STRENGTH AND STIFFNESS REDUCTION OF LARGE NOTCHED BEAMS

Transcription

1 STRENGTH AND STIFFNESS REDUCTION OF LARGE NOTCHED BEAMS By Joseph F. Murphy 1 ABSTRACT: Four large glulam beams with notches on the tension side were tested for strength and stiffness. Using either bending net section beam theory or shear formula to calculate crack propagation critical load is very unconservative. A linear elastic fracture mechanics approach, taking into account the high tension stresses perpendicular to grain and shear stresses at the notch reentrant comer, conservatively predicts the critical load. The data corroborate the substantial analytic effect of size predicted by fracture mechanics for notched beams. Results quantify the observed behavior of bending of beams with notches on the tension side. The strength reduction is so severe for large beams that substituting a beam having the net depth of the notched beam is preferable. Removing material would remove the stress concentrator and would increase the strength up to net section theory prediction at the notch location. Using an effective notch length (actual notch length + notch depth added to each end of the notch) and variable moment of inertia, beam theory accurately predicts the notch beam deflection under load. OBJECTIVE AND BACKGROUND In current practice, notches are cut into wood beams to allow for clearance and to adjust the top surfaces to desired levels. Bending stress values calculated at the notch by dividing the applied bending moment by the net section modulus are not accurate in predicting beam strength because the notch reentrant corners cause additional shear stresses and tension stresses perpendicular to the grain. This also applies to using net section shear stress in predicting beam strength. These stresses cause crack propagation at loads lower than the breaking load determined by net section (6). This paper compares fracture mechanics predicted stength, bending net section predicted strength, and a modified shear net section predicted strength to experimental crack initiation/propagation (failure) load for large glued-laminated wood beams. This comparison was used to find the most accurate method to quantify and predict notched beam strength and to check the rational recommended practice avoid notches in beams (12); notching on the tension side of single beam in the center of the span is not recommended (11); and notching of bending members should be avoided, especially on the tension side of the lumber (7). A modified net section approach to predict beam stiffness is shown to give good engineering answers. Murphy (6) analyzed crack propagation loads for notched beams (notch on tension side) using linear elastic fracture mechanics theory on small wood beams. However, he used a (conservative) linear combined mode 1Research Engr., Forest Products Lab., Forest Service, U.S. Dept. of Agriculture, Madison, WI Note.-Discussionopen until February 1, To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on October 12, This paper is part of the Journal of Structural Engineering, Vol. 112, No. 9, September,

2 failure criteria, Mode I and Mode II. Mall et al. (5) studied the combined mode failure equation for wood and corroborated Wu's (13) experimental and Hahn's (2) theoretical crack propagation equation. This localized failure equation accounts for the shear stresses and stresses perpendicular to grain at the notch corner. In linear elastic fracture mechanics theory, the strength of material with a sharp crack is proportional to the square root of the crack length. If one normalizes the geometry by the beam depth and compares the strength of two beams which are similar in all dimensionless length ratios, then their failure stresses are inversely proportional to the square root of their respective depths. Fracture mechanics analysis therefore predicts a substantial analytic effect of size for a sharp slit-notch beam. This can be contrasted with the smaller statistical size effect that Bohannan (1) found for clear wood beams Whereas Bohannan found a "weakest link" statistical size effect for large clear unnotched beams (in which failure is equally likely to occur in the regions of maximum stresses), fracture mechanics theory predicts analytically a substantial effect due to size for large beams (in which failure is likely to occur only at a sharp crack tip or corner). Bohannan's size effect is statistically based on clear unnotched beams and the fracture mechanics effect of size is theoretically or analytically founded for notched beams. Hirai and Sawada (3) experimentally determined the maximum failure moments of square-notched wood beams. From small beams (the largest beam tested was 1-1/2 by 4 by 60 in.) they derived an empirical equation to predict crack propagation load of notched large beams. ANALYSIS PROCEDURE There are presently three separate checks to calculate the critical load of a notched beam: (1) Bending strength at the net section; (2) shear strength at the net section; and (3) fracture propagation at the net section. Beam Geometry. -The beam geometry used in the experimental test procedure is given in Figs. 1(a) and (c), while the assumed beam geometry used in the analysis [Figs. 1(b) and (d)] is slightly different for the rectangular notched beam. The slit or notch depth, a, is taken as 0.3 times the overall depth, d, in both test and analysis. For the slit beam test and analysis, the slit is located 2.5d from the left end of the beam Figs. 1(a) and (b). The notch length in the notch beam test is located starting at 2.5d and ending at 5.5d for a total notch length of 3.0d [Fig. 1(c)]. In the notch beam strength analysis, however, a slit is assumed located at 5.5d [Fig. 1(d )]. The beams are simply supported at each end and are loaded with a concentrated load two-thirds of the span from the left end of the beam. Two beam depths, 12 and 18 in., and a beam width equal to in. are used. Note that relative notch size and location are both restricted in design manuals (7, 11) and in this study the notch size and location are outside the bounds of these restrictions. Net Section Bending Strength.-The equation for net section bending strength for the configuration shown in Fig. 1 is: 1990

