H 62 54287

H 62 54287https://ntrs.nasa.gov/search.jsp?R=19930082943 2018-02-15T02:42:19+00:00Z

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL NOTE 2287

AN INVESTIGATION OF PURE BENDING IN THE PLASTIC RANGE

WHEN LOADS ARE NOT PARALLEL TO A PRINCIPAL PLANE

By Harry A. Wifliams

SUMMARY

Rectangular beams and I-beams of aluminum alloy 75S-O were tested in pure bending in the plastic range with the plane of loading at angles of 00, 300, 600, and 900 to the minor principal axis of the cross section. It was found that Cozzone's method of plastic bending analysis gave rea-sonable correlation between theoretical and experimental bending moments for the plastic range beyond the yield strength. The analysis is approxi-mate since it assumes that the neutral axis does not rotate or translate. The latter occurs in all cases and the former• occurs when the loads are not parallel to a principal plane.

Similar tests of an exploratory nature were made with angle cross section beams. A modification of Cozzone's method gave reasonable results for some positions of loading but not for others.

INTRODUCTION

Numerous solutions have been proposed for the problem of pure bending in the plastic range when the loads are parallel to a principal plane of bending. The earlier work of Saint Venant is presented in ref erenOe 1. Solutions based on an exponential relationship between stress and strain, have been used by a number of investigators (refer-ences 2 to tj). Semigraphical solutions are presented in references S and 6.

• When the problem of plastic bending arises in practice, it is usually not the simple case treated in the references. The plane of loading may not coincide with a principal plane of bending and shear or axial loading is often present. Hence, the present investigation was undertaken to determine whether a reasonably simple solution could be found for the particular problem of pure bending when the loads were not parallel to a principal plane.

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This work was conducted at Stanford University under the sponsor-ship and with the financial assistance of the National Advisory Committee for Aeronautics.

SYMBOLS

a distance to midpoint of neutral axis, inches

Aac change in ac, inches

c one-half depth of symmetrical beam, inches

c, Ct distance from neutral axis to outer compressive and tensile fibers, respectively, inches

eN distance from rotated neutral axis to outer fibers of symmetrical beam, inches

e general symbol for strain, inch per inch

ec, et outer fiber compressive and tensile strain, respectively, inch per inch

e1 to e 4 strains determined experimentally, inch per inch

f normal stress, psi

f, f proportional limit stress and yield strength, respectively, from average stress-strain curve, psi

outer fiber bending stress, psi

tine, mt outer fiber compressive and tensile stress, respectively, in angle beams, psi

f0 Cozzone's intercept stress, psi

roe' rot intercept stress for compressive and tensile area, respectively, of angle beams, psi

k Cozzone's beam section constant

kc,kt beam section constant for compressive and tensile areas, respectively, of angle beams

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Xb, X.fldistance to point on rotated neutral axis, inches

xn, yn j

E Young's modulus of elasticity, psi

I moment of inertia of cross section with respect to neutral axis, inches

Ix I,-, I- moment and product of inertia of cross section with respect to x- and y-axes, inche&4

1x ly moments of inertia of angle beam cross section with respect to principal axes, inchesL

'Nc' 'Nt moments of inertia of compressive and tensile areas, respectively, with respect to neutral axis of angle beam cross sections, inches

Kf plastic bending factor based on average stress-strain curve, psi

L moment arm, inches

experimental bending moment, inch-pounds

Mth bending moment computed analytically, inch-pound

Mth', Mth" bending moments computed analytically for compressive and tensile areas, respectively, of angle beams, inch-pounds

statical moment of one-half of syrmnetrical beam cross section, inches3

inc ' mt statical moments of compressive and tensile areas, respectively, of angle beams, inches3

W total load on beam, pounds

angle between principal axis and neutral axis, degrees

change in angle• , minutes

S angle between plane of loading and y-axis, degrees

0 angle between plane of loading and normal to rotated neutral axis, degrees

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TEST SPECINENS AND APPARATUS

Test specimens.- The test specimens were from a single shipment of annealed aluminum alloy 7S-O bars, which were 1 inch thick arid 2 inches deep. The rectangular specimens were merely cut to length from the full bar. The I-beams and angle beams were machined to the dimensions shown in figure 1. The maximum variation from the dimensions shown in the figure was 0.0O inch. Since this variation changed the analytical results by less than 1 percent, the section properties were based on the nominal dimensions throughout this report. Photographs of typical speci-mens after testing are shown in figure 2.

• It was decided to use the 7S-O material after investigation of other materials. It had been originally planned to use 2tS-T aluminum alloy, 1 but preliminary experience indicated that there was considerable difference between the stress-strain curves in tension and compression. In order to eliminate this variable as far as possible and also obtain a material which would reach the plastic range as soon as possible, the aluminum alloys 61S-0, 6lS-W,- and 7SS-0 were all sampled. In all cases the compression stress-strain curve was above the tension curve, but the 75S-0 had curves which were more nearly alike. However, when the ship-ment arrived, samples from it showed a greater variation between the two curves.

In selecting the annealed aluminum alloy for the test specimens, it was realized that they could not be bent to rupture. This had two advantages. There would be no risk to gaging equipment as a result of such failure and the distortion of bent specimens could be studied niore easily if the specimens were still in one piece.

