to the x-axis and y-axis, resectively. The global mass matrix M, given in Eq. (8), can be obtained through consideration of dynamic equilibrium and alication of D Alembert s Princile. With reference to Fig. b, the global mass matrix for a single story structure is given by: m myc M = m mx c (9) 2 2 myc mx c I + m(x c + yc ) where m is the mass of the diahragm, (x c, y c ) are the coordinates of its center of mass from an absolute reference and I is the mass moment of inertia about the center of mass. As resented, Eq. (9) only reresents the contributing mass from the diahragm; mass associated with each shear wall can be reresented by a discrete mass that includes rotatory inertia and added aroriately to Eq. (9). The global viscous daming matrix C, given in Eq. (8), accounts for all sulemental energy dissiating mechanisms in the structure other than the hysteretic daming roduced in the shear wall elements. In this study a Rayleigh daming model is assumed. To advance the solution of Eq. (9), from time t to t+ t, the equations of motion are rewritten in incremental form: M D& + C D& + F sw = Mr & x g (t) () with ( ) = ( ) t + t ( ) t. Within each time ste, the structural resonse is assumed linear. As a consequence, the increment in the global restoring force is given by: Fsw = K T D () where K T is the global tangent stiffness matrix given by Eq. (3) and evaluated at time t. Newmark s Method, utilizing constant average acceleration within a time ste, is used to integrate Eq. () over the time domain. As resented, this incremental solution strategy does not guarantee that equilibrium is achieved at the end of each time ste. To minimize the accumulation of error over time, the relative acceleration at time t + t is determined by enforcing dynamic equilibrium of Eq. (8) at the end of each time ste. In turn, this adjusted value of acceleration is used in the next time ste of the incremental solution strategy. Also, energy balance calculations are erformed at the end of each time ste to monitor the accuracy of this solution strategy as well as to assess the energy absortion caacity of the shear wall elements of the structure. The rocess of integrating the equations of motion is generally aborted if an energy balance is not satisfied within a secified tolerance. Pushover Analysis The governing equations of motion for a three-dimensional structure when subjected to a general dynamic force excitation are given by: ( D(t) ) (t) M D & ( t) + CD& (t) + Fsw = P (2)

8 where Eq. (2) is identical to Eq. (8) excet that the right- hand side of the equation has been relaced by the global alied forcing function P(t). In order to erform a uni-directional quasi-static ushover analysis arallel to the x-axis or y-axis, using Eq. (2), P(t) is decomosed as follows: P( t) = (t) F (3) with (t) reresenting the temoral variation of a slowly varying, monotonically increasing, alied ushover load, while F determines the distribution of nodal forces acting on the structure. The time varying load (t) takes the form: t (t) cos π = Pmax t T 2 T (4) where P max reresents the magnitude of the maximum lateral load alied to the structure and T is the duration of the alied loading. To minimize the contribution from inertia and daming effects in Eq. (2), the duration of loading T is set to one hundred times the comuted fundamental elastic eriod of the structure. The distribution of the lateral loading alied at the global degrees-of-freedom is determined through F. First off, F reresents nodal loads equivalent to ushover loads alied at the center of mass of each floor of the structure. Three otions on the distribution of loading over the height of the structure are considered in this analysis: a uniform distribution, an inverted triangular distribution and a modal adative distribution [3]. In general, secification of an aroriate lateral load distribution is a weak oint in the ushover methodology. At least being able to utilize the three different distribution noted above can hel determine the resonse sensitivity of a given structure to alied lateral loadings. SAWS Comuter Program The model formulation resented above has been incororated into the comuter rogram SAWS Seismic Analysis of Woodframe Structures [3]. This comuter rogram has analysis otions to redict the dynamic characteristics, quasi-static ushover and seismic resonse of light-frame wood buildings. MODEL VERFICATION Descrition of the Verification Test Structure In seeking to validate the SAWS numerical model, it must be noted at the outset that very few tests have been conducted in North America on full-scale light-frame wood buildings. As art of the CUREE- Caltech Woodframe Project a simlified full-scale two-story single-family house was tested on a uniaxial earthquake facility [9]. This extensive exerimental study considered different construction hases of the test structure. Only the fully engineered Phase 9 test structure, as described below, will be used in the verification study of the SAWS model. Figure 3 shows lan and elevation views of the test structure along with the identification of the major structural comonents used in its construction. The Phase 9 test structure included only the bare wood framing; interior and exterior wall finishes were not included nor were door and window details. The design of the Phase 9 test structure was based on the engineering rovisions of the 994 edition of the Uniform Building Code [] for a seismic zone 4 and common design ractices in California. The design assumes a force-reduction factor R w of 8, for a lateral load-resisting system consisting light-frame wood shear walls. The seismic weight of the structure was kn and the eriod of vibration estimated by the code was.8 seconds. The resulting design base shear in the shaking (north-south) direction was 5 kn. The ultimate base shear caacity of the test structure was estimated to be 75 kn, according to the FEMA 273 guidelines [].

