UNITED STATES DEPARTMENT OF AGRICULTURE. FOREST SERVICE. FOREST PRODUCTS LABORATORY. MADISON, WIS A STUDY OF THE STRENGTH OF SHORT AND INTERMEDIATE WOOD COLUMNS BY EXPERIMENTAL AND ANALYTICAL METHODS January 1964 FPL-028
FOREST SERVICE REGIONAL EXPERIMENT STATlONS AND FOREST PRODUCTS LABORATORY PROGRAM LOCATIONS HEADQUARTERS OF REGIONAL FOREST EXPERIMENT STATIONS FOREST PRODUCTS LABORATORY WASHINGTON - BELTSVILLE Forest Service regional experiment stations and Forest Products Laboratory
A STUDY OF THE STRENGTH OF SHORT AND INTERMEDIATE WOOD COLUMNS BY EXPERIMENTAL AND ANALYTICAL METHODS 1 By JOHN E. FAIRKER, Engineer 2 Forest Products Laboratory, Forest Service U.S. Department of Agriculture ---- Summary Wood columns of various lengths in two species were evaluated for strength and the results analyzed by several methods. Experimental stress-strain data were obtained over a 6-inch gage length from specimens 10 inches in length. The data thus obtained were used to predict buckling loads given by the tangent modulus theory. In one species, the average experimental results for short; and intermediate columns of the grade used agreed very closely with the tangent modulus predictions. In the other species, average experimental results went above tangent modulus predictions for the short columns and below for the intermediate columns. Several analytical approximations were developed to predict tangent modulus buckling loads, The Ramberg-Osgood equation for stress versus strain and Timoshenko s equation for tangent modulus versus strain yielded Satisfactory results. A hyperbolic equation for stress versus strain did not yield satisfactory results, The approximate theoretical tangent modulus buckling loads of the columns with L/d equal to 15 and 20 were determined by Southwell s method. The results that were obtained by Southwell s method agreed very closely with theory for intermediate columns of one species, but were too high for intermediate columns of the other species. 1 This study was made in partial fulfillment of the requirements for the Master of Science degree in Civil Engineering from the University of Wisconsin. 2 Maintained at Madison, Wis., in cooperation with the University of Wisconsin. FPL - 028
General Considerations of Elastic and Inelastic Buckling 3 The Euler equation ( 3 ) for predicting the buckling load of columns that are not stressed above the proportional limit gives the smallest load at which the original straight form of equilibrium of a centrally loaded ideal column becomes unstable. The problem can be formulated mathematically by applying an infinitesimally small disturbance to the originally straight column and then investigating whether this bent form of equilibrium can be maintained by the axial load acting alone when the disturbance is removed. The idea of applying and removing a small disturbance is not important in elastic buckling. The time when the disturbance is applied and when it is removed have no effect on the Euler buckling load. However, for inelastic buckling, disturbances applied and removed at different levels of stress produce different buckling loads ( 12 ). Engesser ( 2 ) and Considère ( 1 ) were the first to recognize the problem of inelastic buckling in 1889. However, the load at which a column fails by inelastic buckling is still open to question. Shanley ( 7 ) recognized the paradox in the reduced modulus theory of inelastic buckling and showed that the load predicted by this theory could not be realized in actuality. He concluded that the tangent modealus theory was correct for predicting the maximum load at which a perfect centrally loaded column will remain straight. Wang ( 12 ) defines the buckling load of a column stressed beyond the proportional limit as the smallest axial load at which the bent form of equilibrium of an originally straight and centrally loaded column, resulting from the action of a small disturbance, becomes stable. The application of a small disturbance requires further discussion. A disturbance can be a small force or moment which is applied and removed at will. If the disturbance is applied before the axial load, P, reaches the buckling load and is then removed, the ideal column will return to its original straight form when no fiber is stressed beyond the proportional limit. When some or all of the fibers are stressed beyond the proportional limit, the column will assume a bent form different from the bent form realized while the disturbance is acting. Thus the original bent form taken during the action of the small disturbance is not a stable form of equilibrium. If the disturbance is applied when P is equal to the buckling load and then removed, the bent form will remain unchanged because it is a stable form. 3 Underlined numbers in parentheses refer to Literature Cited at the end of this report. FPL-028-2-
