Dynamic Analysis and Modeling of Wood-framed Shear Walls

Dynamic Analysis and Modeling of Wood-framed Shear Walls Yasumura, M. 1 ABSTRACT Dynamic performance of wood-framed shear walls was analyzed by means of the non-linear earthquake response analysis and the equivalent linear method. The simulated results were compared with the pseudo-dynamic test results. ysteresis model for the non-linear analysis consisted of envelop curve up to and beyond the maximum load, unloading stiffness toward the x-axis and reloading stiffness toward the previous peak. These parameters were determined using the experimental results of reversed loading tests of shear walls. The maximum displacement response was calculated by means of the equivalent linear response method. The capacity spectrum of the structure was obtained from the load-deformation relationship of the structure. Linear response was calculated with the equivalent viscous damping ratio of shear walls in founction of the natural period. The maximum displacement response was obtained from the intersection of the transition curve and the capacity spectrum. Pseudo-dynamic test results were compared with the simulations. The displacement response obtained by means of the non-linear analysis agreed comparatively well with the pseudo-dynamic test results, but the equivalent linear method tended to overestimate the displacement response when the equivalent viscouse damping was kept 1%. INTRODUCTION Wood-framed structures have a mechanism that the permanent and snow loads are supported by the vertical members and that the wind and seismic loads are resisted by the shear wall system. Therefore, the structural behavior of wood-framed construction against horizontal loads can be estimated simply by knowing those of shear walls. Structural performance of wood-framed shear walls are mainly determined by the mechanical properties of sheathing materials, joints connecting sheathing materials to wooden frames and those connecting shear walls to the foundation and other horizontal members. It is generally dominated by the elasto-plastic behavior of nailed joints connecting sheathings to wooden frames if the joints connecting shear walls to the foundation and horizontal members have sufficient stiffness and strength. Design of woodframed shear walls is based on non-damage requirements under the moderate earthquake motions and non-collapse requirements under the severe earthquake motions. Non-collapse requirements are based on the ultimate strength and displacements. Equivalent viscous damping ratio is one of the factors to determine a maximum story drift response under earthquake motions. In this study, the methods to estimate the static and dynamic performance of wood-framed shear walls were investigated by comparing the simulation by non-linear and equivalent linear earthquake response analysis with the experimental results of pseudo-dynamic tests. SPECIMENS Specimens had wooden frames of 1.82m or 2.73m length and 2.44m height sheathed with 9.mm thick spruce plywood on one side as shown in Fig.1. Sheathing materials were connected to frames consisting of nominal two-by-four lumbers of S- P-F Standard with IS A8 CN nails (.8mm length and 2.87mm diameter). Studs were spaced 4mm and connected to bottom and double top plates with CN9 nails (88.9mm length and 4.11mm diameter). The outline of specimens is shown in Table1. Specimens N1, N2, N12 and B12 were shear walls of 1.82m in length, and two plywood sheets of 182mm by 122mm were sheathed horizontally. No blocking was applied to the joints between upper and lower plywood sheets except for the specimen B12. Nail spacing was 1mm in N1 and 2mm in N2. For the specimens N12 and NB12, nails were spaced 1mm in the perimeters of a sheet material and 2mm on the central support. Two hold-down bolts were applied to both ends of walls connected to the studs with three bolts of 12mm diameter. In the specimen V12, two plywood sheets of 91mm by 244mm were sheathed vertically on the frame. Specimens W2, W12, WN12 had an opening of 91mm in width and 1mm in height at the center of the wall, and specimens D12 and DN12 had an opening of 91mm in width and 18mm in height. Specimens W2, W12 and D12 had hold-down bolts at the both ends of shear walls connecting studs to steel foundation with four lag-screws of 12mm diameter. In the specimens WN12 and DN12, hold-down bolts were not attached to the studs besides the opening. 1 Assoc.Professor, Dept of Forest Resources Science, ShizuokaUniversity, Ohya, Shizuoka 422-829, apan

