Predicting racking performance of walls sheathed on both sides

Transcription

1 Predicting racking performance of walls sheathed on both sides Marcia Patton-Mallory William J. McCutcheon Abstract This paper extends a previously developed wall racking model to the case of walls sheathed on both sides with dissimilar materials. (The previous model considered sheathing on one side only.) Four types of curves representing fastener load-slip data predict wall load-displacement behavior. Comparing theoretical computations to data from small wall tests, we find that asymptotic fastener curves give the best predictions of shear wall performance. The results of this study should be of interest to researchers in light-frame wood construction and to building code authorities. Wood-frame buildings consist of a number of structural elements, including floors, roofs, and walls, which interact to resist the variety of forces the buildings must withstand. Walls parallel to the forces, shear walls, provide wood-frame buildings with their resistance to earthquake, wind, and other lateral loads. Walls resist shear loads primarily through the phenomenon of racking, wherein the wall panel distorts from its rectangular shape into that of a parallelogram as the sheathing-to-frame connectors absorb the external energy applied to the wall. Both ultimate strength and the load-displacement relationship are important considerations in determining a wall s racking acceptability. This paper extends a previous model for shear wall performance that considered the case of sheathing on only one side of the wall, to the case where a wall is sheathed on two sides with dissimilar sheathing materials. We will show that lateral nail tests can accurately predict the racking performance of sheathed walls. The results of this study should be of interest to researchers in light-frame construction and to building code authorities. Background Typical shear walls in light-frame structures consist of sheathing panels attached to wood studs (usually 2 by 4) by means of mechanical fasteners such as nails or screws. The racking performance of a shear wall depends primarily upon the relative movement between the sheathing and the studs. This movement, in turn, depends upon the load-slip characteristics of the fasteners. Earliest wall racking theories (12,13,17) empirically related wall racking strength to the lateral strength of the individual fasteners. The theories were limited to specific wall geometries and combinations of materials. Recent research has sought general analytical models to predict racking performance based upon wall geometry, fastener placement, and properties of the fasteners and other materials (3,5-10,15,16). The relationship between lateral load and slip is highly nonlinear for typical mechanical fasteners. Because wall performance depends primarily upon fastener performance, wall racking load-displacement behavior is also nonlinear. McCutcheon (10) presented a general model for predicting nonlinear wall behavior as a function of nonlinear fastener performance. The model can accommodate sheathing on one side of the wall and a variety of definitions for the nail load-slip relationship. Patton-Mallory et al. (14) investigated the racking performance of small walls sheathed on both sides. They showed experimentally that the stiffness and strength of such walls, with similar or dissimilar materials on the two sides, can be determined by summing the performances of single-sided walls sheathed with corresponding materials. This paper extends the aforementioned model (10) to account for dissimilar sheathings on the two faces of a wall, and uses the experimental data from Patton- Mallory et al. (14) to verify the method. The authors are, respectively, General Engineer and Research General Engineer, USDA Forest Serv., Forest Prod. Lab., One Gifford Pinchot Dr., Madison, WI This paper was received for publication in June Forest products Research Society Forest products J. 37(9): FOREST PRODUCTS JOURNAL Vol. 31, No. 9 27