Loading apparatus.- The arrangement of the loading apparatus used in the tests is shown in figure 3. A cross beam was mounted on tcp of a 20,000-pound Emery capsule which was in turn mounted on the upper or stationary head of a 200,000-pound-capacity Riehle testing machine. The capsule was supported on ball bearings and could be operated by a chain-driven screw so that the whole assembly moved laterally, thus keeping the suspender rods in a vertical plane if necessary. The loading jig in which the specimen was placed is shown in figures )4 and . It was mounted on the upper side of the moving head of the testing machine and, as the head was lowered, the jig applied third-point loading to the specimen. Provision was made for movement of the interior supports toward each other as the beam curvature became large by having each sup-port mounted on ball bearings and actuated by a centering device (not shown in the figures), which maintained symmetry with respect to a center line.

'New temper designations for alloys given are: .2LS-TL for 2tiS-T and 61S-Tt for 6LS-W. -

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it win be noted that the third-point knife-edge assembly was arranged to rotate about one axis. When the beam was to be tested at an angle to the vertical, this whole assembly was rotated and bolted in place at this angle.. The construction used then aflowed the beam to rotate the knife edges as it bent about the weaker axis and considerable restraint was thus eliminated. After the rectangular beams were tested, ball bearings were inserted at the knife-edge spindles in order to eliminate friction.

The fittings which connected the ends of thebeam to the suspender. rods are shown in figure 6. It will be noted that these fittings allowed for three degrees of freedom. The action of the loading jigs and end fittings can be seen in the photographs of figureè 7, 8, and 9. The photograph of figure 10 shows the general arrangement.

Strain apparatus.- Strains were recorded automatically by the Brown strip-type potentiometer which is shown on the table at the left in fig-ure 10. Strains iere measured at the outer corners of all beams by special electrospring-type gages which can be seen in the photographs. These gages consisted of two hardened steel knife edges, 2 inche's apart and connected at the top by a piece of spring steel 0.I38 inch wide and 0.016 inch thick. SR-t-e1ectric strain gages were cemented to the top and bottom sides of the spring steel. Since the resistance change was of opposite sign in the two gages, they acted together to unbalance a Wheatstone bridge and thus increase the sensitivity. The electrospring gages were mounted as shown in figure 9. Their accuracy will be dis-cussed later in the report.

SR-)4 electric strain gages cemented to the specimens were used rather sparingly and only for checking purposes in the elastic range when early experience indicated that they were liable to go out of action any time aTter the strain reached 1 percent. A comprehensive test in which 10 strain gages of type A-l2 were used is shown in fig-ure 9 for' beam C-i, which was a rectangular beam in the 600 position.

TEST PROCEDURE

Stress-strain characteristics of material.- The 7S-0 aluminum alloy bars were shipped in 12-foot lengths. The bars used in the inves-

tigation were lettered from A to G. Six beam specimens l9 inches long and two 13-inch sections from which tension and compression specimens could be taken were ctit from each bar. The first 13-inch section was taken,39 inches from one end of a bar' and the other, l9 inches from

the other end. Three tension specimens and two 'compression specimens

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were taken from each 13-inch section. The tension specimens were 8 inches long and were machined to 1/8 by 1/2 inch at the reduced sec-tion. One specimen was cut from the outside and the other two were cut from the interior of the bar. The compression specimens which were finally used were l/16 inch by 1 inch by 3 inches in length and might be referred to as the block type of compression specimen.

The tension specimens were tested in the conventional manner with an Ames dial extensometer having a 2-inch gage length. The block com-pression specimens were carefully squared and placed in the testing machine so that the compression stress was as uniform as possible. The deformation was measured with a 1-inch gage length Ames dial extensometer. Readings were taken until the specimen had buckled considerably, which was usually when the strain had reached 8 to 10 percent.

The results of all stress-strain tests are shown in figures 11 and 12. All specimens from bars A, B, and C were tested, but, since the scatter was not large, only a few specimens from the remaining bars were tested to check the uniformity of the shipment.

A number of compression tests were made on strip specimens which were 1/8 inch by 1 inch by 2 inches. The specimens were supported laterally by two pieces of cold-rolled steel between stiff springs. The specimen was 1/8 inch shorter than the cold-rolled steel side pieces and the load was applied by means of a steel strip approximately 0.11 inch thick. Strains of the order of 6 percent could be obtained by this method before the specimen buckled. However,'the stress-strain curve obtained from these tests was about 2000 psi higher than the corresponding curve obtained from the block compression tests. Since it was believed that this difference resulted from friction and lateral pressure, it was decided that the block compression specimens gave a better average value of the properties of the material. Hence, tests of the strip-type specimens were discontinued.

Calibration of strain gages.- The electrospring gages were mounted on the apparatus shown in figure 13 and calibrated against a 0.0001 inch Ames dial. When the Ames dial readings were plotted against divisions on the recorder chart, the relationship was very close to linear. The maximum variation in strain was 0.000 inch per inch for range 2 of the recorder (11 in. on the chart equals 10-percent strain) and was about the same for range 3 (11 in. on the chart equals 5-percent strain). The recorder chart contained 200 divisions in the 11-inch width. Readings were estimated to the nearest tenth of a division.

The gages were calibrated seven different times during the course of the tests. The maximum variation from the average of all calibrations for anyone gage was 3. percent.