214 42 12 188 227. 4 4 4 227. 214 42 12 188 227. 4 4 4 227. 3376 286 227. 4 227. 91 227. 4 227. 9 24 9 38 2336 38 38 4 4 9 24 9 38 2336 38 38 4 4 24 9 38 3838 2336 CN at 1/2 9 6 1 814 1 4 4 4 4 1 212 1 4 4 4 4 1 212 Fig.1 Shematic diagram of specimens 4 4 4 4 4 4 33 TEST METODS The bottom plates of wall panels were connected to 89mm by 89mm sill and steel foundation with four bolts of 16mm diameter. Double top plates were connected to 89mm by 89mm girder with four bolts. The monotonic and reversed loads were applied at the end of the girder by the computer-controlled actuator, and horizontal and vertical displacements of wall were measured by electronic transducers. First of all one or two specimens were subjected to the monotonic loading, and then the reversed loading based on the loading protocol as shown in Fig.2 was conducted (1997). Specimen Length of wall (mm) Openings Nail spacing (mm) old down N1 182 N.A. 1 2-3M12 N2 182 N.A. 2 2-3M12 N12 182 N.A. 1/2 2-3M12 B12 182 N.A. 1/2 2-3M12 Sheathing application orizontal without blocking orizontal without blockingl orizontal without blockingl orizontal with blocking V12 182 N.A. 1/2 2-3M12 Vertical W2 273 Window 2 4M12 Vertical S2 273 Slit 2 4M12 Vertical Test method monotonic, monotonic, monotonic, monotonic, monotonic, pseudo-dynamic monotonic, pseudo-dynamic monotonic, pseudo-dynamic W12 273 Window 1/2 4M12 Vertical monotonic WN12 273 Window 1/2 4M12 No.D. at the opening Vertical monotonic D12 273 Door 1/2 4M12 Vertical monotonic DN12 273 Door 1/2 Table 1 Outline of specimens 4M12 No.D. at the opening Vertical monotonic

12Dy Pmax.9Pmax Pmax Pu.8Pmax.7Dy.2Dy.Dy Dy 2Dy 4Dy 6Dy Py.4Pmax Py 8Dy.1Pmax Dy Fig.2 Loading protocol Fig.3 Definition of yield load (Py) Fig.4 Definition of ultimate load (Pu) Du Pseudo-dynamic tests were conducted on the specimens V12, W2 and S2. The earthquake accelerograms based on the records of N-S components of 199 MA Kobe and 194 El Centro were excited with the specimen V12. The accelerograms were linearly scaled to have the maximum acceleration of.4g. For the specimens W2 and S2, the input earthquake accelerograms were based on the records of N-S components of 199 MA Kobe and 194 El Centro, respectively. MA Kobe NS and El Centro NS accelerograms were linearly scaled to have the maximum acceleration of.612g and.432g, respectively. The damping was kept 2% for all the specimens. EXPERIMENTAL RESULTS Table 2 shows the yield load, yield displacement, maximum load, maximum displacement, ultimate load and ultimate displacement obtained from the experiments. The yield load was determined by the intersection of two lines on the loaddisplacement relationships. The first line was determined by the drawn through the points on the load-slip curve corresponding to.1 Pmax and.4 Pmax. The other was determined by the drawn as a parallel line having the inclination through the points on the load-slip curve corresponding to.4 Pmax and.9 Pmax as shown in Fig.3. Ultimate load and displacement were determined by the bi-linear approximation of the load-displacement relationships as shown in Fig.4. Ultimate displacement was that corresponding to 8% of the maximum load after the peak. The ultimate load was obtained so that it had the same energy dissipation as that of the original curves up to the ultimate displacement (Yasumura et al, 1997,1998). INFLUENCE OF BLOCKING Table 2 shows