The two extreme cases of inelastic buckling will now be considered. If a column is loaded to its critical, stress and then the disturbance is applied and immediately removed, there is a reversal of strain and the buckling load is given by the double or reduced modulus theory. At the other extreme, if the disturbance is applied before the compressive stress reaches the proportional limit and is removed at a value of load such that the bent form of the column is stable, there is no reversal of strain and the buckling load is given by the tangent modulus theory. Test results have shown that the buckling loads of short columns approach the double modulus load and that the buckling loads of long columns we closer to the tangent modulus load. Approaching the problem realistically, disturbances in the testing machine will always be present during the testing process. A short column is relatively insensitive to disturbances, so that it is probable that the column will not bend until the proportional limit is exceeded, For this case, a reversal of strain takes place and the buckling load approaches the double modulus load. A slender column is more sensitive to disturbances, so that bending will take place before the proportional limit is reached, For this case, there is no reversal of strain and the buckling load is given by the tangent modulus theory. The possibility of using the tangent modulus theory to determine the effect of strengthreducing characteristics on wood columns of intermediate length was investigated by Byars. 4 One purpose of this study was to determine if the tangent modulus theory could be used to predict the maximum loads of short and intermediate wood columns of structural grade with such characteristics as hots and initial eccentricity, Theoretically, the tangent modulus theory is valid only for centrally loaded, homogeneous, straight columns. Approximations to the theory can be obtained if the point of application of the load is adjusted to compensate for the nonhomogeneity of the material and the initial curvature. However, in this study an attempt was made to predict the maximum loads of wood columns loaded on the geometric center, since this is the reference point used in design work, Because of the variation of tangent modulus values obtained in tests of structural grade columns, another purpose of this study was to determine if a mathematical expression with certain parameters could be used as an approximation for the experimental compressive stress-strain curves. The mathematically expressed curve does not have to approximate the actual stress-strain curve very closely, but the two curves must have approximately the same slope at the same level of stress, so that the predicted buckling loads will be close to the actual maximum loads. 4 Byars, Edward F. Analytical and Experimental Determination of Buckling Loads of Wood Columns by the Tangent Modulus (Engesser) Theory. Ph. D. Thesis, Department of Theoretical and Applied Mechanics, University of Illinois, 1957. FPL-028-3-
A third purpose of this study was to determine if Southwell s method of plotting deflection divided by load versus deflection for each column tested could be used to predict the critical (buckling) load of the corresponding straight column loaded so that the eccentricity is zero. Experimental Procedure Preparation and Matching of Material The column material was Construction Grade Douglas-fir and No. 1 SR (stress-rated) southern pine. All material received had been air dried, and the pieces were 4 by 4 inches by 12 feet, nominal. A 1-inch section was cut from each end of the 12-foot pieces for the purposes of determining specific gravity and moisture content. Assuming a linear variation of moisture content and specific gravity along the length of the 12-foot pieces, an effort was made to match the different lengths of columns for specific gravity and size and location of knots. Each species was cut into six sets, which contained lengths of 10, 18, 36, 54, and 72 inches. Test Procedure Compression Specimens.- The compression specimens were 4 by4 by 10 inches. Before testing, the ends of each specimen were dried in an oven for 2 to 3 minutes to reduce the possibility of the failure occurring outside of the gage length. The compression measuring devices used on all four sides were Marten s mirrors, with a 6-inch gage length (fig. 1), calibrated so that the sum of the two scale readings of the compression of opposite sides gave the average compression for the two sides. The two sets of values thus obtained were averaged to obtain the average compression. The end supports used were double-roller devices, so that bending could take place about any plane. The point of loadingwas taken as the geometric center as a start and then adjusted until approximately equal compression readings were recorded up to a load of 15,000 pounds. Once the approximate elastic center was located, the specimen was tested to failure, with compression readings being taken on all four sides at 5,000-pound increments up to 30,000 pounds and then 2,000-pound increments to failure (tables 1 and 2). The loading machine was hydraulic with a 100,000-pound range. The rate of loading was set at a constant compression rate of 0.003 inch per inch of length per minute. FPL-028-4-