3 FIG. 1. -Test Geometry in Units of Beam Depth; Width = b = in.; Depth = d = 12 or 18 in., d n = d - a (1) in which MOR = modulus of rupture; M = bending moment at distance x from left end of beam; b = beam width (3.125 in.); d = beam depth (12 or 18 in.); d n = notched beam net depth (0.7d ); x = notch end closest to load, measured from left end of beam (2.5d or 5.5d); R = left end reaction (= P/3), in which P = applied load; and a = notch depth (0.3d). This gives (2) 1991

4 with a/d = 0.3; x/d = 2.5 or 5.5; and b = in.; yields P = (MOR )d for the slit beams, and (MOR )d for the notched beams. For this study MOR was calculated using the ASTM D a (9) clear wood design stress in bending (for dense Douglas fir) published value of 3,500 psi divided by 1/2.1 [duration of load and manufacture and use factors (4)] and multiplied by 0.85 [for scarf joint efficiency (12)] to get a 5th percentile short-term ultimate strength. MOR = 6,250 psi; P = 1,910 d (slit); and P = 870 d (notch) in which P is in pounds and d is in inches. Net Section Shear Strength. -The modified equation (7,11) for net section shear strength of a notched beam for the Fig. 1 configuration is: in which V = beam shear = P/3; and F v = allowable shear stress, giving, for this concentrated load case (i.e., independence from notch location x), and with a/d = 0.3 and b = in. yields P = (F v )d. For this study F v was calculated using the ASTM D a (10) published value 165 psi divided by 1/4.1 (4) to get a 5th percentile short-term ultimate strength of F v = 675 psi and P = 2,070 d. Fracture Mechanics Strength.-Similar to stress concentration factors describing maximum stresses around holes, stress intensity factors describe the stress field around the tips of cracks. Theoretically stresses approach infinity as the reciprocal of the square root of the distance r from the crack tip (i.e., in Fig. 2). The stress intensity factors, K,, describe the stress conditions in a small region around the (3) (4) FIG Coordinate System around Crack Tip (Two-Dimensional Geometry for K 1, K II) 1992

5 crack tip, and are useful for describing crack extension. These factors are linearly dependent on boundary conditions (e.g., load and displacement), are highly dependent on geometry, and are slightly dependent on orthotropic parameters (for finite bodies). When the stress intensity factors reach critical values (a material property), crack propagation occurs. The stress intensity factors as a function of notch depth can be found in Murphy (6) and the combined mode fracture initiation equation in Mall et al. (5). From Murphy (6) for a/d = 0.3 one should use the following to determine the effective stress intensity factors K I, K II : (5) (7) (8) in which K I, K II = Mode I, II stress intensity factors; and K I, K II = effective stress intensity factors on the imminent fracture plane (6). To better understand the role of effective stress intensity factors, refer to an x, y coordinate system at the crack tip (Fig. 2): (6) (9) (10) Eq. 9 describes the theoretical stresses on a plane coplanar to the slit and Eq. 10 describes the theoretical stresses on the imminent fracture plane, perpendicular to the coplanar plane. From Mall et al. (5) the combined mode failure criterion is in which subscript c refers to critical factors that are material properties. Substituting Eqs. 5-8 into Eq. 11 yields: (11) (12) K IIc is taken as 1,600 psi (from J. F. Murphy, Mode II Wood Test Specimen-Beamwith Center Slit, submitted to J. of Testing and Eval 1993