9 C4 D4 C6 D6 C8 D8 Figure 3: Plan and elevation views of the test structure, with major structural comonents identified. During Phase 9, the test structure was subjected to an extensive series of shake table motions, which included harmonic, white noise and seismic ground motions. The white noise was used to determine frequencies and mode shaes of the structure under low amlitude vibrations. The harmonic inut at the natural frequency of the structure was used to ascertain its daming characteristics. The seismic ground motions were used to determine the inelastic resonse of the structure under increasing intensities of shaking. Five levels of seismic tests were erformed on the test structure. The inut ground motion for seismic test levels to 4 reresents a scaling of the 994 Northridge Earthquake recorded at Canoga Park, as shown in Fig. 4. The inut ground motion for seismic test level 5 reresents the unscaled 994 Northridge Earthquake recorded at Rinaldi, also shown in Fig 4. Seismic test level 4 reresents a ground motion with a hazard level of % robability of exceedance in 5 years. Seismic test level 5 reresents a ground motion with a hazard level of 2% robability of exceedance in 5 years. As art of the test rogram, one seismic test was reeated once the maximum transient inter-story wall drift exceeded.5% of its height. For Phase 9 this occurred at level 3. Figure 4 resents the comlete sequence of acceleration records alied to the Phase 9 shake table test structure. Over Phase 9 the test structure was subjected, in turn, to each one of these ground motions without any reair or modification to the structure. The test structure was monitored with nearly three hundred digital instruments to measure forces, dislacements and accelerations in the structure during the shake table tests. For the evaluation of the SAWS model redictions, resonse quantities of interest will be limited to relative dislacement and absolute acceleration time-histories at the roof level. Data acquisition sensors to measure dislacement (C4, C6 and C8) and acceleration (D4, D6 and D8) were laced at roof level along the east wall, center-line and west wall of the structure, as shown in Fig. 3.

10 Acceleration (g) g.6g.24g.24g.35g.8g -.8g -.22g -.35g -.35g -.52g -.65g Level Level 2 Level 3 Level 3R Level 4 Level Time (s) Figure 4: Sequence of acceleration records alied to the shake-table test structure. SAWS Model of the Test Structure The SAWS model for the test structure is schematically shown in Fig. 5. It is comosed of eight zeroheight shear wall sring elements and two rigid diahragms; one for the second floor and one at the roof level. The force-deformation resonse of each shear wall sring element requires the secification of ten hysteretic arameters, as discussed reviously. These arameters were determined using the analysis rogram CASHEW - Cyclic Analysis of SHEar Walls [6]. As an examle from this analysis ste, Fig. 6 shows the CASHEW model redictions of the cyclic resonse of the first and second level east shear walls of the test structure. As determined by the CASHEW rogram, the calibrated hysteretic arameters required to describe the cyclic resonse of the eight Phase 9 shear walls are given Table. In Fig. 5 and Table, the eight shear sring elements have been identified using the notation S xi and S yi, with i =,..4 and the subscrit x and y indicating that an element is aligned either along the x or y axis. y y S X2 C A 7 S X4 C A 7 N 2 ND FLOOR DIAPHRAGM ROOF DIAPHRAGM S X S Y Zero-Height Shear Wall Sring Element, Ty. Fixed Suort, Ty. S Y2 x S X3 S Y3 Connects to 2 nd Floor Diahragm, Ty. S Y4 x Figure 5: SAWS model of the test structure.

11 Force (kn) Force (kn) Dislacement (mm) (a) st floor east shear wall Dislacement (mm) (b) 2 nd level east shear wall Figure 6: CASHEW redictions of the cyclic resonse of the test structure shear walls. Table : Hysteretic arameters for the shear wall sring elements in the test structure. Sring Element S X Level East Wall S X2 Level West Wall S Y Level South Wall S Y2 Level North Wall S X3 Level 2 East Wall S X4 Level 2 West Wall S Y3 Level 2 South Wall S Y4 Level 2 North Wall K r r 2 r 3 r 4 F u α β (kn/mm) (kn) (kn) (mm) F I SAWS Model Predictions Dynamic and Seismic Analysis of the Test Structure The SAWS model of the Phase 9 shake table test structure redicted a fundamental frequency of 3.28 Hz. This rediction is based on assigned seismic weights of 62 kn to the second floor diahragm and 48 kn to the roof diahragm. At the start of Phase 9 testing, the exerimentally measured fundamental frequency was 3.96 Hz [9]. The SAWS model under redicts the exerimental result by 7%. Potential sources for this difference are the inherent simlicity of the SAWS model and the CASHEW redictions of the initial stiffness of each shear wall element.