Short and Intermediate Columns.- The short and intermediate columns were 18, 36, 54, and 72 inches long with a 4- by 4-inch cross section, nominal. All the columns were tested in a hydraulic loading machine at a constant rate of strain of 0.003 inch per inch of length per minute. The load was applied at the geometric center, and the lateral movement of the column at midheight was obtained by the use of Ames dials mounted on the machine (fig. 2). For the 18-, 36-, and 54-inch lengths, four dials with a maximum travel of 1/2 inch were used one dial on each side. For the 72-inch length, two dials with a maximum travel of 1 inch were used on adjacent faces. Deflection readings were taken at 5,000-pound increments to failure. The maximum load was also recorded. The end supports were the same as those used for the compression specimens, The initial curvature of each column was obtained by holding a straightedge on the curved face and measuring the offset at midheight. Determination of Tangent Modulus from Experimental Results The values of tangent modulus at different stress levels were determined for each compression specimen by dividing an increment of stress by an increment of strain. For example, a value of the tangent modulus at a stress corresponding to a load of 36,000 pounds was obtained by calculating the difference in stresses corresponding to loads of 38,000 pounds and 34,000 pounds, and dividing this value by the difference in strains obtained at loads of 38,000 pounds and 34,000 pounds. Only values of load and compression recorded in tests were used to obtain tangent modulus values, so that it was assumed that the slope at any test point could be approximated by the slope of a straight line drawn through the two test points immediately above and below the test point at which the tangent modulus was desired. From the curves of tangent modulus versus stress for each compression specimen, average values of tangent modulus versus stress were obtained for each species (figs. 3 and 4). For southern pine, average values of tangent modulus were obtained at each selected stress level (table 3). For Douglas-fir, the individual stress-strain curves varied so widely that at a selected stress level near the average maximum stress, one specimen curve yielded a considerable value for the tangent modulus and another curve yielded no value at all because the selected stress level was greater than the maximum stress for that specimen. For this reason, better average values were obtained by averaging the stress at selected values of tangent modulus (table 4). FPL-028-5-
With average values of stress versus tangent modulus, Euler s generalized formula, (1) where P = load, A = area, E = tangent modulus, L/d = slenderness ratio, was used to T calculate values of slenderness ratio (L/d) versus stress (figs. 5 and 6 solid lines). The two curves predict the buckling loads of straight columns of comparable material, loaded so that eccentricity due to nonhomogeneity is zero and strain reversal does not occur. Determination of Tangent Modulus from Analytical Approximations Ramberg-Osgood Equation for Stress versus Strain Ramberg and Osgood ( 6 ) proposed an analytical expression of the type (2) where e = total strain, S = stress, E = modulus of elasticity, (e p ) = maximum plastic max strain, S = maximum stress, and B = the measured slope of a straight line drawn max through the experimentally determined points of log (S/S ) versus log (e ). max p Approximating the plastic strain of structural materials as a power function has as its basis the fact that it gives results whichapproximate a probability function. Shanley ( 8 ) shows that statistics can be a very useful tool in analyzing the results of a number of tests. Expressing the plastic strain in terms of a power function will approximate actual results very closely if the amount of slip (plastic strain) depends on certain factors which have a random distribution. If equation (2) is differentiated and inverted, an expression for the tangent modulus in terms of stress is obtained as follows: FPL -028-6-