6 uation for publication) and from Petterson and Bodig (8) K Ic is calculated from their generalized equation (13) in which for a specific gravity (S g ) of 0.54 and moisture content of 12% (i.e., M = 12) (14) Substituting these values, and b = in. and x/d = 2.5, 5.5 into Eq. 12, yields P = 1,050 d 1/2 (slit) and P = 550 d 1/2 (notch) Hirai and Sawada (3) derived the following equation for square-notch softwood beams from small beams (< 1-1/2 by 4 by 60 in.). For the present study's beam geometry, Hirai and Sawada's formula with eight empirical constants reduces to (15) in which F v = average shear value. For a 15% coefficient of variation (CV) (10), (16) in which P is in pounds and d is in inches. Beam Stiffness. -Beam stiffness was calculated theoretically assuming the moment of inertia was reduced to the net section along the effective notch length. In the analysis the effective notch length is equal to the actual length of the notch plus the depth of the notch added onto each end (12). Using this procedure, for the slit beams the effective notch length starts at 2.2d and goes to 2.8d for an effective length of 0.6d. For the notched beam tests the effective notch length is 3.6d starting at 2.2d and going to 5.8d. EXPERIMENTAL PROCEDURE Four large glulam beams were provided by the American Institute of Timber Construction. The laminations for these beams were hand selected so that no knots, grain deviation, or scarf joints occurred in the critical beam section, making the beams essentially clear Douglas fir. The four beams were to be tested each three times with the load and support configuration shown in Fig. 1. The three tests comprise uncut, 1994

7 slit cut, and notch cut. The uncut beam tests were used to find the beam modulus of elasticity used in stiffness comparisons. The relative dimensions and locations are also shown in Fig. 1. Two of the beams were 12 in. deep and two 18 in. All beams were in. wide with 1-1/2-in. lamination thickness. TEST PROCEDURE The test procedure consisted of loading the uncut beam to about 8,000 lbs (18-in. depth) or to 4,500 lbs (12-in. depth), recording strain gage and LVDT readings at 100-lb increments. These loads were in the range of the predicted crack propagation loads for the slit and notch beams. For the uncut beams, the extreme fiber stress produced by these loads is well below the elastic limit assuring no damage during these tests. In the second round of the tests, the beams had a saw slit made on the tension side a distance of 2.5 beam depths from the far support to FIG. 3.-Photograph of Crack initiation from Slit Notch; also Shown Are Displacement Gage Locations FIG. 4.-Photograph of Propagated 12-ft Crack 1995

