Pr 32.413/3:1828 RESTRICTED . OCT-4 1943 NATIONAL DEFENSE RESEARCH COMMITTEE ARMOR AND ORDNANCE REPORT NO. A-216 (OSRD NO. 1828) DIVISION 2 THE BEHAVIOR OF LONG BEAMS UNDER IMPACT LOADING by P. E. Duwez, D. S. Clark, D. S. Wood and H. F. Bohnenblust Copy No. 32 RESTRICTED LIBRARY UNIVERSITY OF WASHINGTON NOV 14 1950NATIONAL DEFENSE RESEARCH COMMITTEE ARMOR AND ORDNANCE REPORT NO. A-216 (OSRD NO. 18?8) DIVISION 2 THE BEHAVIOR OF LONG BEAMS UNDER IMPACT LOADING by P. E. Duwez, D. S. Clark, D. S. Wood and H. F. Bohnenblust i Approved on September 19h3 for submission to the Division Chief D. £. Clark, Official Investi- gator for Contract OHvIsr-3i|8 Approved on September 1.9, 19b3 for submission to the Committee Merit P. White, Secretary Division 2 /fccrf , -y d ,'^it 'c i t dbhn E. Burchard, Chief Division 2 Structural Defense and Offense/ % . ) ■ . ■ - ; A ' ; IPreface The work described in this report is pertinent to the projects designated by the Navy Department Liaison Officer as NO-11 and NS-1O9 and to Division 2 project P2-3O3. The work was carried out and reported by the California Institute of Technology under Contract OHIsr-3h8. Heretofore, th§ study of impact loading has been limited to the case of tension in rods, although the results of preliminary tests on centrally loaded plates have been presented, The fundamental theory underlying tension impact has been considered extensively, and a large amount of experimental work has been done which, in general, substan-' tiates this theory. This work is being continued in order to account for certain minor discrepancies between theory and experiment and to present the influence of impact on the tensile properties of specific metals and alloys. It is logical to extend this work to include the study of the behavior of beams subjected to an Impact load. For this purpose, it is preferable to consider first the case of very long freely.supported beams loaded at the center. To this end, experimental investigations have been made on small rectangular, freely supported beams 10 ft long, and a theoretical analysis has been made on infinitely long beams. The present report, which presents the results of this work, is divided into two'parts. The experimental Investigation is described in Part I by Duwez, Wood and Clark while the theoretical analysis is given in Part II by Bohnenblust. Each part may be read independently, and suitable cross references are given. Acknowledgment. Prof. H. F. Bohnenblust wishes to express his thanks to J. V. Charyk for his help in the computation of all numerical results of Sec. 6.60 and in making the drawings. Initial distribution of copies of this report Nos. 1 to 23, inclusive, to the Office of the Secretary of the Com mittee for distribution in the usual manner^ No. 26 to R. C. Tolman, Vice Chairman, NDRCj No. 27 to R. Adams, Member, NDRCj No. 28 to F. B. Jewett, Member, NDRCj No. 29 to J. E. Burchard, Chief, Division 2j No. 30 to W. Bleakney, Deputy Chief, Division. 2; No. 31 to W. F. Davidson, Office of the Chairman, NDRCj No. 32 to R, A. Beth, Member, Division 2; No. 33 to H. L. Bowman, Member, Division 2j No. 3h to C. W, Curtis, Member, Division 2jNo. 33 to C. W. Lampson, Member, Division 2; ' No. 36 to W. E. Lawson, Member, Division 2; No. 37 to H. P. Robertson3 1 No. 38 to F. Seitz, Jr., Member, Division 2; No. 39 to A. H. Taub, Member, Division 2; No. J4O to S. 3. Wilson, Jr., Member, Division 2; Nos. hl and h2 to R. J. Slutz, Technical Aide, Division 2; No. b3 to the Army Air Forces (Brig. Gen. B. W. Chidlaw); Nos. bb and b3 to the Corps of Engineers (Col. J. II. Stratton, Lt. Col. F. S. Besson, Jr.); No. b6 to the Ordnance Department (Col. S. B. Ritchie); No. b7 to M. P. White, Technical Aide, Division 2; No. I4.8 to Watertown Arsenal (Col. H. H. Zornig); No. b9 to Frankford Arsenal (Lt. Col. C. H. Greenall); Nos. 30 and 31 to Aberdeen Proving Ground (R. H. Kent, 0. Veblen); Nos. 32 and 33 to Bureau of Ordnance (Lt. Comdr, T. J. Flynn, A. Wertheimer); No. 3b to Naval Proving Ground (Lt. Comdr. R. A. Sawyer); No. 33 to'David Taylor Model Basin (Capt. W. P. Roop); Nos. 36 and 37 to Bureau of Ships (R. W. Goranson, E. Rassman); No. 38 to Bureau of Yards and Docks (War Plans Division); < No. 39 to Naval Research Laboratory (R. Gunn); No. 60 to Rear Adm. R. S. Holmes, California Institute of Technology; No. 61 to H. F. Bohnenblust, Consultant, Division 2; No. 62 to J. G. Kirkwood, Consultant, Division 2; No. 63 to N. M. Newmark, Consultant, Division 2; No. 6b to D. S. Clark, Consultant, Division 2; No. 63 to Th. von Karman, Consultant, Division 2; No. 66 to A. Nadal, Consultant, Division 2; No. 67 to P. W. Bridgman, Consultant, Division 2. The NDRC technical reports section for armor and ordnance edited this report and prepared it for duplication.ABSTRACT 1 Section 1 .00 2.00 0.00- ' ' ' boo £.00 6.