(3) Figures 7 and 8 are typical examples of the relation of stress ratio to plastic strain; they show that the experimental results obtained approximate straight lines, except for values of stress close to the maximum, The average values of the quantities in equation (3) that were used to compute values of tangent modulus versus stress are shown in tables 5 and 6. Equation (3) yields a family of curves for different values of B with values of tangent modulus at the maximum stress which are greater than zero for all finite values of B. However, the experimental stress-strain results show that the slope at maximum stress approaches zero in most cases. Tables 7 and 8 show that the values of tangent modulus obtained from the Ramberg-Osgood formula for levels of stress near maximum are higher than values obtained directly from experimental curves. Timoshenko Equation for Tangent Modulus versus Stress An analytical expression for the tangent modulus to be used in Euler s generalized formula is given by Timoshenko ( 11 ): (4) where ds/de denotes the derivative of stress with respect to strain, E is the tangent T modulus, E is the modulus of elasticity, S is the maximum direct stress, S is the max stress at which the tangent modulus is desired and c is a parameter which is a constant for each material, The value of c for pine wood suggested in the reference is 0.875. However, a closer approximation to the experimental results was obtained by taking c = 0.80 for the southern pine that was tested. For the Douglas-fir that was tested, a value of c = 0.90 was found to yield an approximation close to the experimental results. The only way that is known to determine the value of the parameter, c, that will approximate the test results is by trial and error. An approximate value for c can be obtained by observing a typical stress-strain curve for the material. As c approaches 1.0, the stress-strain curve approaches Hooke s law, S = Ee. The typical stress-strain curve for Douglas-fir has a sharper bend at the knee than the one for southern pine, so that c has a higher value for Douglas-fir (compare tables 9 and 10). FPL -0 2 8-7-
The equation for tangent modulus versus stress as given by Timoshenko was integrated to determine the equation for stress versus strain: Equation (5) is seen to be complicated. However, it does give a good approximation to the average experimental stress-strain curves, as shown in tables 9 and 10. (5) Hyperbolic Equation for Stress versus Strain A rectangular hyperbola of the form, (6) with asymptotes, S = Ee and S = S, was developed to approximate the experimental remax sults. The value of the parameter, k, determines the degree of curvature. As k approaches zero, the hyperbola approaches its two asymptotes. Taking the derivative of the above equation, an equation for tangent modulus versus strain is obtained (7) The use of this analytical relationship in Euler s generalized formula requires a lengthy procedure. If the tangent modulus is desired at a selected value of stress, the strain that corresponds to the selected stress must be obtained from the hyperbolic relationship of stress versus strain. The value of strain is then used in equation (7) to determine the value of the tangent modulus at the selected value of stress. Southwell s Method of Determining Buckling Loads and Eccentricities Theoretical Basis Southwell ( 9 ) developed an important procedure for determining the buckling load of a perfect column using the load versus deflection data from a test of an imperfect column. FPL-028-8-
If the strut is not straight initially, and if y is the initial deflection of the centerline, o the condition for equilibrium is (8) If the deflected shape can be expressed as a Fourier sine series and if the load, P, is a considerable fraction of the buckling load, P, then the deflection at the center can be cr approximated as (9) where a is the deflection reading at midheight referred to the initial position and a is the o deflection at midheight. Rearranging equation ( 9 ), a linear equation of the form (10) is obtained. A plot of a/p versus a yields a straight line with an inverse slope which is a measure of the theoretical buckling loadandanx-axis intercept which is a measure of the initial deflection. Timoshenko ( 11 ) shows that Southwell s method can be applied to a column loaded eccentrically. The approximate expression that is obtained for the case of initial curvature and some eccentpicity in the point of application of the load is (11) where y is the deflection at midheight referred to the initial position, a is the initial c o deflection at midheight, and e is the eccentricity at midheight of the load which may be produced by an error in centering the load and nonhomogeneity of the material. Wang ( 12 ) shows that Southwell s method can be applied to the case of inelastic buckling. If P is a large portion of P such that EI approximately a constant expressed as cr (12) FPL-028-9-
