DISTRIBUTION NEGATIVE MOMENT OVER BLOCK SUPPORTS CONTINUOUS WOOD LAMINATED BEAM

Transcription

1 U.S. DEPARTMENT OF AGRICULTURE FOREST SERVICE FOREST PRODUCTS LABORATORY MADISON, WIS. U. S. FOREST SERVICE RESEARCH NOTE FPL-060 September 1964 STRESS DUE TO DISTRIBUTION NEGATIVE MOMENT OVER BLOCK SUPPORTS IN A THREE-SPAN CONTINUOUS WOOD LAMINATED BEAM

2 Summary Present design procedures for a continuous wood beam use a design moment (negative moment over interior support) calculated by the method of three moments. The design of a reinforced-concrete continuous beam uses a reduced span and correspondingly reduced moment to account for the width of support. The purpose of this study was to determine whether the design moment (and hence the stress) over the support of a three-span continuous wood beam could be reduced. Three southern pine laminated beams were tested, and loading was accomplished by a concentrated load every 2 feet along the length of the beam to approximate a uniform load. All stresses were kept below the proportional limit. Although stress, strain, and moment distributions varied somewhat from theory, the agreement of maximum theoretical and experimental values in this research indicates current design procedures do produce accurate results. This study also showed that stress and strain distributions are essentially linear, indicating that elementary theory can be used over the support.

3 STRESS DISTRIBUTION DUE TO NEGATIVE MOMENT OVER BLOCK SUPPORTS IN A THREE-SPAN CONTINUOUS WOOD LAMINATED BEAM 1 By LAWRENCE A. SOLTIS, Engineer 2 Forest Products Laboratory, Forest Service U.S. Department of Agriculture ---- Introduction A three-span continuous beam under uniformloading has the maximum moment occurring over the interior supports. In the current design of the three-span continuous beam, this maximum negative moment is found either from the 3 theorem of three moments ( 4 ) or from moment coefficients found in many handbooks, such as the Timber Engineering Company s handbook ( 6 ). These coefficients are shown in figure 1. The moment coefficients, calculated from the theorem of three moments, are based upon the conditions that the beam has equal uniform load in each span, constant cross section (and hence constant moment of inertia), equal length spans, knife-edge supports, and no particular length-depth ratio. Many of the present structural designs satisfy the conditions of having equal uniform loads in each span, constant cross section (and hence moment of inertia), and equal length spans. Structures having unequal loads, variable cross sections and span lengths can be designed using the method of three moments: however, the conditions of a knife-edge support is not generally satisfied in practice. Beams (simple or continuous) generally rest on column caps, masonry walls, or some type of support that resembles a block support. This is recognized in reinforcedconcrete design where the width of support is used to reduce the maximum negative moment because of effective reduction of span. 1 This study was conducted at the Forest Products Laboratory for a thesis in partial fulfillment of the requirements for the degree of Master of Science at the University of Wisconsin. Special acknowledgment is made to Prof. L. W. Crandall of the University and to the Koppers Company, Inc., Unit Structures Division, who provided the study material. 2 Maintained at Madison, Wis., in cooperation with the University of Wisconsin. 3 Underlined numbers in parentheses refer to Literature Cited at the end of this report. FPL-060

4 The outermost fiber stress used as the design stress is calculated by the conventional beam formula σ = My, where σ = stresss, M = moment, y = distance I from neutral axis, and I = moment of inertia about the neutral axis. However, the validity of this formula is limited in the vicinity of the support where St. Venant s principle and the theory of elasticity govern. This study considers the negative moment and the stress distribution over various-sized block supports. A study by Cowan ( 2 ) of shear stress over a block support in a two-span continuous wood beam showed the shear stress to differ from the stress calculated by the conventional formula for parabolic shear-stress distribution. Paralleling Cowan s work, the bending stress, strain, and moment over a support in a three-span continuous beam were investigated in this study. Description of Material Three glued-laminated structural-lumber beams of southern pine, 3-1/4 inches by 6-1/2 inches by 30 feet 7 inches, were studied. The size was chosen to approximate a practical problem A continuous basement girder of three 10-foot spans supporting 11-foot joists spaced 2 feet center to center in a single-story wood-frame building is similar. Characteristics of each beam were as follows: Grade: Southern pine No. 1 Dense, in each of the four 1-5/8-inch laminations Glue: Interior-type casein Joints: 1:10 plain scarf joints at random locations Slope of grain: Maximum 1 in 20 Moisture content: 9 to 11 percent during experiments Southern pine Dense-Structural wood supports were used, measuring 3-1/4 inches wide, 2 inches deep, and 3-1/4-, 5-1/4-, or 7-inches long; also, a steel support measuring 8 inches wide, 1/2 inch deep, and 6 inches long. FPL