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Tests of beams.-. Each beam specimen was scribed and measured with a,inicrometer to the nearest 0.001 inch and placed in the loading jig with the third-point supports set at the desired degree of rotation. The third-point loading was used in all but a few cases in which the moment arm L (fig. 1) was 9 inches instead of 8 inches. When the angle e was 300 or 600, the beam had a tendency to rotate about a longitudinal axis. Thi3 was prevented by wedging the beam in the interior supports. Under these conditions, restraint at the interior supports was largely eliminated by the rotation of the knife-edge assembly. There was, however, always an indeterminate amount of restraint against bending with respect to the x-axis of th beam.

The electrospring gages were attached to the corners of the beam as shown in figure . 9. The knife edges were set on previously scribed lines.approximately 2 inches apart. The gages were set at zero on the recorder chart after a small initial load had been applied to the beam, and an additional load was usually applied and removed to check the recorder.

The normal testing procedure was to apply an increment of load, sto'p the testing machine, and immediately start the recorder. The latter step was particularly important in the plastic range because of creep.

As the plastic range was reached, the curvature of the specimen decreased both the distance between the suspender rods and the distance between the interior load points. Changes in the latter were measured to the nearest 0.01 inch by means of the Ames dial shown at the lower left in figure 8. The changes in the suspender-rod distance were meas-ured to the nearest 0.01 inch by means of the longitudinal scale shown in the same figure. The change in the length of the moment arm L was computed from these values.

The moving head of the testing machine was operated at a speed of 0.1 inch per minute until the strain was of the order of 0.02S0 inch per inch, after which the speed was changed to 0.! inch per minute.

Specimens were removed when the strain was between t and 10 percent in the outer fiber, depending on the depth of the beam. Tests were usually continued intil the curvature was so great, that the interior knife edges were in danger of slipping toward the center or until other parts of' the apparatus threatened to interfere with the normal action of the jig.

When the angle e was 300 or 600 for the symmetrical beams and at all angles for the angle cross section beams, the outer ends of the specimens moved laterally- as much as 2 inches. However, the suspender

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rods were kept in a nearly vertical plane by operating the pull chain from, time to time, thus moving the capsule assembly laterally.

In the case of the angle beams, the rotation of one leg was meas-ured by clamping an arm to the longer leg. A protractor and plumb bob at the end of this arm enabled the rotation between the interior load points to be read to the nearest 0.2°.

ANALYTICAL PROCEDURE

When the loads are not parallel to a principal plane and the outer fiber stresses are in the plastic range, it is obvious that the principal of superposition ordinarily used in the elastic range is no longer suit-able. However, the problem can be solved by treating It as one in pure bending with respect to the true neutral axis. This is an alternative solution in the elastic, range, also, and is illustrated in a number of textbooks on elementary strength of materials.

The approach to the problem was to determine the location of the neutral axis in accordance with the conventional methods used for the elastic condition. The bending moment with respect to this neutral axis is equal to the cosine component of the resultant bending moment. It was recognized that, if the neutral axis translated or rotated as the plastic range was reached, it would be necessary to make a new solution for every value of the bending moment. However, it was determined during the investigation that large errors were not introduced by neg-lecting this factor and the solution in the plastic range could be based on the assumption that the neutral axis remained 'stationary.

Several methods of relating stress to bending moment with respect to the neutral axis were 'investigated. The use of a plastic bending. factor (reference 6) was discarded because a solution by this method. would require a large amount of tedious computation since the method is primarily applicable to rectangular cross sebtions which are perpen-dicular to the neutral axis.

A solution based on an exponential relationship between stress and strain was next investigated. It was found that when the product fe was plotted on logarithmic paper against the strain e, a straight-line relationship was obtained for that portion of the stress-strain curve beyond the yield strength. 'This relationship could be expressed in the form fe = aenl However, the expression for bending moment contained

the quantity ['yn cIA instead of the usual term for moment of inertia.

This' led to a simple approximate solution for the rectangular beams

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when the plane of loading coincided with a principal axis, but otherwise the approximate result obtained hardly justified the labor involved.

Further investigation indicated that the method presented by Cozzone (reference ) offered the best possibility for a solution which would be reasonably simple to apply, although the result would be approximate. Cozzone's expression was

Nth = • L[f + f0(k - l

(1)

• 2Q where k=

IN/c bending factor

The intercept stress f0 is related to the plastic.

Kf (reference 6) as follows:

fO = 6Kf

(2)

As presented by Cozzone in reference , the method is based on the assumption that the stress-strain curve for tension is identical with that for compression and the neutral axis is an axis of symmetry. Actually, equation (1) is approximate for any- cross section which is not rectangular even when the stress-strain curves are identical (see under "Discussion," reference 6). Since in the plastic range the stress-strain curves in tension and compression are not identical for most materials and the neutral axis must continuafly change position accord-ingly, equation (1) results in an approximate solution even for a rec-tangular cross section. However, the method gives reasonable results because of compensating error5 and because in the plastic range the restraining moment is not sensitive to even large errors in strain.

When the beam cross section is symmetrical but the plane of loading does not coincide with a principal plane of bending (figs. lL and l), equation (1) becomes .

Nth = cNcos [m+ f0( - 1)]

It is apparent from figures lIt and l that the neutral axis is not an axis of symmetry. However, the tension and compression areas of these beams contribute equally t . the restraining moment with respect to the neutral axis as long as stress-strain curves- in.tension and compression are identical. This further implies that-the neutral axis does not translate or rotate from its elastic-range position.

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It might be noted that, if the exa'ct location of the neutral axis is known, it is possible to modify equation (3) to obtain a more accurate solution. However, a designer would find the computation required for this determination extremely tedious. It would be highly impractical to locate the neutral axis by analytical methods.