where E is the tangent modulus corresponding to the stress on the convex side and E is 1 2 the tangent modulus corresponding to the stress on the concave or opposite side, then EI in equation (8) can be replaced by EI and equation (9) is a close approximation to the theory where P for the case is given by the expression cr If the tangent modulus is assumed to be approximately constant over the stress range existing across the section, then equation (13) can be written in the form known as the tangent modulus, or Euler s generalized formula, equation (1). Wang ( 12 ) states, The assumption of constant tangent modulus is usually justified by arguing that the bending is only slight; however, it would be interesting to see the influence of such an approximation on the magnitude of the buckling load. Wang ( 12 ) summarizes the importance of Southwell s method as: Actually, Southwell s method represents a much greater achievement than is usually recognized, This is because, for imperfect columns, the buckling load is not defined, and all actual columns are more or less imperfect; thus, in the strict sense, actual testing of buckling is impossible. Southwell s method, however, provides a theoretically sound basis for analyzing the experimental data--from the test results of an imperfect column the buckling load of the corresponding perfect column can be estimated. Application to Experimental Results Southwell s method agrees with theory only if the curvature is small because equation (8) is valid only for small curvature. However, deflection readings for a small amount of curvature are small, so that the ratio of deflection to load is not determined with a high degree of accuracy. Also, if the curvature is expressed as a series, the first term, given by equation (9), of the series will not necessarily dominate the expression for deflection. However, Southwell states ( 9 ), Trial alone can reveal whether the method will be successful in any particular instance. If the problem is approached from an analytical viewpoint, it can be shown that Southwell s method will predict the approximate buckling load, P, if the load-deflection cr curve approximates a rectangular hyperbola with asymptotes (14) (15) FPL-028-10-
The equation of the hyperbola is seen to be where k is a constant. If the curve given by equation (16) passes through the origin, it can be shown that equation (16) can be rearranged and reduced to (16) (17) which is the same as equation (10). Therefore, the approximate theoretical buckling load can be obtained by this method if the load deflection curve approximates a rectangular hyperbola. The deflections that were obtained in the experimental part of this report were not the true deflections due to buckling alone. The total movement of each column at midheight, which included movement of the ends as they rotated on the semicylindrical surfaces, was measured. Therefore, the deflection readings obtained for this report were greater than the deflections due to buckling. As the deflectionat midheight increased, the movement of the ends increased also. Southwell s method was applied to the 54- and 72-inch columns that were tested. Many of the columns which contained knots failed in such a way that the knot acted as a hinge. For this type of buckling, Southwell s method was not valid because the curvature could not be expressed as a sine function. An idea of whether Southwell s method was valid can be obtained by observing figures 9 to 12 representing typical test columns. If the load-deflection curve approaches asymptotically the value of load given by the inverse slope of the straight line, then Southwell s method was a good approximation of the theoretical buckling load. Figures 9 and 11 show that for southern pine, Southwell s method yields values for buckling loads which are much too high. Figures 10 and 12 show that Southwell s method yielded a fairly close approximation of the buckling loads of the Douglas - fir intermediate columns. Tables 11 and 12 show the amount of initial deflection at midheight that was measured and the approximate eccentricity due to nonhomogeneity that was obtained by Southwell s method for each column of intermediate length. The effect of initial curvature and eccentricity may be observed by comparing the maximum test load with the approximate buckling load. FPL-028-11-