5 Experimental Procedure Loading Method and Reaction Determination The beam was set up as a three-span continuous beam with supports spaced 10 feet center to center. A concentrated load was applied through a block 3-1/4 inches in width by 1-5/8 inches in length every 2 feet along the length of the beam. This loading system corresponds to joist loading of a basement girder and approximates a uniform load on a beam. The entire beam-loading arrangement is shown in figure 2. To provide the required 15 load points on the beam, a loading system comprised of yoke-like pin-connected assemblies was devised. The yokes were connected to pulleys over which two cables ran. Winding of the cables on a drum caused a uniform tension in the yokes. Due to the large number of load points, a relatively small tension in the cables provided a large total force on the beam. It was found that the load could be closely controlled and held for a period of about 15 minutes without loss. Losses due to stretching of the cable were negligible. The amount of load was determined by extending the end yoke so that a load point rested on a 20,000-pound-capacity load cell (fig, 2). The amount of load was checked by a strain gage at the bottom of the beam at the center of the middle span. Loads were applied at four levels--100, 200, 300, and 400 pounds per foot. This kept the stresses below the proportional limit. Total reaction forces were determined by placing load cells under each size of block support at the start of testing each beam. The reactions were numbered from the north end of the beam. The load point resting on the load cell was reaction 0; the first exterior reaction was reaction 1; the next two interior reactions were reactions 2 and 3; and the last exterior reaction was reaction 4. The beams were tested under three support conditions: 1. Block supports resting on load cells, permitting rotation of support. 2. Block supports resting on rollers, permitting lateral movement but no rotation. 3. Block supports permitting no rotation or lateral movement. FPL

6 Measurement of Strain Strains were measured in the horizontal direction parallel to the grain at reaction 2 by electric strain gages with 1/2-inch gage length. Although Cowan ( 2 ) did not use electric gages in his study of shear over a support, Walker 4 reports an electric gage to be well suited toparallel-to-grain readings, though a serious error may result in measurements perpendicular to the grain. A Tuckerman optical gage was used to take vertical (perpendicular-to-thegrain) readings and to take supplementary horizontal readings. The Tuckerman gage, collimator, and holding device are shown in figure 3. The electric strain gages were mounted in five vertical lines over the support, with six gages in each vertical line. The vertical lines were 1-3/8 inches apart in the first and second beams and 1-3/4 inches apart in the third beam The center vertical line coincided with the centerline of support. The gages were spaced 1.3 inches apart in each vertical line. Consequently, there were 30 gages mounted on each side of one interior support of each beam (fig. 2). The gages were numbered consecutively, starting at the top nearest the other interior support, and were read horizontally. Therefore, the top row was numbered 1 through 5, the second row 6 through 10, etc.; the first vertical line nearest the other interior support was numbered 1, 6, 11, 16, 21, and 26. There were two check gages mounted on the bottom at the centerline of the middle span as a check on the applied loads. Strain readings from these gages remained relatively constant for the same loads in different runs. Each pair of gages located on opposite sides of the beam at the first interior support were wired in series, thus giving the average strain over the width of the beam at that particular point. The two check gages were also wired in series to give the average strain at the bottom of the centerline of the middle span. Vertical Tuckerman-gage readings were taken 1/8 inch from the bottom of the beam at reaction 3 to show the approximate reaction distribution. Readings were taken at 11/16-inch intervals, from the centerline of the support to 4-1/8 inches on each side of the centerline for the 3-1/4- and 5-1/4-inch supports; readings were taken at 7/8-inch intervals, from the centerline to 4-3/8 inches on each side of the centerline for the 7-inch support. Small grooves in the wood caused by the knife edges of the Tuckermangages limited the number of vertical readings taken. Therefore, beam No. 1 was used to find the approximate reaction distribution of the 3-1/4-inch-long support; beam No. 2 for the 5-1/2-inch support; and beam No. 3 for the 7-inch support. 4 Walker, J. N Interpretation and measurement of strains in wood. Ph. D. Thesis, Purdue University, Lafayette, Ind. FPL