Equation (3) can be modified for an urisynimetrical cross section such as the angle beam of figure 16(a). Additional errors are introduced by the rotation of the beam in the plastic range. Neglecting this factor, a rough solution may be made by dividing the cross section into a com-pression area and a tension area. The resisting moment of the compres-sion area is obtained by assuming the symmetrical cross secton shown in figure 16(b). The resisting moment of this beam is computed and divided by two. This is equivalent to computing 'Nc for one-half the cross

2Qm Qm

section only. Then kc = c = Cand the moment becomes

2INc/cc INC/cc

?Ith' = cos 0[fmc + f c(kc - i)] 4) Cc

A similar approach for the tension area (fig. 16(c)) results in the equation:

'Nt _____ = - 0[fmt + f(kt - 41 (S) Ct Co

The final -restraining moment is

Nth = Mth t + Nth t? (6)

Further investigation may develop a simpler solution.

It is easier to compare experimental arid theoretical bending moments in the plastic range than to compare outer fiber stresses; hence, the general procedure used in this investigation has been to take the experi-mental strain corresponding to a given load, determine the necessary - stress values from a representative stress-strain curve, and then insert these in the appropriate formula to compute the so-called "theoretical bending moment."

In order to check the experimental procedure as a whole, the shifting- of the neutral axis was taken into account when computing the theoretical bending moments by themethod of reference 6 for the

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rectangular beams in the 0°-and 900 positions. Since it was deemed impractical to take into account the shift and rotation of the neutral axis for the other cases, in which Cozzone's solution was used, it was necessary to decide whether the computations should be based on the tension curve only, as proposed by Cozzone, or on an average of the tension and compressioncurves. It was reasoned that if an average curve gave acceptable results, it would be safest to use this approach for materials having.decidedly different properties in tension and com-pression. It was apparent that solutions based on the tension curve could give satisfactory results only because of compensating errors. (Some comparisons between the two approaches are made later in the report.) Hence, the curves of figures 17 and 18 were constructed on this basis., Strains were plotted to an enlarged scale in . figure 17 to facilitate reading of values when strains were small.

The curves of figures 17 and 18 were constructed originally from the coupon results for bars A and B. It was found, however, that results would be affected only about l.S percent if the same curves were used for all bars.

RESULTS MJD DISCUSSION

Strain measurements.- Typical load and strain data are recorded in tables 1 to 3. The total load registered by the weighing apparatus is given in the first column. The moment arm is given in the second column, and it will be noted that it decreases as the curvature of the beam becomes larger. The experimental bending moment is the product of the moment arm times one-half the total load. The measured strains have been computed by multiplying the number of divisions on the recorder chart by the appropriate calibration factor.

The electrospring strain gages were designed to measure large strains and as a result could not be relied upon to give suitable accuracy in the elastic range. This deficiency was partially overcome by plotting strain readings for each gage against the experimental bending moment.. Selected results are shown for each group of tests in figures 19 to 22. The procedure adopted was to use the values from a straight line which was representative of the plotted points if the - resulting computed moments were within about 10 percent of the experi-mental moment. If the discrepancy was larger, it was assumed that the gages had not operated correctly and a line having approximately the correct theoretical slope was substituted. An average value of

E = lO.2 x 106 psi was -used in computing the theoretical slope.

The justification for this procedure is that the investigation was primarily concerned with the plastic range beyond the yield strength of

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the material. A difference of even 0.0010 inch per inch in the strain had negligible effect on the results in this range. Results for that portion of the range between the proportional limit and the yield strength were usually changed 2 to percent by the procedure. However, no great accuracy can be expected in this range as long as the computa-tion is based on an average stress-strain curve.

The corrected strains given in tables 1 to 3 were obtained from curves of Nt against e such as those in figures 19 to 22. These strains were recorded to the nearest tenth of a ten-thousandth when read from the curves. When they were recorded to two figures to the right of the decimal point for the elastic range, the values were com-puted by formula. The figure beyond the decimal point was dropped arbitrarily when the strain reached 100 x 10 4 inch per inch in the interests of consistency. Obviously, such figures ceased to have sig-nificance long before.

Strain readings were of the same order of magnitude in general for the specimens of the same group. The main exceptions to this were usually near the end of an experiment when the curvature had become quite large. The discrepancy was probably due to some extent to an inability to duplicate exactly conditions in each test. In the plastic range, relatively small differences in loading conditions may result in large changes in outer fiber strain. The strain gages were a factor also because it was found that certain gages sometimes became unstable when the curvature became large. The gages were attached in pairs and, if the spring tensions were higher on one side of the beam than on the other, the pair of gages occasionally rotated slightly. The test was always continued with them in this position since it was believed that the error would be more serious if the gages were readjusted to their original position. Such discrepancies did not greatly affect the com-puted bending moment because even large changes in outer fiber strain did nbt-a.ffect the stress factors after the stress-strain curves had flattened out.

The strain gages obviously measured the change in length of the chord distance between the knife edges. When the strain was 10 percent for a 2-inch-deep beam or percent for a 1-inch-deep beam, the dis-crepancy amounted to about O.O01 inch per inch. This amount should be added to the tension strain and subtracted from the compression strain. Since such discrepancies would have a negligible effect on the computed bending moment when the strains were of this order of magnitude, this factor was neglected.