Discussion of Results The southern pine specimens were No. 1 SR (stress-rated) Grade. The material was fairly uniform with many of the specimens containing the pith. All of the southern pine specimens contained knots with a maximum size of about 1.5 inches, The Douglas-fir specimens were Construction Grade. In contrast with the southern pine, the Douglas-fir specimens had a great deal of variability in rate of growth and specific gravity. The majority of the Douglas-fir specimens contained at least one knot with a maximum size of about 1.5 inches, but a few of the specimens contained no knots at all, The stress-strain results for southern pine obtained from compression specimens 10 inches in length showed very little variation. Figure 3 shows that no definite proportional limit was observed and that the tangent modulus was not constant for small values of stress. The low average value of the modulus of elasticity for southern pine was considered to be caused indirectly by the knots. The slope of grain around most of the knots was considerable. An idea of how much the slope of grain affected the modulus of elasticity can be obtained by observing table 1 which shows the distance from the geometric center to the point of application of the load. Each specimen was loaded first with the load acting at the geometric center. However, due to the slope of grain around the knots, the strains observed on the sides that contained knots were greater than the strains observed on the sides with negligible slope of grain. Therefore, the point of application of the load had to be moved away from the sides with the knots to obtain equal strain readings. Sunley ( 10 ) also found that knots decreased the value of the modulus of elasticity. The analytical approximations yielded results which were very close to the results obtained by using the average experimental tangent modulus values (tables 7 and 8, figs. 5 and 6). The Ramberg-Osgood approximation for stress versus strain was the only analytical expression in which all the parameters were determined from the experimental results. The method requires a large amount of work to determine the value of the parameter, B, for each species, but once a suitable value is found for a given material, the method yields satisfactory results. The Ramberg-Osgood empirical stress-strain expression follows the average experimental stress-strain curve very closely until the stress nears its maximum value, Therefore, the method is very accurate for predicting the strength of centrally loaded intermediate columns, but it predicts failure loads of short columns (L/d<10) that are too high. Timoshenko s empirical expression for the tangent modulus in terms of stress required fitting the expression to experimental values by assuming values for the parameter, c. FPL-028-12-
Once suitable values for the two species were found, the empirical expression yielded results which were easily and rapidly obtained and which approximated the experimental tangent modulus curves very closely for all values of slenderness ratio (figs. 5 and 6). The rectangular hyperbolic formula for stress versus strain required fitting the equation to the experimental values by assuming values for the parameter, k. However, the determination of values of tangent modulus at different levels of stress from equations (6) and (7) is a lengthy process, and the results were not considered to be satisfactory. Values of the parameter, k, were found for each species such that the results approximated those obtained from experimental values for intermediate columns but were too conservative for short columns (figs. 5 and 6). If a value of k is chosen such that equations (6) and (7) are good approximations for short columns, the predictions for intermediate columns will be too high. Theoretically, the maximum loads that were observed for short and intermediate columns should have been less than those predicted by the tangent modulus theory because nearly all of the columns were tested with an eccentricity due to nonhomogeneity and curvature (tables 11 and 12). Figure 5 shows that the average results for the southern pine short and intermediate columns fall very close to the tangent modulus curve. One explanation may be that the average tangent modulus values were higher for the short and intermediate columns than they were for the compression specimens. Each set of columns of different lengths was matched for specific gravity and moisture content, as shown in tables 1, 2, and 13 to 16. An effort was made to include the same size and number of knots in each specimen of a set regardless of the length of the specimen. A 1-inch knot within the 6-inch gage length for which strain was measured may cause a slope of grain over a considerable portion of the gage length. Therefore, the tangent modulus obtained is affected considerably by the strain taking place near the knot. For a longer specimen containing the same size knot and degree of slope of grain, the amount of strain near the knot is a small portion of the total strain, so that the tangent modulus values for the longer specimens should be higher. The above explanation is believed to be the reason for the results obtained for southern pine. However, no direct proof of the matching for tangent modulus for the different lengths was obtained. Knots were found to have little effect on the tangent modulus for the Douglas-fir specimens. A low specific gravity was the major factor in reducing the average value of modulus of elasticity for Douglas-fir. Figure 6 shows that the average results for Douglas-fir intermediate columns fall below the.tangent modulus curve. Figure 6 also shows that the average buckling stress determined by Southwell s method for the intermediate columns falls very close to the results FPL-028-13-