7 Moisture Control Moisture readings were taken with a resistance-type moisture meter before and during testing. The moisture content of the beams varied from 11 to 13 percent when the beams were received. After a few days, the moisture content dropped to 9 to 11 percent, with some surface checking occurring. During loading, the moisture content remained constant at 9 to 11 percent, depending upon the distance from the surface of the beam. The change in moisture content of the outer fibers caused only minor changes in the strain readings. Upon completion of testing, a sample was cut and the moisture content determined by the ovendrying method to check the accuracy of the electric moisture meter. The ovendried sample was 10.7 percent, compared to 10.5 percent by the electric moisture meter. It was concluded that the accuracy of the moisture meter was acceptable and that the moisture content of the beams stayed approximately constant, with only small variations in the outer fibers which had no significant effect on the results. Modulus of Elasticity The modulus of elasticity was determined from compression tests of small specimens cut from the beams over reaction 2. The specimens were cut so that the same electric strain gages used to measure horizontal strains were also used to measure compressive strains in the modulus of elasticity tests. In this manner, the modulus of elasticity was determined at each point where the horizontal bending strains were measured. Following standard design procedure, the modulus of elasticity was assumed to be the same in compression as in tension caused by bending. The electric strain gages were again wired in series for the modulus of elasticity determination to give the average strain across the width of the beam at a particular point. The modulus of elasticity varied considerably among laminations. Extremes 6 ranged from 1.85 x 10 6 to 3.74 x 10 pounds per square inch in beam No. 1; 1.80 x 10 6 to 3.20 x 10 6 pounds per square inch in beam No. 2; and 1.82 x 10 6 to 2.84 x 10 6 pounds per square inch in beam No. 3. Variations along the length of lamination ranged from 2 to 50 percent, due to the scarf jointing. A list of moduli of elasticity corresponding to the numbering system used for strain measurements at reaction 2 is given in table 1. FPL

8 Analysis of Data and Discussion of Results The effects of shear strain were neglected and the primary strains were assumed to act in only one direction. Hooke s Law (σ = EE ) was used to relate stress and strain, where σ = stress, E =modulus of elasticity, and E = strain. Bending stresses were calculated on the basis of experimentally measured strains and moduli of elasticity. The tensile and compressive resultant forces of the stress volumes and the negative moments over the support were calculated by digital computer. The data for the three beams at all load levels were in close agreement and were used in formulating the conclusions. However, beam No. 3 under a load of 300 pounds per foot was used for illustrative purposes because it was representative of the entire data; the only exception was the use of beam No. 1 under a load of 300 pounds per foot for one moment-diagram illustration to point out a particular occurrence. Strain Results and Discussion Strains plotted across the depth of beam 3 for the three sizes of woodblock supports and a steelplate support, under various support conditions, are shown in figures 4 and 5. The following phenomena were observed: 1. Tensile strains remained linear functions of the distance from the neutral axis at all positions across all four sizes of supports. 2. Compressive strains remained linear functions of the distance from the neutral axis at positions not directly above the support. Above the support the strain distribution had a point of discontinuity about midway between the neutral axis and outer compressive fiber. This discontinuity was more distinct over the centerline of the 3-1/4-inch-block support than over the centerline of the 7-inchblock support. The discontinuity was similar to that found by Talbot ( 5 ) in his study of stresses under a load on a railroad track and was the result of a stress concentration due to the vertical bearing load. The degree of severity of the discontinuity was dependent on the distance from the centerline of the block support (most severe at centerline) and the size of support (more concentrated for small supports and more distributed for large supports). The support conditions of rotation and lateral movement did not have any influence on the discontinuity. FPL