-00' - ' APPENDIX • 7 .‘DO Figure- • 1. 2. 3- h-£. 6-7. 8-9. 10. 11. 12. PART I. EXPERIMENTAL INVESTIGATION Introduction......................................... 1 Testing procedure ................................... 2 Summary and discussion of the theory................... 9 Comparison of experimental results vn.th theory ...... 1£ Summary and conclusions ............................. h2 PART II. A THEORY OF THE PLASTIC.DEFORMATION OF ........ 'BEAMS OF'INFINITE'LENGTH UNDER IMPACT Introduction ...................................... hh Hysteresis behavior of the bending moment-curvature ' 'curve ............................................ £7 'Hysteresis’ assumption for the stress-strain curve ... £8 • • - ■ ■ pp sp of Fi gur e s .......................................................... Pag Beam testing assembly ............................... I4 Photographic record of a beam before and at end of impact............................................ £ Schematic deflection curves of an infinitely long beam during and after impact for different impact velocities ........................................... 11 Tension and compression stress-strain curves and bending moment-curvature curve for cold-rolled steel beams .. 12 Deflection curves fur cold-rolled steel beams ........ 12, Angle © and distance x versus vB for cold-rolled steel beams .................................. 19 Angle © versus impact velocity for cold-rolled steel bearns .•.••••••••••••••••••.••••.19 Deflection curves for cold-rolled steel beams .. 23 Final shape of deflection curves for cold-rolled steel beams at’ various impact velocities .... 2h Deflection curves for cold-rolled steel beams at end of impact and after beam has come to rest . 2hFigure Page 1h-l£. Tension and compression stress-strain curves and bending moment-curvature carves for annealed copper beams ................................................. 28 16-17• Deflection curves for copper beams ..............,..... 28,29 18. Distance versus for annealed copper beam....... 31 19. Deflection curves for copper beams .................. 33 20-21. ' Tension and compression stress-strain curves and bending .moment-curvature curves for hot-rolled steel beams .................................................. 3h 22-23. Deflection curves for hot-rolled steel beams .......... 3h,37 2g-2^. Angle 9 and distance x0 versus for hot-rolled steel beams ....................................... 38 26. Angle 9 versus impact velocity for hot-rolled steel beams ............................................ 38 27. Deflection curves for hot-rolled steel beams ........... 39 28. Bending moment-curvature curve and hysteresis effect. I4.6 29. Simplified moment-curvature curve ..................... £1 30. The function S ........................................ £1 31 • Distribution of H and k along the beam ............. 5>1 32. Function S for c " 0 .................................. 5'h 33. vJi and tan 9/Vb VQkQ versus ViAe ...................... 5?THE BEHAVIOR OF LONG 3MS UNDER IMPACT LOADING Abstract Part I presents the results of static and dynamic tests on beams of cold-rolled and hot-rolled low-carbon steel and annealed copper. These tests were made in order to check the theory presented in Part II. Deflection curves at the end of impact and after the beam has come to rest have been obtained. The results indicate that the principles set forth in the theory are correct. It has been shown that for materials such as low-carbon steel for which plastic deflection is localized, the observed deflection curve is closely approximated by considering the elastic behavior in the theoretical case. Two approximations of the moment-curvature curve are presented by which the deflection characteristics may be computed for certain cases. ... This investigation establishes the fundamental