8 a relative depth of The slit and notch dimensions were chosen so that after crack propagation in the slit beam test, a rectangular notch extending well past the propagated crack could be cut in the same beam. The point of crack initiation in both the slit and notch tests was sufficiently far from both load and reaction points. The notch depth was chosen so that the crack would initiate in wood, not in a bond line, and would be at least two lamination thicknesses from the beam edge. Also, a relative notch depth of 0.3 would test a large difference between net section theory predictions and fracture mechanics predictions. Load was applied and recorded like the uncut beams until a crack initiated and passed a strain/lvdt gage (Fig. 3). If the propagating crack arrested itself (as manifested by the load dropping off), the test was stopped and the beam was unloaded. (Beam 2 did not cooperate with crack arrest and the crack propagated explosively over 12 ft. See Fig. 4.) In the third round, a notch was cut with a length equal to three times the beam depth and with the closest end to the load at a distance 5.5 beam depths from the far end of the beam (Fig. 1). The beam was loaded until the crack initiated at the notch comer and propagated past the strain/ LVDT gages. Beam deflection was measured under the load point for all tests. RESULTS AND ANALYSIS The crack propagation load was determined from the strain/lvdt gage readings (e.g., Fig. 5, from the rightmost LVDT in Fig. 3). The seven critical crack propagation loads (the beam in Figs. 3 and 4 did not have a notch counterpart) are graphed in Fig. 6, along with the three predicted critical load equations for both slit and notch beams. This information is also summarized in Table 1. The straight lines in Fig. 6 are from net section theory and use fifth percentile material properties, while the curved lines use fracture mechanics theory and expected material property values. The larger net depth 0.70 x (18) = 12.6 in. is slightly above 12 in. so the size effect of 1/9 (1) would not contribute much and not at all below d = 17.1 in. (net depth = 12). Net section theory, either bending or shear, is very unconservative for estimating the critical load. Using expected material properties would make the predicted lines even more unconservative! The reader should be reminded that the critical load is defined as the load at which the crack propagates from the notch or slit. In small, clear straight-grained specimens, as well as these special glulam beams (i.e., essentially clear Douglas fir), the crack propagates leaving a beam with a net section. In production lumber and glulam beams, propagating cracks can and do run into knots, grain deviation, and other discontinuities. Where the crack runs determines whether the beam can carry the load. Therefore the design strength of a notched beam should be based on crack propagation load since there is no way of assuring that a net section will remain. Square-notch fracture mechanics theoretical predictions fall above the experimental data. Two reasons for this bias might be: (1) The theory uses six constants empirically derived from small beams (ranging from 3/4 by 2 by 30 in. to 1-1/2 by 4 by 60 in.) to account for width variation 1996

9 FIG Load/Displacement Curve from LVDT as Crack Passes; Crack Propagation Load is 5,750 Ibs FlG Crack lnitiation Load of Large Slit and Notched Beams and Theoretical Predictions, for x/d = 2.5, 5.5; S = Shear Equation; B 2, B 5 = Beam Equations; F 2, F 5 = Fracture Equations; N 2, N 5 = Square-Notch Equations besides two other empirical constants; and (2) the theory assumes the critical notch stress intensity factors are multiplicative factors times the shear block strength of the species. Sharp slit fracture mechanics theoretical predictions fall below the experimental data. Probable reasons for this are: (1) The beams have higher critical stress intensity factor values than the assumed average value; and (2) the simplifying assumption of the sharp-slit geometry yields rather than the more complex for a square notch. 1997

10 TABLE 1. -Beam Strength Comparisons TABLE 2. - Beam Stiffness Comparisons, at 2,000-lb Loads The notch beams have about one half the strength of the slit beams. Fracture mechanics theory predicts this tendency. Beam deflection calculations (at the applied load) of the uncut, slit, and notched beams are summarized in Table 2 together with experimental observations. Beam modulus of elasticity was calculated for each beam from the uncut beam tests. This beam modulus of elasticity was used in subsequent calculations for the corresponding slit and notched beam tests. Both tests and analysis show that for this geometry there is a 2% increase in deflection due to the slit and a 20% increase due to the notch. CONCLUSIONS Fracture mechanics methodology can conservatively predict the critical crack propagation load of large notch beams with notches on the tension side if the notch is modeled as a slit. Because there is no way of assuring 1998

11 that a net section will remain after a crack starts propagating, net section bending or shear theory is unconservative in estimating this load for large beams. The reduction in strength is so severe that substituting a beam having the net depth would avoid such a large strength reduction-that is, removing material would remove the strength concentrator and would increase the strength up to the net section theory prediction at the notch location. In other words, bending members should not be notched on the tension side. Beam theory with a variable moment of inertia, and an effective notch length equal to actual notch length plus a notch depth added on each end, predicts beam deflection under load quite accurately. APPENDIX I. -REFERENCES APPENDIX II. -NOTATION The following symbols are used in this paper: a = notch depth; b = beam width; d = beam depth; 1999

12 notched beam net depth; allowable shear stress; mode I, II stress intensity factors; effective stress intensity factors on imminent fracture plane; bending moment at distance x from left end of beam; modulus of rupture; applied load; left end reaction; beam shear; and notch end closest to load. 2000