9 3. At positions where both tensile and compressive strains were linear functions of the distance from the neutral axis, the slope of the tensile-strain distribution was steeper than that of the compressive strain. 4. The outer-fiber strains in compression and tension were equal at positions not over the support. At positions over the support, the tensile strain was greater than the compressive strain. However, this was dependent on support size. The outer-fiber strains in tension and compression were equal over the centerline of the 7-inch support, but the tensile strain was about 25 percent greater than the compressive strain over the 3-1/4-inch support. This difference in strains was believed to be caused also by the effects of the vertical bearing load. 5. The neutral axis was shifted below the center of gravity of the cross section toward the compressive side for all positions over the support. The neutral axis shifted down about 3 percent when not over the support and about 10 percent at the centerline of support. This held for all sizes of supports. 6. There was no noticeable increase in strain at the centerline of support as compared to the strains 3-1/2 inches from the centerline. No comparison could be made between experimental and theoretical strains due to the large variation in the modulus of elasticity. 7. The effect of a scarf joint is seen in figures 4 and Type of support condition, whether permitting rotation and lateral movement or restricting rotation and lateral movement, had no significant effect. Some of these phenomena have been observed in studies of the positive moment regions of simple beams. Comben ( 1 ), using a simple span with third-point loading, found: (a) the strain to be a straight line to about one-half the distance between the neutral axis and compressive edge, beyond which the slope of the compressive portion of the strain curve decreased slightly; and (b) found the neutral axis to be unpredictable. The increase in strain at the compressive edge corresponds to the discontinuity mentioned under (2) of the observed phenomena. It is not known whether bearing load alone or bearing load plus this natural occurrence, as reported by Comben ( 1 ), produced the discontinuity found in this study. Some of these phenomena contradicted findings made by Dietz ( 3 ) in his study of a simply supported Douglas-fir beam. Dietz found that: (a) the neutral axis was shifted toward the tension edge where it remains stationary; and (b) the tensile strains were slightly greater than the compressive strains. Comben ( 1 ) found that compressive strains were slightly greater than the tensile strains. FPL

10 The most significant conclusion drawn from this study is that sections which are plane before bending remain essentially plane after bending. This suggests the use of elementary-mechanics theory over the support, regardless of St. Venant s principle. Stress Results and Discussion Stress is assumed to be related to strain by Hooke s law. Stern 5 concluded that stress was proportional to strain for beams of symmetrical physical structure. Comben ( 1 ) also assumed a linear stress-strain relationship but made no comment as to the structure of the material. Dietz ( 3 ) stated that linear stress-strain relations were exhibited in his study of Douglas-fir beams. The distribution of stress across the depth of beam No. 3 is shown in figures 6 and 7A for the woodblock supports, and in figure 7B for the steelplate supports. The effect of scarf joint on stress is also shown in figure 6A. The following phenomena were observed: 1. As with strain, the tensile stresses remained a linear function of the distance from the neutral axis. 2. The compressive stresses remained a linear function of the distance from the neutral axis at positions not over the support. Above the support, the discontinuity about midway between the neutral axis and outer compressive fiber appeared more pronounced than indicated by strain results. 3. The value of the compressive stress in the outer fibers was from 10 to 40 percent greater than that of the tensile stress. 4. The neutral axis shifted toward the compressive edge by 3 percent when not over the sppport to 10 percent at the centerline of support. 5. The total compressive force, found from the stress volume, was equal to the total tensile force away from the support--a necessary equilibrium condition. However, at the centerline of the 3-1/4-inch support, the tensile force was about twice the compressive force, but at the centerline of the 7-inch support the compressive and tensile forces were about equal. This is attributed to the friction force between the support block and the beam caused by the vertical bearing load. This means that friction transmitted a portion of the 5 Stern, E. G The elastic and strength properties of wood in flexure. Ph. D. Thesis. Pennsylvania State University, University Park, Pa. FPL