relations that will be employed in the investigation of the energy absorption in beams dynamically loaded. It is hoped that this will be directly applicable to the design of structures. Part II contains a theory of the plastic deformation of an infinite beam that.is .subjected to a concentrated transverse impact of constant velocity. Plane cross sections of the beam are assumed to remain plane and the bending moment is assumed to depend on the curvature of the deflection curve, not to vary linearly as is the case for the elastic deformation but according to a function that is obtained from the stress-strain curve of the material. The theory neglects both the lateral displacement of the cross sections against each other due to the shearing force and the rotary kinetic energy of the motion of the beam-. The experimental results which were obtained agree, well enough with the theory and show that the simplifications made were justified. PART I. EXPERIMENTAL INVESTIGATION 1.00 Introduction The purpose of this part of the report is to present the results of dynamic tests made on long freely supported beams and to relate these results to the theoretical' analysis presented in Part II. The dimensions of the beams and the experimental conditions were chosen in order to be as close as possible to the hypotheses used in the theory. Since the theory was developed for infinitely long beams, it was necessary to test beams for which the ratio of length to depth was very large. The tests -were made in such a way that the impact velocity remained- 2 - essentially constant for a predetermined period of deflection referred to as the duration of impact. This period was sufficiently small for only the central portion of the beam to be influenced by the impact. In this way, the effect of the supports was reduced as much as possible. In the theoretical analysis, approximations to the true bending moment-curvature curve were studied. Tests have been made with two metals in order to illustrate both approximations. The bending moment-curvature curve of cold-rolled low-carbon steel may be approximated by an extended elastic line terminating in a horizontal line in the plastic region. This is one of the approximations made in the theory. The ■ other approximation to the moment-curvature curve, which consists of an elastic line terminating in a straight line with an upward slope, may be used for copper. Neither of these metals shows a yield point, therefore, in order t-o complete the investigation, tests on hot-rolled low-carbon steel were made. The results are compared w/ith the theory based on an approximated moment-curvature curve of the first type. In order to study separately the influence of the impact velocity and the duration of impact on the deflection characteristics, two series of experiments were made on beams of each of the three materials. One series of tests wjas made with an impact velocity of 100 ft/sec with the duration of impact ranging from 0.5> to £ millisec. In another series, the duration of impact was held constant at 1.2f> or 3*3h millisec and the impact velocity was varied from 25> to 1^0 ft/sec. 2.00 Testing procedure 2.10 Beam impact testing machine. — A vertical tension Impact ~ 1 / testing machine described in a previous report— was used in modified form for the present investigation. A drawing of the beam testing assembly is showm in Fig. 1. The beam specimens wvere of rectangular cross section, 3/8 x 3/h in. or 3/8 x 1 in., and were 10 ft longs they 1/ P. E. Duwez, D. S. Wood and D. S. Clark, The propagation of plastic strain in tension, NDRC Report A-99 (OSRD No. 931 A- 3 - were tested in a position such that the long side of the rectangular cross section was in the plane of the motion of the beam. The ends of the beam were restricted from moving vertically by pins passing horizontally through the beam and resting in slots parallel to the beam, thus permitting horizontal motion and rotation about the pin. In this way the bending moment at the ends of the beam was zero. The impact of desired velocity and duration was applied by means of a 5~3/k lb hammer striking the beam in the center. The length of contact of the hammer along the beam was 3 A in. The hammer was accelerated to the desired velocity by means of rubber bands, and the veloc- 2/ ity measured in the manner previously described.