that the yield and ultimate loads of the specimen B12 were about twice as high as those of N12. This indicates that the blocking plays an important roll to transmit the shear forces between horizontally applied plywood sheets. The yield and ultimate loads of the specimen N1 were approximately 1. times higher than those of N2. Thus, the shear strength of shear walls with horizontal sheet application without blocking is not proportional to the nail spacing, and shows more complicated mechanical properties than those with blocking. Fig. demonstrates the deformation of shear walls with and without blocking simulated by the finite element method when the story drift is 6mm. Significant bending of studs can be observed in the shear wall without blocking, while those remains almost straight in the shear wall with blocking. Fig. 6 compares the experimental load-deformation curves with those obtained from the simulation by the finite element method. In the analysis, the nails connecting plywood to frames were modeled as non-linear spring from the single shear test of nail joints, and the rotation of wall was submitted from the story drift. These figures show that the simulated loaddeformation relationships agreed quite well with the experimental results, and the non-linear finite element method is an appropriate tool to analyze horizontally sheathed shear walls with and without blocking. SIMPLIFIED METOD Shear strength of shear walls (Pw) with vertical sheathings or horizontal sheathings with blocking can be estimated from the shear strength of nailed joints (Pn) by using the simplified method expressed by the following equation(yasumura et al., 1986, 1992). A P w = 1+ ( Ah Ba) P 2 n ; A = n 2 + 3mn + 2 ; B = m 2 + 3mn + 2 3n 3m [1]

Where, a and h are the length and the height of a shear wall consisting of a single sheet of sheathing, and m and n are number of nail spacing along the length and height of a sheathing. Shear strength of shear walls with an opening can be obtained by the following equation. 1 α P = [2] 1 α + αβ P where, P is the shear strength of wall panel with openings, P is the shear strength of wall panel without openings, andα and β are ratios of the length and height of an opening to those of the wall panel. orizontal displacement (Dw) of wood-framed shear walls can be calculated as the sum of the displacement due to the shear deformation of sheathing material (Ds) and that due to the slips of the nailed joints connecting sheathing materials to wooden frames (Dn). Specimen N1 N2 N12 B12 V12 W2 S2 Loading method Yield load (kn) Yield displacement (mm) Maximum load (kn) Maximum displacement (mm) Ultimate load (kn) Ultimate displacement (mm) Monotonic 9.26 13.61 16.7 63.73 1.33 11.8 7.78 12.91 14.6 3.36 13.7 74.21 Monotonic.872 11.64 11.18 6. 1.17 96.2.366 1.99 1.12 1.29 8.891 73.73 Monotonic 6.422 1.6 12.4.8 11.47 81.8 7.3 9.36 12.99 4.37 11.67 67.32 Monotonic 13.28 1.77 26.71 9.97 24.1 124. 13.83 14.69 2.6 8.37 23.12 86.98 Monotonic 1.98 13.9 21.64 88.11 19.82 138.2 11.37 14.48 21.6 82.82 19.12 114. Monotonic 1.41 1.78 19.9 67.3 18.33 12.6 9.38 9.273 17.11 4. 1.74 19.8 Monotonic 6.742 12.63 12.22 71.31 11.33 18.8 Table 2 Experimental results of static tests 6.64 11.27 11.19 62.3 1.46 12.9 W12 Monotonic 17.18 14.3 29.82 64. 27.76 7.87 WN12 Monotonic 13.86 11.87 26.3 48.79 24.2 117.7 D12 Monotonic 13.82 12.96 26.3 9.36 23.92 118.6 DN12 Monotonic 1.23 11.77 19.76.1 17.89 7.4