11 compressive stress to the block support, depending on the length of the support. The vertical bearing load is more concentrated in the smaller support; thus, the friction force is inversely proportional to the length of support, and more friction occurs with short supports than with long supports. 6. The tensile stresses computed by the standard beam formulawere generally 5 to 10 percent greater than the experimental tensile stresses. The theoretical and experimental compressive forces over the support differed or coincided, depending on the amount of stress transferred by the friction force. The theoretical compressive stress at the outermost fiber was about 10 to 30 percent less than the experimental stress at the outermost fiber for positions outside the support. This difference was greatest over the smaller-sized support. The outer-fiber compressive stresses (experimental. and theoretical) were generally about equal at the centerline of all supports. 7. Type of support condition did not yield any significant stress differences. Figures 6 and 7 correspond to results shown by Talbot ( 5 ), who found the effects of the vertical bearing load to extend beyond the center of depth; therefore, (a) the neutral axis was shifted about 15 percent toward the compression edge, (b) a discontinuity was observed near the compression edge, and (c) the maximum tensile and compressive stresses did not differ much from calculated bending stresses. These same phenomena were observed in this study and it is therefore concluded that the theoretical stress in the outer fibers is comparable to the experimental results. Moment Results and Discussion The negative bending moments over support No. 2, at a loading of 300 pounds per foot, were calculated from the couples formed by the compressive and tensile forces as found from the stress volumes. The experimental moment results were not as consistent as the strain and stress results. Figures 8 and 9 show the negative moment diagrams over the four sizes of supports of beam No. 3. Figure 10 shows the negative moment diagram over the 5-1/4-inch block support of beam No. 1. A comparison of experimental with theoretical moments is difficult due to the magnification of experimental discrepancies. The load could not be controlled to increments smaller than about 5 pounds per foot. A change of 5 pounds per foot in load application caused a change of 50 foot-pounds in moment. Limitations in the accuracy of strain measurements and in the determination of moduli of elasticity caused stress variations of about 25 pounds per square inch. This FPL

12 variation was too small to appear in the stress analysis, but a change of 25 pounds per square inch in outer-fiber stress corresponded to a change of 50 foot-pounds in moments. An error of 50 foot-pounds was only about a 2-percent error but was magnified on the large scale used in figures Therefore, while trends in moment diagrams could be observed, no conclusions could be made regarding small variations. The moments were dependent on the distribution of the reaction over the support. Those distributions were approximated by vertical strains measured 1/8 inch above the bottom of the beam, as shown in figure 11. The experimentally determined reactions nearly coincided with the theoretical reactions, except under the lowest load level, as shown in table 2. Under this low-load level the beam may not have been seated properly on the supports. The moment diagram over the 3-1/4-inch support in beam No. 3 (fig. 8A) had two apexes corresponding to the two apexes of the reaction distribution. This occurred in every loading; however, the depression between apexes was much greater in some cases. This depression was believed to be dependent on the amount of friction between the support and the beam The two apexes were located near the edges of the support, in contrast to the theoretical apex which occurs at the centerline; however, the levels of the experimental and theoretical apexes were approximately equal. The moment diagram over the 5-1/4-inch support (fig. 8B) approximated the theoretical moment diagram in most instances; however, the apex of the negative moment diagram was not as near the edge of support as the apex of the reaction distribution curve. In a few instances the moment diagram over the 5-1/4-inch support of beam No. 1 (fig. 10) had two apexes, as occurred over the 3-1/4-inch support. The different moment diagrams over the 5-1/4-inch support were believed caused by the variability of the seating of the beam on the support. The seating varied for two reasons, namely: (1) slightly inaccurate shimming, which resulted in slightly different elevations of the four supports; and (2) slight twisting of the beam due to the equipment used in loading. The moment diagrams over the 7-inch-block suport and the 6-inch-steel plate (fig. 9) were generally similar to those for the 5-1/4-inch support. The theoretical and experimental moment diagrams were similar, except for the experimental moment apex being shifted toward the support edge on the exterior span side. This shift was not as near to the support edge as that of the apex of the reaction distribution. FPL