— The duration of the impact was controlled by adjusting the distance D between the top of the test beam and the stopping anvils. The stopping anvils were so arranged that they stopped the motion of the hammer, but permitted the test beam to continue its motion downward due to its inertia after the impact load had been released. The impact velocity was measured within an error of +2 percent, and the duration of impact is known to +6 percent maximum error for short durations and to greater accuracy for longer durations of impact. 2.20 Measurement of the deflection curves. — In this investigation, the deflection curves of the beam at the end of impact and after the test is completed are measured. The deflection curve at the end of impact, that is, at the instant the hammer hit the stopping anvils, was obtained by means of a spark photograph. For this purpose, a l,Strobolux1’ (gas-filled electric discharge tube) was used which gave a single flash of light of about 30 psec duration. The tube was fired ■ by the hammer’s crushing and grounding an insulated wire placed on the stopping anvils. The shutter of a camera equipped with an fsi 1 lens was opened manually just prior to the test and closed just after the test. Agfa SSS Panchromatic film was employed and developed in Eastman DK76 developer. The side of the beam toward the camera was painted 2/ Reference 1. os /a3/r n O $ i *£ S'** 5 sn Us <9 ■HQ nyv i v wiawviF ^y>/vz/zzx/ i ucunj/oo /a&.^ A/^./. Bearn /'ejfrrjg assembly.- $ - Fig. 2. Photographic record of a beam before and at end of impact. This record is for an annealed copper beam, 3/8 x 3/U in. in cross section, tested at an impact velocity of 100 ft/sec for a duration of impact of milliseq.optical blacky and a thin white lino was painted on the side and adjacent to the top of the beam. A piece of heavy black cardboard with white horizontal and vertical inch-scales was placed directly behind the beam. Figure 2 is a typical photograph. In taking the photographic record, an exposure was made with the beam in the static position, and then a second exposure was made when the hammer came in contact with the anvils. The photographic negatives wore projected at an enlarged scale and the curves traced for purposes of analysis. The tracings for the curves are given in Figs. 8, 11, 18, 20, 2f and 26. About 1 in. of the beam from the center is obscured in the picture by the machine, and in the tracings in Figs. 8, 11, 18, 20, 2f and 26 the deflection curves were extrapolated to the point of impact. For this, reason and because of the elasticity of the hammer and the anvil a discrepancy is sometimes found between the.distance D measured on the testing machine and the distance D measured on the tracings. Til Tables III, IV, VIII, IX, XI and XII the distance p is measured on the machine and does not always agree exactly with the distance, p on the corresponding tracings. Another discrepancy exists, in certain cases for the point at which the deflection curve crosses the horizontal axis (called x ) as listed in the tables and read from the tracings Figs.8, 11, 18, 20, 2f and 26. The value of xQ listed in the tables is the accurate one since it was taken directly from the negative. This discrepancy is due to the fact that all fracings for the sake of comparison were put on the same page and with a common horizontal scale. Actually each tracing should have its own scale since the'distortion of the scale depends upon the posi-•tion of the camera. Furthermore, it should be remembered that a very slight error in the tracing introduces a large error in the value of xQ due to the small angle at the point of intersection. The deflection curve of the beam after the test — that is, the permanent deflection in the beam after it had come completely to rest — was determined by means of a simple mechanical beam-deflection measuring instrument. The instrument .consisted of a 60-lb/yd railroad rail mounted on concrete pillars. The head of the rail was machined to give a plane surface on which a rider fitted with ball bearings could be rolled- 8 - from, one end of the rail to the other. The test beam -was suspended above the rail, and the spindle of a dial gage mounted on the rider made contact with the underside of the beam. The deflection of the beam was determined at intervals