Theoretical and experimental bending moments.-. The correlation

between the theoretical and experimental bending moments is shown graphically in figures 23 to 2S. The differences are computed from the

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f Mth \ expression - 1) x 100 percent. The elastic, semielastic, and

plastic ranges are shown in these graphs by indicating on the moment scale S the approximate values of the experimental moment at which the proportional limit f and the 0.2-percent offset yield stress were reached in the outer fibers of the beam in question.

The theoretical bending moments were computed as shown in tables t to 6. The quantities in bhe formulas which depend upon section prop-erties are given in figures lt to 16.

An inspection of figures 23 to 2 shows that the theoretical bending moment was within 10 percent of the experimental moment in the plastic range beyond the yield strength and was usually within l per-cent between the proportional limit and the yield strength. Some of the angle beams and one group of I-beams (e = 300 ) , where twist was involved, were excptions to this statement.

The twist mentioned above results from the fact that, when the beam

curvature is not in the same plane as the applied moment couple L, this

couple can be resolved intocomponents concerned with bending and a torsion component at. any point on the curved beam. The torsion is inaxi-mum at the support and decreases to zero at the center. When the maximum moment of inertia of a beam cross section is large compared with the minimum value and the torsional rigidity is small, the torsion component becomes an important factor in the failure f some beams through loss of stability.

The beams used in this investigation, with the exception of one rectangular beam, failed by excessive twisting when the angle e was 30°. It is quite probable that the beams in the 0° position did not fail in the same way because the tests could not be carried far. enough. This conclusion is consistent with the known behavior of deep, narrow beams. Had longer spans been used, in the investigation, torsional insta-bility would havebeen more prominent.

Alternate method of comjuting moments.- The theoretical bending moments were computed for the rectangular beaths and the I-beams on the basis that the outer fiber strain was an average of the maximum tensile and compressive strain. However, some comparisons were made on the assumption that the outer fiber tensile strain governed and that the compression stress-strain curve was identical with the tension curve, in accordance with Cozzone's original proposal. It developed that the

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computed bending moment was between and 10 percent higher for that region where the tensile stress-strain curve is above the compression curve. Through the middle region where the curves crossed there was little to choose between the two methods. In the later stages of plastiô bending, the tension curve resulted in moments which were between 2 and percent lower. Hence, it would be more conservative for the designer to use the average curve in the earlier stages of plastic bending and the tension curve in the later stages. Obviously, the percentages given above apply only to the particular material used in this investigation.

Deflections.- No particular attempt was made to measure the deflec-tions of the beams tested, although some approximate observations were taken. However, it is obvious from the photographs of tested specimens (fig. 2) that the final deflections were large. Deflection rather than resistance to plastic bending may govern many designs.

Change in position of neutral axis.- Dimensions from which the neutral axis could be located (see figs. lb to 16) were computed for all beams. Sample results are shown in the second and third columns of tables b to 6, primarily as a matter of interest. The distances were computed from the appropriate corrected strains by simple propor-tion. The results are rough approximations only, because these dis-tanc,es are quite sensitive to small changes in strain. Ihen a gage was near the neutral axis and the strains were relatively small, the effect of errors in the strain was quite pronounced. However, the results for a given group of tests indicated trends when considered collectively. Dimensions ar given in the tables to 0.001 inch merely to indicate the trend when variations are small.

Inspection of the values of cc and ct for the symmetrical beams in the 00 and 900 positions (figs. lb and 1) showed that the neutral axis moved toward the tension side of the beam as the stress exceeded the proportional limit. This trend continued until the yield strength had been passed, when the trend was reversed and the movement was toward the compression side. Theoretically, this was to be expected. When the compression stress-strain curve was below the tension curve, the neutral axis had to move down and put more of the beam cross section in compres-sion in order that the total compressive force on one side of the axis would balance the total tensile force on the other side.

The sample curves in figure 26 show the rotation and translation of the neutral axis for two of the rectangular beams in the 300 position. The lack of symmetry in this case results in both a rotation anda translation to balance the compressive and tensile forces on the cross

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section. The change in angle for the symmetrical beams (figs. lL and 15) may be computed from the expression

p=(9o0)_t1Xb+l(7)

The change in the midpoint distance ac is equal to

- Xb - xu 2

These expressions are derived from simple geometrical relationships.

The 600 rectangular beams followed the trend indicated in figure 26. The maximum clockwise rotation was of the order of 20, and the curve did not recross the axis. The maximum decrease in ac was about 0.03 inch, and the maximum increase was of the order of 0.02 inch. The I-beams at 30°and 600 followed much the same pattern, and the magnitudes were of the same general order.

The unsymmetrical angle beams rotated and translated for all load positions. The geometry- of figure 16 gives the.following expressions f or the changes:

= (j0 - p ) - tan (9) yn

= 0.261 - 0.760 + X (10) yn

In general, 'rotations were all clockwise. The order of magnitude was about 10 for the 0° position, 2° for the 30° position, and 3° for the 600 and 90° positions. The distance ac increased about 0.07 inch for the 0° position but showed little change for the 30° position. It increased 0.01 inch and 0.03 inch in the 60 0 and 90° positions, respectively.

Rotation of angle beams.- The rotation of the 2-inch leg of most

of the angle beams was measured. Typical results are shown in table 6.

(8)

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The rotation was clockwise (see fig. 16), and the maximum values were of the following order of magnitude:

Angle e (deg)

Rotation (deg)

0 3

30 7

60 2.