predicted by the tangent modulus theory. However, figure 5 shows that the average buckling stress determined by Southwell s method for the southern pine intermediate columns falls very close to the Euler curve. The fact that Southwell s method agrees with theory for the Douglas-fir intermediate columns and not for the southern pine requires an explanation. One explanation may come from the observation of the curvature of the columns as they buckled. Many of the southern pine columns buckled such that a knot acted as a hinge and the curvature approximated two straight lines. For this case, the theory used to obtain Southwell s method is not valid. In contrast, the majority of Douglas-fir columns buckled such that the curvature could be expressed as a sine function, and for this case the theory behind Southwell s method is valid. Conclusions The conclusions that were drawn from this study are as follows: (1) The tangent modulus theory predicts values of maximum loads that are too high for intermediate, structural grade, wood columns loaded at the geometric center. The discussion suggests a possible reason why the experimental results for the southern pine intermediate columns fall close to the tangent modulus curve. (2) The Ramberg-Osgood expression for stress-strain can be used to obtain accurate values of tangent modulus at all levels of stress, except near ultimate, once suitable values of the parameters are determined by experiment. (3) Timoshenko s expression for tangent modulus versus stress can be used to obtain accurate values of tangent modulus at all levels of stress, but it requires fitting to experimental results by trial and error. (4) The rectangular hyperbolic formula for stress-strain did not fit the experimental stress-strain curves satisfactorily at all levels of stress. (5) Southwell s method is valid for predicting the critical loads of the corresponding straight and homogeneous columns if the curvature can be expressed as a sine function such that the load-deflection curves approximate rectangular hyperbolas which become asymptotic to the horizontal lines, P = P cr (6) Predictions of the maximum loads attained by structural grade wood columns of intermediate length, loaded at the geometric center, require the use of formulas which take into account eccentricity of loading ( 4, 5, 10 ). FPL-028-14-
Literature Cited FPL-028-15-
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Figure 1.--Test setup for compression specimens, showing Marten s mirrors on a 4- by 4- by 10-inch specimen. Z M 123 766 FPL-028
Figure 2.--Test setup for short and intermediate columns. Four Ames dials, one on each side, were used with columns of 18, 36, and 54 inches. For the 72-inch length, two dials were used on two adjacent faces. FPL-028 Z M 123 765
Figure 3.--Average stress-strain and stress-tangent modulus curves for the sample of southern pine, No. 1 SR grade.
M 125 723 Figure 4.--Average stress-strain and stress-tangent modulus curves for the sample of Douglas-fir, Construction grade.
M 125 724 Figure 5.--Curves of stress versus slenderness ratio showing a comparison of test results with experimental and analytical tangent modulus predictions for a sample of southern pine. No. 1 SR grade.
M 125 725 Figure 6.--Curves of stress versus slenderness ratio showing a comparison of test results with experimental and analytical tangent modulus predictions for a sample of Douglars-fir, Construction grade.
M 125 726 Figure 7.--The Ramberg-Osgood equation of stress ratio versus plastic strain as applied to specimen of southern pine, No. 1 SR grade (No. P-2-10-SK).
Figure 8.--The Ramberg-Osgood equation of stress ratio versus plastic strain as applied to specimen of Douglas-fir, Construction grade (No. F-8-10-C).
M 125 728 Figure 9--Load versus deflection and deflection/load versus deflection for specimen of southern pine. No. 1 SR grade (No. P-14-72-LK).
M 125 729 Figure 10. --Load versus deflection and deflection/load versus deflection for specimen of Douglas-fir, Construction grade (No. F-8-54-C).
M 125 730 Figure 11.--Load versus deflection and deflection/load versus deflection for Compression specimen of southern pine, No. 1 SR grade (No. P-4-54-SK).
M 125 731 Figure 12.--Load versus deflection and deflection/load versus deflection for specimen of Douglas-fir, Constuction grade (NO. F-4-72-LK).
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