of T in. before and after testing. In this way the elastic deflection curve of the beam under its own weight could be subtracted from the final deflection curve to give a curve showing only the permanent plastic deflection in the beam. The error in the deflection measurements is due to the limitation of the equipment and is within ±0.002 in. 2.30 Materials tested. — The materials used in this investigation consisted of rectangular bars of cold-rolled low-carbon steel, annealed bus-bar copper and hot-rolled low-carbon steel. The copper beams were annealed at 1200°F by passing through a furnace 12 in. long at the rate of 9 in./hr. The chemical analysis of the steels is given in Table I. Table'I. Chemical analysis of the steels. Component (percent) 'Cold-Rolled Steel Hot-Rolled Steel Carbon - 0.19- O..23 Manganese 1.03 .66 Phosphorus 0.018 .O1£ Sulphur .029 .026 Silicon .3h .oa 2.I|0 Static tension and compression tests. — Two static tension tests were made on each of the materials tested for which the nominal gage length of the specimens was 8 in. and the nominal diameter was 0.3 in. For each specimen, these dimensions were- measured within an error of ±0.001 in. The static tests were made following the technic 3/ previously described.— 3/ Reference 1.The compression tests were made on cylindrical specimens 0. 3o in. in diameter and 0.7 in. long nominally, that is, the ratio of length to diameter was 2. The dimensions were measured within an error of + 0.001 in. Since the specimens were so short, the compression stress-strain diagrams were obtained in several steps. The length of the specimen was measured with a micrometer, and it was then compressed with a particular load and the length measured again. This procedure was repeated for successively higher loads. The apparent, or engineering, stress was obtained by dividing the load by the initial cross-sectional area. Fran these data an apparent stress-plastic strain curve was plotted, and from this the elastic limit was determined. The elastic recovery was th.n added to each permanent strain to obtain the total strain and thus a compression stress-strain diagram was constructed. 3.00 Summary and discus si on of the theory Since the results of this investigation will be discussed in relation to the.theory presented in Part II, it is' desirable to consider some,of the most pertinent features of the theory as they relate to the experimental work. The theory gives a complete solution for the case of an infinitely long beam subjected; to a .concentrated impact of constant velocity. The notations used in this report are as follows. x Distance along the beam measured from the point of impact. x0 Point where deflection curve crosses the horizontal axis, y Deflection. k Curvature of the deflection curve. ke Curvature at the elastic limit. p Density of the beam. .. • A Cross-sectional area of the beam. • • I Moment of Inertia of the cross section. E Modulus of elasticity. CQ ■Strain at the elastic limit. Maximum velocity of impact without plastic deformation; for convenience, this will be. called the elastic veloc- iiz- ... ....... Vx Impact velocity.10 - cQ Velocity of propagation of the elastic wave, t Time. The maximum curvature that occurs at the point of impact is given for small velocities by the formula,^ k = viV^a7ei, which shows that the curvature is proportional to VT. This formula is valid up to a velocity for which k is equal to the elastic limit curvature kQ. For higher velocities the beam will react plastically. The minimum impact velocity at which a elastic curvature wri.ll occur in the beam will be denoted by Vg and is given by the formula.,— ve = kev®/£S. (1) The shape of the deflection curve depends on the relative value of VT with respect to Vg. 3*^0 Case 1 j Vi < Vg. — When the impact velocity Vi is less than the maximum velocity Vp for elastic behavior., the beam remains in the elastic state and the deflection curve is wavy, similar to curve 1_ in Fig. 3» In this case, the tangent at the point x = 0 is always zero.' The position of the point xn where the deflection curve crosses u A / the horizontal axis is given by the formula3 — Xo = 2.13vO7a vpy (2) This shows that whereas the deflection at the point of impact is proportional to the time, xQ is proportional to the square root of the time. After the impact has ceased, the beam returns to its original position since no point has undergone any plastic deformation (curve 2, Fig. 3). h/ Eq. (18), Part II. £/ Eq. (19), Part II. 6/ Eq. (20), Part II.