13 Conclusions The following conclusions are made from this study: 1. Tensile and compressive strains are each linear, but the slope of the tensile strain-distribution line is steeper than that of the compressive line. Thus, the neutral axis is shifted toward the compression side; however, the maximum values at the outer fiber in tension and compression are equal. 2. Tensile and compressive stresses are each linear, but the slope of the tensile stress-distribution line is steeper than that of the compressive line. Thus, the neutral axis is lowered and the outer fiber stresses in compression are 10 to 40 percent greater thanintension. The variation in moduli of elasticity causes the compressive stress to be greater than the tensile stress, but the compressive strain equals the tensile strain. 3. Reaction distribution for block supports could be approximated by a concentrated force near each edge of the support, with the larger concentrated force on the exterior span side of the support. 4. Maximum theoretical and experimental moments over the supports are approximately equal; however, maximum experimental moment is shifted slightly toward the exterior span side of the support or occurs at two points corresponding to the reaction distribution. 5. A straight line connecting the outer-fiber strains in tension and compression is comparable to the strain distribution from positive moment in a simple span; a straight line connecting outer-fiber stress in tension and compression is comparable to the stress distribution calculated from elementary theory. Strain, stress, and moment over a support can be analyzed by elementary theory. 6. A friction force between support and beam is developed due to the vertical bearing load. The friction force transmits some compressive bending stress to the support, depending on the length of support. Friction force is inversely proportional to the length of support. 7. Although stress distributions and negative moments near the support vary somewhat from the theoretical, their maximum values indicate that current design procedures are satisfactory. FPL

14 Literature Cited FPL

15 FPL-060

16

17 M Figure 1.--Shear and moment diagrams for a three-span continuous beam using handbook coefficients.

18 Figure 2.--Loading arrangement of study beams. M

19 Figure 3.--Devices used for vertical strain measurement: A, Tuckerman gage: B, collimator; and C, holding device. M

20 Figure 4.--Strain distributions through the depth of beam No. 3 with a load of 300 pounds per foot of length:, 3-1/4-inch-block support; B, 5-1/4-inch-block support, FPL-060

21 Figure 5.--Strain distributions through the depth of beam No. 3 with a load of 300 pounds per foot of length: A, 7-inch-block support; B, 6-inch-steeplate support.

22 Figure 6.--Stress distributions through the depth of beam No. 3 with a load of 300 pounds per foot of length: A, 3-1/4-inch-block support; B, 5-1/4-inch-block support. FPL-060

23 Figure 7.--Stress distributions through the depth of beam No. 3 with a load of 300 pounds per foot of length: A, 7-inch-block support; B, 6-inch-steelplate support.

24 Figure 8.--Negative moments over support No. 2 of beam No. 3 with a load of 300 pounds per foot of length: A, 3-1/4-inch-block support; B, 5-1/4-inch-block support. FPL-060

25 M Figure 9.--Negative moments over support No. 2 of beam No. 3 with a load of 300 pounds per foot of length: A, 7-inch-block support; B, 6-inch-steelplate support.

26 M Figure 10.--Negative moments over 5-1/4-inch block support No, 2 of beam No, 1 with a load of 300 pounds per foot of length. FPL-060

27 M Figure 11.--Vertical strains across support No. 2 with a load of 300 pounds per foot of length: A, 3-1/4-inch-block support, beam No. 1; B, 5-1/4-inch-block support, beam No. 2; C, 7-inchbloack support, beam No. 3.

28 PUBLICATION LISTS ISSUED BY THE FOREST PRODUCTS LABORATORY The following lists of publications deal with investigative projects of the Forest Products Laboratory or relate to special interest groups and are available upon request: Box, Crate, and Packaging Data Chemistry of Wood Drying of Wood Fire Protection Fungus and Insect Defects in Forest Products Glue and Plywood Growth, Structure, and Identification of Wood Furniture Manufacturers, Woodworkers, and Teachers of Woodshop Practice Logging, Milling, and Utilization of Timber Products Mechanical Properties of Timber Pulp and Paper Structural Sandwich, Plastic Laminates, and Wood-Base Components Thermal Properties of Wood Wood Finishing Subjects Wood Preservation Architects, Builders, Engineers, and Retail Lumbermen Note: Since Forest Products Laboratory publications are so varied in subject matter, no single catalog of titles is issued. Instead, a listing is made for each area of Laboratory research, Twice a year, December 31 and June 30, a list is compiled showing new reports for the previous 6 months. This is the only item sent regularly to the Laboratory s mailing roster, and it serves to keep current the various subject matter listings. Names may be added to the mailing roster upon request.

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