2 Theoretical analysis McCutcheon (10) presented a general model for predicting the racking load-displacement curve of a wall sheathed on one side. The model accounts for nonlinear load-slip in the fasteners and for shear deformation in the sheathing. The model is summarized as follows. The first step in calculating the racking response of a wall is to express the slip of each fastener, 6, in terms of the wall s horizontal racking displacement due to fastener slip, D f : δ = KD f [1] Figure 1. Panel geometry used to describe fastener placement. K is a strictly geometric constant whose value for each fastener depends upon the location of that fastener on the panel. The subscript f indicates that we are initially considering only displacement of the wall due to fastener slip. Next, the external energy of the racking load R, acting through the wall s racking displacement due to fastener slip, f, is equated to the internal energy absorbed by all the fasteners. This yields a relationship between the wall racking load (R) and the wall racking displacement due to fastener slip: R = f(κ, f ) fasteners The exact form of Equation [2] depends upon the fastener load-slip relationship and the values of the geometric constants K. In addition to the displacement due to fastener slip there is wall displacement due to shear deformation in the sheathing: s = wall displacement due to shear deformation in the sheathing R = wall racking load H = vertical dimension of each piece of sheathing N = number of individual pieces of sheathing L = horizontal dimension of each piece of sheathing G = shear modulus of the sheathing t = thickness of the sheathing [2] [3] In general, points on the wall racking loaddisplacement curve can be computed by specifying a wall displacement due to fastener slip, D f, computing the wall racking load R from Equation [2], calculating the additional wall displacement due to sheathing shear, A, from Equation [3], and adding D f and D s, to give the total wall displacement, D t : t = f + s [4] The preceding procedure will give values of total wall racking displacement, D t, and racking load, R, for specified values of displacement due to fastener slip, f, If one wishes to compute the racking load at specified values of total displacement, t, an iterative technique can be used: 1. Estimate the value of f ; 2. Compute the racking load R from Equation [2]; 3. Calculate s from Equation [3]; 4. Calculate the total wall displacement from Equation [4]; 5. If t is not the desired value, multiply f by the ratio of the desired value of t to the one computed in step 4 and return to step 2. In the analyses carried out in this study, we considered the procedure to have converged when the values of t from one cycle to the next differed by 0.5 percent or less. For a wall sheathed on both sides, Equations [2] to [4] are valid for each side of the wall, i.e. add the subscript 1 or 2 to all the variables in those three equations to describe the performance of each side. For two-sided walls, we assume there is no twisting of the lumber frame and, therefore, both sides of the wall go through the same total racking displacement. Experimental observations support this assumption. There- 28 SEPTEMBER 1987

3 [11] [12] The hyperbolic tangent forms are: [13] [14] Thus, the load-slip behavior of an individual fastener (Eqs. [7],[9],[11], or [13]) is used to predict the racking performance of a shear wall (Eqs. [8],[10],[12], or [14]). Figure 2. Small-scale wall racking test. Test panel is 22 inches high by 6 feet long. Racking load is applied at the lower right corner of frame. fore, two additional equations apply for walls sheathed on both sides: [5] [6] R = total wall racking load R 1 and R 2 = portions of the load resisted by each side of the wall D t = total racking displacement D t1 and D t2 = total displacements computed for each side of the wall To compute the response of a two-sided wall with dissimilar sheathing materials and/or fastener geometries, one must carry out the iterative procedure for each side. Specifically, one must 1) specify the total wall racking displacement, D t ; 2) use the iterative procedure to compute R 1, the resistance from side 1; 3) similarly, compute R 2, the resistance from side 2; and 4) compute the total wall racking force, R, as the sum of R 1 and R 2 (Eq. [5]). In the following analyses, we will consider four different equations in defining the fastener load-slip relationship: a simple power curve, a logarithmic form, an asymptotic form, and a hyperbolic tangent. A power curve is of the form: p = Ad B [7] p = lateral load on the fastener d = slip of the fastener A, B = constants This relationship gives for the wall racking equation (Eq. [2]): R = AD B f å K B+1 [8] The logarithmic equation for fastener slip and the corresponding racking equation are: p = A ln(1 + Bd) [9] R = A å K ln(1 + BKD f ) [10] The asymptotic equation and the resultant racking equation are: Experimental methods The data evaluated in this study are from a set of 200 small walls (14), which were 22 inches high and of four different lengths: 2, 4, 6, and 8 feet (Fig. 2). Wall configurations were: plywood on one side, plywood on both sides, gypsum wallboard on one side, gypsum wallboard on both sides, and mixed walls with plywood on one side and gypsum wallboard on the other. There were 10 replications of each wall length and configuration. In this study, we used the data from the 2- and 6-foot walls with plywood on one side, gypsum wallboard on one side, and plywood on one side and gypsum wallboard on the other. The wall frames were constructed from nominal 2 by 4 Douglas-fir lumber with single top and bottom plates and studs at 12 inches. The plywood was 1/2-inch C-D exterior, species group 1 (Douglas-fir face plies) 22 inches high by 12 inches wide, fastened with 8d common nails at approximately 5-1/2 inches. The gypsum wallboard was 1/2-inch drywall, 22 inches high by the full length of the wall, fastened with 1-1/4-inch drywall screws, also spaced at about 5-1/2 inches. The wall tests were accompanied by lateral load tests on both types of fasteners according to ASTM D 1761 (2), using lumber and sheathing from the wall tests. Past experience (10) indicates that lateral load tests on tightly nailed plywood specimens may predict unreasonably high initial stiffnesses in shear walls. Therefore, we also conducted identical nail tests with teflon in the slip plane between the plywood and framing material. Shear moduli for each sheathing type were obtained from the literature as G = 105,000 pounds per square inch (psi) for gypsum wallboard (4) and G = 100,000 psi for plywood, rounded up from 90,000 psi (1). Results and discussion We used experimental data from the study of small shear walls (14) to verify our theoretical approach. Specifically, we fit curves of the four types previously discussed (Eqs. [7],[9],[11], and [13]) to experimental fastener load-slip data, and then computed theoretical racking responses for 2- and 6-foot-long walls, 22 inches high, sheathed with plywood on one side only, gypsum wallboard on one side only, and plywood on one side and gypsum wallboard on the other. The one-sided walls provided a measure of the theory s ability to predict wall FOREST PRODUCTS JOURNAL Vol. 37, No. 9 29