12 Figures 7 and 8 resent the SAWS model redictions of the relative dislacement time-histories along the centerline of the test structure at roof level for level 4 and 5 testing along with the corresonding exerimental results. These relative dislacement measurements corresond to sensor C6, as shown in Fig. 3. The SAWS model redictions are based on an assigned equivalent viscous daming of % of critical in the first and second elastic modes of vibration. It is assumed that under the strong motion ortions of shaking of levels 4 and 5, that daming in the structure can largely be accounted for through the hysteretic resonse of the shear wall sring elements. As a consequence, the viscous daming is set to a low value. This modeling aroach is consistent with what has been suggested by other research work [5]. Exerimentally, the equivalent viscous daming for the test structure was calculated to be 4.2% of critical under small amlitude excitation. Relative Dislacement (mm) (a) Exerimental Result (b) SAWS rediction Figure 7: Level 4 relative dislacement time histories at Sensor C6: (a) exerimental result and (b) SAWS rediction. Relative Dislacement (mm) Time (s) Time (s) (a) Exerimental Result Time (s) (b) SAWS rediction Figure 8: Level 5 relative dislacement time histories at Sensor C6: (a) exerimental result and (b) SAWS rediction. With resect to maximum resonse values, over levels 4 and 5, the maximum relative dislacement at sensor location C6 estimated by the SAWS model under redicts the exerimental values by 7.6% and 6.3%, resectively. Table 2 rovides a comlete listing of the maximum resonse values redicted by the SAWS model and obtained exerimentally for all of the sensor locations shown in Fig. 3. Comaring the maximum relative dislacement results given in Table 2, the differences in the numerical redictions and the exerimental values can, in art, be attributed to the SAWS model not fully caturing the torsional resonse of the Phase 9 test structure and also the model s inability to account for the in-lane deformation in the roof diahragm. Over test levels 4 and 5, the maximum torsional resonse resulted in a difference in relative dislacement between the east and west walls of 5.2 mm and.7 mm, resectively. Relative Dislacement (mm) Relative Dislacement (mm) Time (s)

13 Also for levels 4 and 5, the maximum roof diahragm deformations were 5.8 mm and 7. mm, resectively. Table 2: Comarison of maximum relative dislacement and maximum absolute acceleration resonse quantities from SAWS redictions and exerimental results. Sensor Locations C4 & D4 C6 & D6 C8 & D8 Level 4 Maximum Relative Dislacement (mm) SAWS Model Test Result % Error Level 4 Maximum Absolute Acceleration (g) SAWS Model Test Result % Error Level 5 Maximum Relative Dislacement (mm) SAWS Model Test Result % Error Level 5 Maximum Absolute Acceleration (g) SAWS Model Test Result % Error All of the dynamic time-history results were roduced by the SAWS comuter rogram using a time integration ste of. sec. In order to roerly cature the evolution of damage to the structure over the Phase 9 testing regime, the SAWS comuter rogram was run using the train of shake table acceleration inut records from level 3 to 5 (see Fig. 4). The exerimental results showed that very little damage occurred to the Phase 9 structure under level and 2 testing. The numerical analysis was erformed with a secified tolerance of 5% on the error in the energy balance alied to the equations of motion. The comutational time required to erform a given level of analysis on the Phase 9 test structure using the SAWS rogram on a modern deskto comuter was of the order of only seconds. This fast comutational turn-around time is attributable to the fact the SAWS comuter rogram models the twostory shake table test structure using only six degrees-of-freedom. Pushover Analysis of the Test Structure A static ushover test was not erformed on the shake table structure. However, from the seismic test data from Phase 9, a caacity sectrum was generated as shown in Fig. 9. The caacity sectrum is a lot of the maximum base shear (obtained as the sum of the floor and roof inertia forces) versus the corresonding eak roof relative dislacement (measured at sensor location C6) for each test level [9]. With the caacity sectrum, the comuted base shear reresents the force induced in the foundation of the test structure including the nonlinear restoring force and viscous daming force. In a static ushover, only the nonlinear restoring force is activated since the resulting velocity of the structure is negligible during such a test. Thus, it is to be exected that the maximum base shear redicted by a static ushover analysis would be less than that estimated by a caacity sectrum. Figure 9 shows the SAWS model redictions from three static ushover analyses, corresonding to lateral load distributions that are uniform, triangular and modal adative. These three results are in relatively close agreement with each other and redict a