- // - JPur'/'ng /mpcfct / £~/ast/c ctvr-nt eirerp/ po/'rt Pdter Impact 2 /Vo permanent at&d/e cd/on fr) V «? 4 Ptapd/c ang/e a/x=o Permanent ang/e od X-o r/asr/c civrv'atc/re o/ong dhe /Vopenmanee/ cc/rvadc/re a/onq oeqm the bec/rr/ w Pfasd/c ang/e. VQ. — When the impact velocity exceeds the maximum velocity for elastic behavior, plastic deformation takes place in some part of the beam. The theoretical, deflection characteristics are given for two different types of approximations of the bending moment-curvature curve. The first approximation consists of taking a horizontal straight line for the plastic region of the curve (curve 2, Fig. 3). The second approximation consists of taking a straight line with a given slope with respect to the horizontal axis (curve 2, Fig.13). These two approximations will be discussed separately. ■t 7/ 3.21 First approximation of the bending moment-curvature curve,— horizontal line for plastic range. If the bending moment-curvature curve is approximated by a horizontal line in the plastic range, the shape of the deflection curve depends, upon the value of Vx with respect to the quantity "2.08 7 Vg. ' If <'2.O8'7V'e$ the deflection curve, during the impact assumes the shape of a wavy curve as in elastic "bending.— A'localized-plastic deformation appears at the point, of impact - (.curve 3, Fig, 3) .giving rise to an angle § between the. beam and the horizontal. Such a discontinuity in the slope of the deflection curve is the result of .the-particular shape of the approximate bending moment-curvature 9 / curve. The- angle © is given by the equation,— tan 9 = -(3 y/3x)x=o=(V1 - VQ) ;/t 7^7^”. (3) After the impact is stopped, the beam comes to rest in the form of two straight lines meeting at an angle 9 (curve h, Fig,3); The angle 9 is the result of the local permanent strain at the point of impact. 10/ If—z > 2.087Vq, the deflection curve during impact (curve Fig. 3) does not differ essentially from the one obtained for the case 7/ Sec. 6.30 of Part IT. 8/ Sec. 6.31 of Part II. 9/ Eq. (2h) of Part II. 10/' Sec. 6.32 of Part II.-Ih- in which- V-l < 2.087Ve. The angle 9 is still present at the point of impact and, in addition, a permanent plastic curvature takes place near the point of impact. The angle 6 can be determined' by means of 11/ graphs as explained in the theoretical paper.— The exact computation of the motion of the beam after the impact is stopped has not been obtained. A qualitative description of the shape of the beam at rest is obtained by neglecting the kinetic energy 12/ of/the beam,— After the impact is stopped, the part of the beam that received a plastic curvature during impact will ccme to rest forming an arc of a circle. The final shape of the beam, as show by curve 6, Fig.3, will be circular for a distance x prolonged by a straight portion. 1 3 3.22 Second approximation of the bending moment-curvature curve, inclined straight line for the plastic range. If the bending moment-curvature curve is approximated in the plastic range by an inclined straight line, the deflection curve of the beam has the shape indicated schematically by curve 7, Fig. 3. No definite angle appears at the point of impact, but the angle is replaced by a localized plastic curvature upwards at the point of impact. The shape of the deflection curve is wavy as in the preceding case. However, it is possible for a downward plastic curvature to occur at a certain distance from the point of impactand, after the beam has come to rest, two plastic curvatures can exist, separated by an inflection point I (curve 8, Fig. 3). In this case, the position of the point xQ can not be expressed by means of a single formula, but, for a given impact velocity, it is proportional—^ to |/t. V\_/ Eqs. (2h) and (27), Part II. 12/ Sec, 6.7O of Part II. 13/ Sec. 6.hO of Part II. ih/ Eq. (lha) of Part II.