2 1 Load(kN) 1 N1 N1-1 Calculation N1-2.1.2.3.4..6 True shear deformation(rad) Load(kN) Fig. Deformation of shear walls without blocking (left) and with blocking (right) 2 1 1 N2 N2-1 Calculation N2-2.1.2.3.4..6 True shear deformation(rad) Where, G is the shear modulus of rigidity, t is the thickness of sheathing material and d is the displacement of the nailed joints at a corner of the sheathing. Fig.8 shows the comparison between the calculated yield and ultimate load and displacement and the experimental results. These figures show that the simplified method predicts well the strength of shear walls without openings, but tends to underestimate the strength of those with an opening. It also tends to underestimate the yield displacement because it does not includes of the slips between studs and the rotation of wall. Special consideration should be taken when applying this method to calculate the horizontal displacement of structures. EFFECT OF OLD-DOWN Table 2 shows that the ultimate and yield load of WN12 and DN12 were 13-2% smaller than those of W12 and D12. This indicates there is a considerable decrease of strength by omitting hold-down bolts at the end of opening especially in the case of those with door opening, and some reduction of the shear strength is necessary to omit the hold-down connecting studs at the end of opening. DYNAMIC ANALYSIS ysteresis model A hysteresis model as shown in Fig.7 was assumed for the dynamic analysis of wood framed structures. This model includes; 1) loading on the primary curves up to the maximum load 2) loading on the primary curves over the maximum load Load(kN) 2 1 1 N12 N12-1 Calculation N12-2.1.2.3.4..6 True shear deformation(rad) Fig.6 Comparison of test results with the simulation by FEM ( ) D w = D n + D s ; D n = 2 1 + Ah2 Ba 2 ( ) 2 d ; D s = P w h 1+ Ah Ba G t a Load(kN) 3 2 2 1 1 B12-1 Calculation B12-2.1.2.3.4..6 True shear deformation(rad) Fig.7 ysteresis model B12 [3]

Experiment (kn) 2 18 16 14 12 1 8 6 4 B V12 B12 W S D B B Experiment (mm) 2 18 16 14 12 1 8 6 4 B V12 B12 W S D B B 2 2 Experiment (kn) 2 4 6 8 1 12 14 16 18 2 Calculated yield strength (kn) 3 B V12 2 2 1 1 B12 W S D B B Experiment (mm) 2 4 6 8 1 12 14 16 18 2 Calculated yield displacement (mm) 2 18 B V12 16 B 12 14 12 1 8 6 4 2 W S D B B 1 1 2 2 3 Calculated ultimate strength (kn) 3) unloading from the peak on the primary curve 4) reloading with soft spring ) reloading toward the previous peak 6) unloading from the inner peak 7) reloading toward the peak without slips 2 4 6 8 1 12 14 16 18 2 Calculated ultimate displacement (mm) Fig.8 Comparison of test results with the simulation by optimized method The parameters were determined from the experimental results of the reversed loading test of shear walls. The primary curves up to the maximum load (1) and those over the maximum load (2) are expressed as follows; P = (P + c 2 x)(1 e c 1 x p ) ; P = Pmax+ c 3 (x D max ) [4] Specimen V12 W2 S2 Table 3 Parameters for hysteresis models

Displacement response (mm) Displacement response (mm) 8 6 4 2-2 -4-6 Simulated Pseudo -8 2 4 6 8 1 12 Time (sec.) 1 Simulated 8 6 Pseudo 4 2-2 -4 El Centro NS MA Kobe NS -6 2 4 6 8 1 12 Time (sec.) Load (kn) Load (kn) 2 2 1 1 - -1-1 -2 Simulated Pseudo -2-8 -6-4 -2 2 4 6 8 Displacement (mm) 2 2 1 1 - -1-1 El Centro NS MA Kobe NS Simulated -2 Pseudo -2-6 -4-2 2 4 6 8 1 Displacement (mm) Fig.9 Comparison of test results with the simulation by non-linear time-history earthquake analysis (V12) The values of Po, C1 to C7 obtained from the envelop curves of the reversed loading are shown in Table 3. The primary curves over the maximum load was obtained as the straight line determined by the drawn through the points corresponding to Pmax and.8pmax. Unloading stiffens (3) and the reloading stiffness toward the previous peak () were based on the inclination of the straight line determined by the drawn through the origin and the peak on the primary curve (k). Reloading stiffness with soft spring (4) was based on the inclination of the straight line determined by the drawn through the peak on the primary curve and the crossing point of the X-axis. k 1 [] k = c x + 1 ; k 3 4 k = c x + 1 ; k 2 = 1 c k 6 x c 7 Table 4 Comparison of pseudo-dynamic test results with simulation