4 behavior from nail load-slip data (Eqs. [7] to [14]); the two-sided walls assessed the theory s ability to account for different sheathings on the two sides. Single fastener behavior Initially, we evaluated single nail behavior according to ASTM D 1761 (2), with tight connections tested immediately after fabrication. Standard fastener tests do not necessarily represent load-slip behavior of fasteners in a structure. As expected, results from these tests overpredicted the initial stiffnesses of the walls with nailed plywood sheathing. Past research (10) also showed that lateral nail tests on tightly nailed plywood specimens can lead to over-predictions of initial shear wall stiffness. Therefore, we conducted an additional set of nail tests that were identical to the first, except for a layer of teflon in the slip plane between the framing material and plywood. This second set of nail tests gave much better predictions of wall behavior and are the tests reported here (Fig. 3). Gypsum wallboard has a paper facing that reduces friction between the sheathing and framing. Therefore, it was not necessary to modify the lateral load tests of the drywall screws. The data from Patton-Mallory et al. (14) were used without modification to determine the performance of the screws (Fig. 4). We tit power curves (Eq. [7]), logarithmic curves (Eq. [9]), asymptotic curves (Eq. [11], and hyperbolic tangents (Eq. [13]) to the nail and screw data by selecting values for constants A and B that define curves that closely follow the average fastener data (Figs. 3 and 4). The ranges of fastener slip in Figures 2 and 3 correspond to the ranges of wall racking displacements in Figures 5 to 10. Wall behavior The fastener equations were then used to predict the performance of 2- and 6-foot walls sheathed with plywood only, with gypsum wallboard only, and with Figure 4. Test data and curves for 1-1/4-inch drywall screws attaching 1/2-inch gypsum wallboard to lumber. Figure 3. Test data (high, average, and low) and power, asymptotic, logarithmic, and hyperbolic tangent curves for 8d nails attaching l/pinch plywood to lumber. Figure 5. Test data range (high and low) and theoretical curves (corresponding to power, logarithmic, asymptotic, and hyperbolic tangent fastener curves) for P-foot-long walls with plywood on one side. 30 SEPTEMBER 1987

5 Figure 6. Test data range and theoretical curves for 6-footlong wails with plywood on one side. Figure 6. Test data range and theoretical curves for 6-footlong walls with gypsum wallboard on one side. Figure 7. Test data range and theoretical curves for 2-footlong walls with gypsum wallboard on one side. both plywood and gypsum wallboard. In general, we obtained very good estimates of wall racking behavior, as indicated by plots of theoretical curves and experimental data (Figs. 5 to 10) and a comparison of test average load with predicted load for the four types of curves (Table 1). The wall racking equations predict wall displacement to be the sum of displacement due to fastener deformation and displacement due to shear deformation in the sheathing. Shear deformation in the sheathing contributed less than 5 percent of the total horizontal wall displacement at the maximum wall displacements indicated in Figures 5 to 10. Shear deformation in the sheathing contributed 10 to 15 percent of the horizontal wall displacement at to 0.10-inch wall displacement. The power curve (Eq. [7]) is mathematically the simplest representation for fastener load-slip. It was used in previous research (10) to develop the nonlinear racking theory. However, this equation cannot accurately predict nail behavior or racking performance (Eq. [8]) over the full range of loads because it predicts ever-increasing resistance with increasing displacement, whereas actual fasteners and walls tend to level off at a maximum load (Figs. 6, 8, and 10). It is adequate, though, for predicting behavior over a narrow range of loads. Figure 9. Test data range and theoretical curves for 2-footlong walls with plywood on one side and gypsum wallboard on the other. Figure 10. Test data range and theoretical curves for 6-foot-long walls with plywood on one side and gypsum wallboard on the other side. The logarithmic equation (Eq. [9]) also predicts ever-increasing resistance with increasing displacement, and is therefore subject to the same limitations as the power curve. In a few cases (Figs. 5, 6, and 9), the logarithmic form was more accurate than the power curve. FOREST PRODUCTS JOURNAL Vol. 37, No. 9 31