90 .2

Good correlation between theoretical and experimental bending moment was consistent with small rotation.

APPLICATION OF RESULTS TO PRACTICAL DESIGN

So far, plastic bending analysis has been applied, primarily to cast and forged materials. Other applications may be found as plastic bending theory is extended. This investigation is considered as a step in that direction.

The question might be raised as to whether results obtained for a soft material such as aluminum alloy 755-0 are applicable to the same materials inthe hardened state arid to cast materials. The question applies particularly to predicting the ultimate bending strength. There is no reason to believe that the approach suggested in this report will not apply to such materials. A limited number of tests to failure have been made on 2IS-T rectangular and I-beams when the loads were in a principal plane of bending (reference 6). When the loads are not in a principal plane of bending, the neutral axis does not coincide with a principal axis, but otherwise the general behavior is approximately, the same. The experiments reported in this investigation were continued until the stress-stiain curves had flattened out and there was little change in bending moment with increase of curvature. This condition will prevail un'til the beam ruptures, local crippling occur, or deflec-tion becomes excessive. .

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CONCLUSIONS

The following conclusions may be made from the results of the investigation as far as symmetrical cross section beams are concerned:

1. A modification of Cozzone's method of handling plastic bending problems gives approximate but reasonable results when the plane of loading does not coincide with a principal axis. For the particular material investigated, the correlation between computed and experimental bending moments in the plastic range beyond the yield strength was within 10 percent for all but a. few beams. The correlation will vary for other materials, depending on the uniformity of the material and the relationship between the tension and compression stress-strain curves.

2. If the properties of the material are similar to those of the material used in the investigation and the computed bending moment is based on the tensile stress-strain curve rather than on a curve which is an.average of the tensile and compressive curves, the results will be less conservative in the region of the yield strength and more con-servative in the later stages.

3. If the flanges become unstable and buckle along the edges in the plastic range, the ccxnputed moments will be on the unconservative side.

L. This investigation has been restricted to relatively short span beams. Nost of the beams tested in the 30° position failed from excessive twisting. This type of failure will be more pronounced as the span length is increased.

. No generalizations can be made from the results of the exploratory tests of angle cross section beams because the results may be quite dif-ferent for cross sections of different proportions from those used in the investigation.

Stanford University Stanford University, Calif., December 16, l9)7

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REFERENCES

1. Timoshenko, S.: Strength of Materials. Part II - Advanced Theory and Problems. Second ed., D. Van Nostrand Co., Inc., 191i.1.

2. Beilschniidt, J. L.: The Stresses Developed in Sections Subjected to Bending Moment. Jour. R.A.S., vol XLVI, no. 379, July, l9t2, pp.. 161-182.

3. Osgood, William IL: Plastic Bending. Jour. Aero. Sci. vol. 11, no. 3, July l9)4, pp. 213-226.

L1. Rappleyea, F. A., and Eastman, E. J.: Flexural Strength in the Plastic Range of Rectangular Magnesium Extrusions. Jour. Aero. Sci., vol. 11, no. t, Oct. 19U, pp. 373-377.

. Cozzone, Frank P.: Bending Strength in the Plastic Range. Jour. kero. Sci., vol. 10, no. , May l9t3, pp. 137-11.

6. Williams, Harry A.: Pure Bending in the Plastic Range. Jour. Aero. Sci., vol. lt, no. 8, Aug. 19Li7, pp. ).j57-Lt69..

NAGA TN 2287

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Page intentionallY left. blank

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ACA T 227

27

(a) Top view.

(b) Side view.

Figure 2.- Samples of tested specimens.

Page intentionally left. blank


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Page intentionallY left. blank


.4 .,-4 ¼)

<0) z ci)

cd

C

ci)

bD

11 1

.4-, 0

ci)

ho

cI

NACA TN 2287 35,

Page intentiOnallY left. blank


1 1 00 0 0 U0 0 0

ogo0° 80 (90

0

( p _______ 0-8

c8

0

0 '.1

0 0

8

36

2e

24

2C

w

NACA TN 2287

37

o .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 Strain, in./in.

Figure 11.- Tension test results. Material, aluminum alloy 75S-0; results from coupons 0.125 inch thick and 0.500 inch wide at re'duced section.

38

NACA TN 2287

n 44

40

36

32

28

24

C) 20

16

12

00'

• S •0o0O

• S

•

•

_______ _______ _______ oc9

0

• o'Ra° ____ ____ _____ _____ _____ _____- -

• 000

on

CD 0 0 5

S

00

08-S • S

—0-

(9 • •, -

0 0 _____ _____ _____ ____ ____ _____ ____ ____

0 - - S S

-S•• __

0 .01 .02 .03 .04 .05 .06 • .0'! .08 .0 .10 Stra1n in./in.

Figure 12.- Compression test results. Material, aluminum alloy 75S-O; results from block coupons 1 by 15/16 by 3 inches.

NACA TN 2287 39

Figure 13.- Electrospring gages mounted on calibration apparatus.

Page intentiOnallY left. blank


Neutra

(b) a = 30° or 60°.

NACA TN 2287

lane of loading

9=900

x

fc H

C -- ______ NeutralaxiS

± Ct

0 -- 2.00 -

(c) 0 = 900.

(a) e = 00.

Plane of loading

Section properties

I = 0.667 in.4

131 = 0.167 in.4

a 0 'N m cN

3J0 60°

66° 35' 810 47t

36° 35' 21°47'

0.242 .177

0.287 .252

0.852 .640

Figure 14.- Rectangular beam sections.