Time-history earthquake response analysis Time-history earthquake response analysis was conducted on the wood framed shear walls. Single-degree-of-freedom lumped mass model was applied. The input earthquake ground motions were based on the records of N-S components of the 194 El Centro and the 199 MA Kobe. The accelerograms were linearly scaled to have the maximum acceleration of.4g. On the specimens W2 and S2, the input earthquake accelerograms based on the records of N-S components of 199 MA Kobe and 194 El Centro were excited respectively. MA Kobe and El Centro NS accelerograms were linearly scaled to have the maximum acceleration of.612g and.432g, respectively. The damping was kept 2% for all the specimens. Equivalent linear response analysis Maximum displacement response was calculated by the equivalent linear response method (Kawai,1999). Fig.9 shows how to obtain the maximum displacement response. Capacity spectrum of the structure was obtained with the load-deformation relationships of the structure. Then linear response was calculated for different stiffness with the equivalent viscous damping. Maximum displacement response was obtained as the intersection of the transition curves and the capacity spectrum. Equivalent viscous damping was assumed constant and 1% in all the range of displacement in this study. Comparison of results The maximum deisplacement response obtained by using non-linear and equivalent linear analysis is compared with the pseudo-dynamic test results in Table 4. The simulatd values by non-linear time-history earthquake analysis agreed comparatively well with experimental results in the specimen V12, but about 4% error was observed in specimens W2 and S2. The equivalent linear method tended to overestimate the displacement response when the equivalent viscouse damping was kept 1%. CONCLUSIONS 1. The shear strength of shear walls with horizontal sheet application without blocking is not proportional to the nail spacing. Blocking plays an important roll to transmit the shear forces between horizontally applied plywood sheets. The simulated load-deformation relationships agrees quite well with the experimental results, and the non-linear finite element method is an appropriate tool to analyze horizontally sheathed shear walls with and without blocking. 2.There is a considerable decrease of strength by omitting hold-down bolts at the end of opening especially in the case of those with door opening. Some reduction of the shear strength is necessary to omit the hold-down connecting studs at the end of opening. 3. The simplified method predicts well the strength of shear walls without openings, but tends to underestimate the strength of those with an opening. It also tends to underestimate the yield displacement because it does not include of the slips between studs and the rotation of wall. Special consideration should be taken when applying this method to calculate the horizontal displacement of structures. 4. The simulated maximum displacement response by non-linear time-history earthquake analysis agrees comparatively well with experimental results in Specimen V12, but about 4% error is observed in specimens W2 and S2. The equivalent linear method tends to overestimate the displacement response when the equivalent viscouse damping is kept 1%. ACKNOWLEDGEMENTS The auther thaks Daudeville, L and Davenne, L for supplying FEM codes, and Suzuki, M and Uchida, K for their assistance of testing and analysis. REFERENCES 1. Yasumura, M. 1986. Racking Resistance of wooden Frame Walls with Various Openings, CIB-W18, 19th Meeting, paper 19-1-3. 2. Yasumura, M. and Murota, T. 1992. Design Procedures for Wood-framed Shear Walls, Proceedings of The IUFROS.2. 3. Yasumura, M. and Kawai, N. 1997. Evaluation of Wood Framed Shear Walls subjected to Lateral Load, CIB-W18, 3th meeting, paper 3-1-4 4. pren 1212. 1997. Timber Structures - Test Methods - Cyclic Testing of oints Made with Mechanical Fasteners.. Yasumura, M., Kawai N.1998. Estimating Seismic Performnce of Wood-framed Structures, proceedings of th WCTE, Vol.2, 64-71 6. Kawai, N.1999. Prediction Methods for Earthquake Response of Shear Walls, Proceedings of PTEC.