- - b. 00 Comparison of experimental results with theory The comparison of experimental results with the theory is divided into three parts: (i) cold-rolled low-carbon steel, (ii) annealed copper and (iii) hot-rolled low-carbon steel. The first two metals serve for tests of the two types of approximation of the bending moment-curvature curves discussed in the summary. The last one was studied because of its yield point. The results of tests on each metal will be discussed separately and in the order given. b.10 Cold-rolled low-carbon steel beams,.3/6 x 1 x 10 ft. — b. 11 Bending moment-curvature curve. Two static tension stress-strain curves were made on specimens 0.3 in. in diameter and 8 in. long. The two stress-strain curves did not differ by more than 2000 lb/in? In Fig.b, the two curves merge into a single curve because of the small scale used for the'drawing. A compression stress-strain curve -was made following the technic described in Sec. 2.b0 of this report and is presented in Fig. b, together with the tension curve. In both curves, the stress is the apparent, or engineering, stress. From these two curves, the bending moment-curvature curve was computed for a beam'3/8 in. wide 13/ and T in. deep according to the method given by Nadai.— .For this computation the stress-strain curve must be-used for values of the strain which exceed that corresponding to the ultimate stress. For such strains, the value of the stress was taken to be the value of the ultimate stress. This curve is presented.in Fig. 3 The-proportional limit, ultimate strength, percentage strain in tension and in 'compression, and the bending moment and curvature-at the elastic limit are given in Table II. ' The approximate bending moment-curvature curve taken for the con-outation consists of two straight lines as indicated in curve 2, Fig. The plastic part of the curve is approximated by a horizontal straight line. The approximate curve is determined by the values of the bending moment and the curvature at the elastic limit — 8000 in. lb and O.85>3 x -10~2 in."1, respectively. 1f>/ Nadai, Plasticity (McGraw Hill,'1931)j P« 16£.16 - Table II. Results of static tests for cold-rolled steel. Proportional limit in tension (lb/in?) 68 £00 6£ £00 Ultimate strength in tension (lb/in?) ?££00 9£000 Elongation in 8 in. (percent) 2.£ 2.7 Reduction of area (percent) h8 £0 Rockwell S hardness 92 Proportional limit in compression (lb/in?) . . 60000 Apparent stress for 10-percent compression (lb/in?) 100.000 Proportional limit in bending (in. lb) 39hO Proportional limit for approximate moment-curvature curve (in. lb) 8000 Curvature at proportional limit in bending (in.-1) 0.1|2 x 10“2 Curvature at proportional limit for approximate moment-curvature curve (in.-1) 0.8£3x 10“2- h. 12 Comparison between theoretical and experimental deflection curves. The theoretical deflection curve based on the approximate bending moment-curvature curve has been computed for an Impact velocity of 100 ft/sec and a deflection of 1.0h in. at the point x = 0. The duration of impact in this case is 0.87 millisec. The value of the elastic velocity Ve, computed by Eq. (1) and using the approximate bending moment-curvature curve,, is l|0.9 ft/sec. Therefore, in this case, in which the impact velocity is 100 ft/sec, the beam must be plastically deformed. The theoretical deflection curve is plotted together with one experimental curve in Fig. 6. The angle 9 and the distance x^ from this theoretical curve are compared with the experi-mental values taken from two different tests made under the same impact conditions in the following table: Theoretical Angle 9 (percent)—8.0 Distance xo (in.) 12.6 Experimental 7.0 - 7-7 17.0 ■- 1/.8 16/ For simplicity, angle 9, which is small, is expressed as a percenta'ge slope, or 9 (percent) = 100 tan 9.~ 17 - The agreement, between the theoretical and experimental angles is fairly good. However, there is an appreciable difference between the theoretical and experimental values of x0. Two conditions may account for this: (i) the bending moment-curvature curve taken for the computation is an approximation^ and (ii) the beam is not infinite as required by 17/ the theory, hence the supports may have had sone influence.