6 TABLE 1. Average wall racking test loads and predicted loads at various wall deformations. Predicted Wall type Size Wall Test Hyberbolic deformation average Power Logarithmic Asymptotic tangent (in.) (lb.) (lb. (% error)) Plywood one side 2 feet (+24) 590 (+24) 445 (-6) 235 (-51) (-2) 1,010 (+5) 1,020 (+6) 855 (-10).50 1,270 1,250 (-2) 1,290 (+2) 1,345 (+6) 1,450 (+14) Gypsum one side 2 feet (0) 310 (-2) 280 (-11) 230 (-27) (-5) 520 (+2) 500 (-2) 485 (-5) (+3) 625 (+8) 575 (-1) 510 (-12) Plywood-gypsum 2 feet (+22) 900 (+22) 725 (-2) 465 (-37).20 1,475 1,425 (-3) 1,525 (+3) 1,520 (+3) 1,340 (-9).40 1,795 1,765 (-2) 1,845 (+3) 1,855 (+3) 1,845(+3) Plywood one side 6 feet.05 1,670 1,770 (+6) 1,770 (+6) 1,330 (-20) 710 (-57).20 2,910 2,815 (-3) 3,030 (+4) 3,060 (+5) 2,565 (-12).50 3,640 3,745 (+3) 3,875 (+6) 4,040 (+11) 4,345 (+19) Gypsum one side 6 feet ,072 (+23) 1,090 (+25) 1,020 (+17) 890 (+2).20 1,575 1,665 (+6) 1,755 (+11) 1,670 (+6) 1,560 (-1).40 1,885 2,060 (+9) 2,095 (+11) 1,860 (-1) 1,600 (-15) Plywood-gypsum 6 feet.05 2,575 2,845 (+10) 2,860 (+11) 2,350 (-9) 1,600 (-38).20 4,625 4,475 (-3) 4,785 (+3) 4,730 (+2) 4,125 (-11).40 5,455 5,555 (+2) 5,765 (+6) 5,700 (+4) 5,665 (+3) The asymptotic equation does an excellent job of describing fastener (Eq. [11]) and wall (Eq. [12]) behavior (Figs. 5 to 10). This form has an advantage over the others in that the constants A and B have clearer physical meanings than they have in the other forms. A is the maximum (ultimate) load on the fastener and B is the slip at half the maximum load (A/2). In the wall (Eq. [12]), A K is the ultimate racking load. The hyperbolic tangent (Eqs. [13] and [14]) does not fit the fastener or wall data as well as the other forms. It underestimates loads at low slips and displacements and/or overestimates loads at high slips and displacements. A comparison of predicted racking loads versus average test loads (Table 1) shows that the asymptotic equation for fastener performance yields the best predictions of wall behavior. All curve types show larger errors in predicted versus test average at small wall displacements. Also, there is more variability in experimental data at small wall displacements than at larger wall displacements. Conclusion We have shown that it is possible to accurately predict the nonlinear racking load-displacement behavior of a shear wall from the known behavior of the fasteners and the sheathings. The method gives accurate predictions for small test walls sheathed on one or both sides, including walls with dissimilar sheathings on the two sides. The next logical step in this research is to verify the model with full size wall tests. Four different equations were used to represent the nonlinear behavior of the fasteners. Of these, the asymptotic form provided the most accurate predictions of wall behavior over the full range of the data. The power and logarithmic curves can fit the data only over a limited range of fastener slips and wall displacements. The power curve also has the advantage of mathematical simplicity. The hyperbolic tangent does not fit well and probably should not be considered in further research into fastener or shear wall performance. Literature cited 32 SEPTEMBER 1987