14 maximum base shear for the test structure of aroximately kn. For comarative uroses, estimates of design and ultimate maximum base shears are also shown in Fig. 9. Maximum Base Shear (kn) 5 5 Level 2 Level 3 & 3R Level Caacity Sectrum Level 4 Level 5 V ULTIMATE = 74 kn (FEMA 273) Triangular Adative Uniform V DESIGN = 5 kn (UBC 994) Peak Roof Relative Dislacement (mm) Figure 9: Exerimental caacity sectrum and SAWS ushover redictions. MODEL APPLICATIONS Performance assessment and design rocedures that are currently being develoed often require multile structural analyzes to be conducted. Obviously, with these methodologies, the comutational efficiency of the analysis models becomes imortant. The SAWS comuter rogram was develoed with the intention of minimizing comutational overhead in erforming a structural analysis. Consequently, this analysis rogram is well suited for imlementation with these aroaches. Two examle alications are briefly resented here. First, Fig. a shows the variability in resonse of the Phase 9 test structure to a suite of 2 earthquake records, scaled to match the NEHRP design sectrum [7]. In conjunction with this examle, the SAWS comuter model could easily be couled with reliability software to conduct a reliability assessment of the test structure. As a second examle, Fig. b shows the results of an incremental dynamic analysis [2] using the SAWS comuter model alied to the Phase 9 test structure under a scaling of the 994 Canoga Park ground motion. In both of these examles the erformance of Phase 9 test structure is evaluated against the 2% drift limit for life safety []. Cummulative Probability SAWS Predictions Lognormal Fit 2% Drift Limit for Life Safety IM = PGA (g) % Drift Limit for Life Safety..2.. (a) Dislacement max (mm) Figure : SAWS model alications: (a) resonse variability and (b) incremental dynamic analysis of the Phase 9 test structure. (b) DM = Building Drift (%)

15 CONCLUSIONS A simle numerical model to redict the dynamic characteristics, quasi-static ushover and seismic resonse of woodframe buildings has been resented. This numerical model has been incororated into the comuter rogram SAWS - Seismic Analysis of Woodframe Structures. The redictive caabilities of the SAWS comuter rogram have been comared with recent shake table tests erformed on a full-scale two-storey woodframe house as art of the recently comleted CUREE-Caltech Woodframe Project. It was shown in this study that the SAWS comuter rogram rovides reasonably accurate estimates on the dynamic characteristics, quasi-static ushover and seismic resonse of this test structure. Furthermore, the SAWS rogram requires a minimum amount of data inut and rovides a fast comutational turn-around time to analyze a given structure. Consequently, this simle numerical model may be a useful analysis tool for the erformance assessment of light-frame wood building by both racticing engineers and researchers. ACKNOWLEDGEMENT The work resented herein was carried out under financial suort from the Consortium of Universities for Research in Earthquake Engineering (CUREE) as art 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. The financial suort rovided by these funding agencies is gratefully acknowledged. Oinions, findings conclusions and recommendations exressed in this aer are those of the writers. No liability for the information included in this aer is assumed by CUREE, California Institute of Technology, Federal Emergency Management Agency, or California Office of Emergency Services. REFERENCES. Yancy CW, Cheok GS, Sadek F, Mohraz B. A summary of the structural erformance of single-family, wood-frame housing. Technical Reort NISTIR 6224, National Institute of Standards and Technology, Gaithersburg, MD, Kasal B, Leichti RJ, Itani RY. Nonlinear finite-element model of comlete light-frame wood structures. ASCE Journal of Structural Engineering 994; 2(): Folz B, Filiatrault A. SAWS - Version.: a comuter rogram for seismic analysis of woodframe buildings. Reort No. SSRP 2/9, Structural Systems Research Project, Deartment of Structural Engineering, University of California, San Diego, La Jolla, CA. 4. Philis TL, Itani, RY, McLean DI. (993). Lateral load sharing by diahragms in wood-frame buildings. ASCE Journal of Structural Engineering 993; 9(5): Folz B, Filiatrault A. Cyclic analysis of wood shear walls. ASCE Journal of Structural Engineering 2; 27(4): Folz B, Filiatrault A. CASHEW - Version.: a comuter rogram for cyclic analysis of wood shear walls. CUREE Publication No. W-8, Richmond, CA, Krawinkler H, Parisi F, Ibarra L, Ayoub A, Medina R. Develoment of a testing rotocol for wood frame structures. CUREE Publication No. W-2, Richmond, CA, Durham JP. Seismic resonse of wood shearwalls with oversized oriented strand board anels. M.A.Sc. thesis, University of British Columbia, Vancouver, Canada, Fischer D, Filiatrault A, Folz B, Uang C-M, Seible F. Shake table tests of a two-story house, CUREE Publication No. W-6, Richmond, CA, 2.. International Conference of Building Officials: Uniform Building Code 994, Whittier, CA.. FEMA-273: NEHRP Guidelines for the seismic rehabilitation of buildings, Federal Emergency Management Agency, Publication Distribution Facility, Washington, DC, Vamvatsikos D, Cornell CA. Incremental dynamic analysis. Earthquake Engineering and Structural Dynamics 22; 3(3):