Lane of loading

i.00f

II

I Neutral axis

2.00

(c) 8 = 900.

Neutr

142

NACA TN 2287

I

Section properties

= 0.456

= 0.0436

0 'N cN

30° 60°

80° 35' 86°51'

50° 35' 26°51'

0.0546 .0443

0.0847 .0767

0.657 .554

(a) 0 =00.

Plane of loading

4Q

(b) 0 = 30° or 60°.

Figure 15.- I-beam sections.

NACA TN 2287

L3

Plane of loading

'1 0.25 -7. •2.00

7. 0.510

(a) \

,ti I

(b) (c)

Section properties

Rectangular axes Principal axes

= 0.2706 = 0.2854

I = 0.0456 I = 0.0308

= 0.0600

e 'c 't mc mt C Ct

0° 66°33' 52°35' 0.0330 0.0385 0.0929 0.0862 0.670 a0744 30° 83° 37' 390 39t .0167 .0173 .0635 .0622 .507 .518 60° 88° 13' 14° 15' .0140 .0177 .0581 .0558 .457 .572 90 0 91°32' 12°26' .0117 .0194 .0555 .0584 .416 .646

aDistance to lower corner of 2-in, leg.

Figure 16.- Angle beam sections.

NACA TN 2287

cdo -4

0

a)

•.0

0

CD

CD

0

CD

II

* ______'H \\\ fl _____

' \ ' ' ' C'

' \ C'

\ C'

\

\_\\ _____ _____\\ \

\\\\\

_• ______

-CD C .0

•TC ' .0 J 'SS0.X

- 0

U)

a)-4

1-;

D43

0 Cd

4-4

bo

Cd

0 -

.4-' •,-I

Cdcd '-4 a)

0

- a)

o C1)

Cd .4 '4-4

4)

(1)0 'z .•

^

czw 0 -0Cd

C-.

a)U)

•.-4 a) 0

•.- Cd Cd

-'-'a)

o CD

U]0

Cl)

U) flC a)

-

o 4-' 0

Q..

a)U)

0-.

Cl) ci

NACA TN 2287

ksi o 4 8 12 16 20 24 28 32 36 40

I I I I I I

Kf,ksi 0 2 4 6 8 10 12 14 16 18 20

44

40

36

32

28

U)

24

0

20 a) I-.

Co

16

12

8

4

0 .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 Strain, ifl./in.

Figure 18.- Relationship between stress, strain, intercept stress, and plastic bending factor based on average stress-strain curve. Material, aluminum alloy 75S-O.

Compression -- — — — —

Average sess-___ 7 / — — — — — Tension /

___

______/ /__

/

///

I, ,/

I/' — ___ /7 7

/Kf

///

/

NACA TN 2287

________t ___ _____ _______________ ______ _______________ ______ _______________ .4 1¼I\ - ____________ _______________

____—c —____ ____

\ \o—-- ____ ____

L z ——u— —OO—G—_ —(- --• —o—o--2Q-

---o.---o--o-- —°—°---°- 8

______ •10 c'1 0 '-4 '-4

sdpI-uT ' 4JAI 'U3UIOAI

00

Cd a)

Cd

bO

[H

—I a)

..-I ILl

NACA TN 2287

t7

I ____ ____

I ____ ____

______ __JIJ\ \ _-\

\\

\ ___ _ ___ ___ ___ ___ _______ ___ ___ ___ ___ ___ \\ __

\I

-... _______ _______ _______ _______ _______

-cIO.,c -_ --a-0

__________1

_______ _______ _______ 9 XD C.

sdpi- u uuxow upuq uuzpdx -

00

'-4

CD

0 a)

L8

NACA TN 2287

N - __ _______ _______ _______ _

-z--- __ ____C 0

______ _t_c C'1 0

sdpi- u 'uioI;

0

cD

C)

0 I-4

Cl)

C.)

Cd

U)

0

C-)

c';i

0 Q

fr

0) ho

0 4-I

(I]

C']

0)

ho .,-I

NACA TN 2287

L9

c) C'3 '-I 0 sdpI-uT 'w 'weuxoi

NACA TN 2287

10

(

- ic

w C)

i)

C 0

-' —ic -1

10

-10

-0

0p

00 0 Q) 0

0 0 00

0

0

o

p0

000

-:--°

__

-

• 1 ft. _

Rectangular be am

A-i

A-2

A-3

0 5 1015 2025 30 3540

Experimental bending moment, Mt, in.-kips

(a) e = 00.

Figure 23.- Difference between experimental and computed bending moments for rectangular beams.

NACA TN 2287 -

Si

10

-10

a) 0 a) Q 10

CD 0

-10

10

C

-10

0

-( ( 0 0

20

0

_2 2

0

•

0 0

o0 o

0 0 0

IIIIII]IiI

Rectangular beam

A -4

A -6

0 2 4 6 8 1Q 12 14 16 18 20 22 24

Experimental bending moment, Mt, in.-kips

(b) e = 30°.

Figure 23.- Continued.

oI° 1

•o0 000 Q

S

D oo

tr)_

0' 00 0

-0-0-' -

0 y p

0

___

- ____

___

____ --

00

0 00000

••Q_c•• 0p0

0 0

00 )-00C

f

—0--.; ---00 (0

H _

Rectangular be am

B-i

B -2

B-3

G-1

10

0

-10

10

0) 0 0) a'-lO 0 - 0

'-I

10

C

-'C

'C

0

-'C

NACA TN 2287

0 2 4 6 8 10 12 14 16 18 20 Experimental bending moment, Mt, in.-kips

(c) ,e = 60°.

Figure 23. - Continued.

NACA TN 2287

'C

C

-10

w 0

o C '-4

'-I

'C

ri:

-10

Rectangular beam B -4

00 -0-- - -O--

C) 0

0 0

0 00

0

0p

0 -0-----_•_ ____ ___°o°o6

____ ____ ____ ____ f. ____ ____ ____ ____

B-5

B-6

o 2 4 6 8 10 12 14 16 18 Experimental bending moment, Mt, in. -kips

(d) 9 = 900.

Figure 23.- Concluded.

NACA TN 2287

0 0

--

-

__ 0

__

0

__ 0

__ __

. 0 0o

00-o

0C

>0 - Ooç5_O_)p 00

-0-0 0 0C 0 -.0-D0 0 0

____ ____ ____f____ ____ f - ____ ____ ____

0 4 6 8 10 12 14 16 18 20 Experimental bending moment, Mt, in.-kips

10

•0

-10

-1-)

I ': -10

-

10

-10

I-beam

C-i

,

C -4

C -6

(a) e=0°.

Figure 24.- Difference between experimental and computed bending moments for I-beams.

NACA TN 2287

'C

[I

-'C

-4-)

C)

a)0 0 0

-10

10

ri LA

-10

0 I 0 _ _ __ (i\0

00_____

___- 'w'— .Jwqp0-

-00

p

0 0 0 0 0

- 0-

0 oc

• 0

Q)o 0 0 0___OQj2.__ 0—c--

0 00

- p o 0

____ ____ _____ ____ f _____ _____ ____

0 1 2 3 4 5 6 7 8 Experimental bending moment, Mt, in.-kips

I-beam

c-S

D-1

D-3

(b) e = 3Q0


56 NACA TN 2287

I-beam

10 --0------- 0 0

-----o------ • 0,..

-0

___

0 0 ____ ____ ____ C -2 -

-10----

-4-,

0 a—000

____0 0

-

°. 1C 0 '

Q 0 00 0o 0 o 0

--__ D5 0-0---

-10-----____ ____ fy ____

01 34Experimental bending moment, Mt, in.-kips

(c) e=60°.

Figure 24. Continued

NACA TN 2287

__ __ __ 0 __ 0

____

00

____[0 .0 ____

0 0 . 00 0 1Q p pp ____ ____ ____ ____ ____

00 0 00

C 00 o0-0 0

0 0 0 0• °0()co

00•

—0-0

0 f . f

0 1 2 .3 4 5 Experimental bending moment, Mt, in.-kips

(d) e 930•


10

0

-10

a) 0 a)

Q

0

-10

10

0

-10

I-beam

D-2

C -3

Oajx XE30 0 O-(

Oo000000

-____ D00 )Q_ ____

-0--S

____

__.O-

____ -0-0 -0-0

) 1 2 3 4 J 6

U w cz 0 0 '-I

'-4

10

-10

ib

-10

Angle beam

E -2

E -4

NACA TN 2287

Experimental bending moment, Mt, in..-kips

(a) e = 00.

Figure 25.- Difference between experimental and computed bending moments for angle beams.

NACA TN 2287

10

-10

a) C)

a) o 10 Q 0

0

10

0

-10

___ ___

I

° o Q2 ___

Ip.T

___ ___ ___ ___

___•. 0 ___ ___ ___ ___ _ ___ ___

0'0 0- __2

_-o---

_ _ _ _.0 -0-0

H . "__ _____

2 o—_< ___

0

___ ___

00 _______________ ___

f f ____

_

____ ____ HIlL___

Angle beam

E -3

E -5

E -6

0 1 2 3 4 5 Experimental bending moment, Mt, in. -kips

(b) e = 30°.


10

0

a) 0 - ci)

0 0 '—I

'- 10

-10

—--°od-°---_0_A

0

— -oo 000

------ 0(000

-

f f

)I IIII[l L21

NACA TN 2287

Angle beam

F-i

F -2

10

-10

a)

0 0

'—i 10

0

-10

Experimental bending moment, Mt, in. -kips

(c) e=60°.'


-0-0 0— -o--o Q210 -s 2 -

0-

0 00 H —0—

f f

-

LIII IT0 .1 2 3 4

Experimental bending moment, Mt, in. -kips

(d) 8 = 900.


Angle beam

F -4

F-5

NACA TN 2287

61

(0 0]

-1 U]

U] C'] (I)

a)

C))

o

0 (0 a)

C']

---)---- a --IIIIIIiIUhiIYI ----H---

----f ----—----- — I I ----r-1-o- -----

0 0

0 0

a

--

0 0

----- --

----1---

(0

0

(0 (V)

0

CD

Cd a) •0 cd

Cl)

a)

0

0 -I-,

Cd

Cl)

0 4-, cd

C']

a)

ri

o o 0 0 0 0 0 0 0 O co co c'l (0

OStM)TOI3

UtW 'V 'SIXB f.I1fleU O UO1OJ

--0 o o 0 0 0 0 0 C) cC c) C'] '—i —1 C']

Rqo 0. 0•0• 0• I I I

lIT ' 0EV 'UOflTUJL

NACA-Langley - 2-?-5J 975

Documents

H 62 54287