— In Fig.6 is plotted a second theoretical deflection curve which was obtained by considering the beam to act elastically. In such a case, the theoretical angle 9 at the point of impact is obviously zero,, which is not, of course, in agreement with the experiment, Regardless of this difference, it Is interesting to note that the elastic deflection curve is not so far from the experimental deflection curve. The theoretical value of xQ on the elastic basis — 15-2 in. — is even closer to the experimental value than the xQ computed by considering the beam to behave plastically. b-13 Variation of angle 9 and distance xn with time. According to the theory,— the angle 9 at the point of impact and the distance xQ should be proportional to i/t. A series of experiments has been made on 1O-ft cold-rolled steel beams with an impact velocity of 100 ft/sec. The results are given in Table III and the deflection curves are shown Table III. The angle 9 and the distance for cold-rolled steel beams at an impact velocity of 100 ft/sec and for different durations" of impact, ' Distance D (in.) Duration of Impact, t (millisecy (sec2) Distance (in?) Angle 9 (percent) 0.35 0.29 0.017 9.0 5.0 0.68 .57 .02b 13.0 6.5 1 .Oh .87 .029 15.8 7.7 bob .87 .029 n.o 7.0 1.26 1,05 .032 18.0 8.2 bb3 1.16 j - ■ 03b 19.8 9.0 22/ Sec. .6,70 of Part II. 1.8/ Eqs. (2b) and (27) of Part II.0 5 10 15 20 25 30 35 40 I I I I I I I I I I I I I I I I III I I I I I I I I I I I I I I I I I I I I I I Fig. 7» Deflection curves for cold—rolled steel beams 10 ft long at an impact velocity of 100 ft/sec for various durations of impact as indicated on each curve. Both horizontal and vertical scales are in inches. ■'“ /?in Fig. 7. The values of 0 and xQ have been plotted as functions of \/t in Figs.8 and 9. The agreement between experiment and theory is satis-factory. The two solid lines in the x0,v/'f'~ diagram are theoretical. One is computed on the hypothesis that the beam behaves plastically, while the other is computed on the hypothesis that the beam remains elastic indefinitely. Here, as previously shown in comparing the deflection curves, the experimental value of xQ seems to be in better sgneement with the value computed on the basis of elastic behavior of the beam. It is significant that the experimental points in Figs.8 and 9 lie on a straight line, which means that the experimental results fol-/- 19/ low the vt-law obtained in the theory.— The difference between the slopes of the experimental and the theoretical straight lines can be attributed to the approximation made for the bending moment-curvature curve. h.lb Variation of angle Q with impact velocity. From the theory it is possible to compute 9 as a function of V^ employing the approximate bending moment-curvature curve (given in Fig.3) for which the moment at the proportional limit is 8000 in. lb. The angle 9 is zero for impact velocities up to which in this case is h.0.9 ft/sec. From Ve to 2.087Vo — that is, between h0.9 and 83.3 ft/sec — 9 varies linearly with the impact velocity Vx [Eq. (3)]. For impact velocities above 2.087V„, the function 0 is determined by means of 20/ graphs.— The theoretical value of 0 as a function of V± is given in Fig. 10. To check these theoretical considerations, a series of experiments has been made with the impact velocity ranging from 23 to 130 ft/sec and the duration of impact about 1.22 millisec. The results are given in Table IV, and the deflection curves are given in Fig.11. The experimental values of the angle 9 are plotted together 19/ References 1h and 18. 20/ As explained in Part II, Fig. 33 •Table TV. The angle 9 and the distance xn for cold-rolled steel steel beams for a duration of impact of approximately 1.22 millisec and for different impact velocities. Impact Velocity (ft/sec) Distance D (in.) Durati on of Impact,, t (millisec) Distance xo (in.) Angle © (percent) 2f> 0.37 1.18 18.£ 0 (approx £0 0.7b 1.23 19.0 1.6 7^ 1.10 1.22 18.0 h.6 100 1 .h8 1.23 19.0 8.0 ' 12£ 1.85 1.23 17-5> 12.1 l£0 ' 2.^ 1,2£ , 18.0 1/.2 with the theoretical curve in Fig. 10. The measurement of the angle © from the experimental deflection curve of Fig. 11 is not accurate for impact velocities under 7/ ft/sec because the angle is too small. For 2£ ft/sec, the tangent to the deflection curve" at the point of. impact seems to be horizontal and the value of © is taken as zero. For 5>0 ft/sec, however, there is a measurable angle. The agreement obtained here between the' experimental and theoretical relations of © and V-l is not as satisfactory as the agreement between the experimental and the theoretical relations of © and t (Sec. h.13). For impact velocities up to 100 ft/sec, the discrepancy might be due to experimental errors. However, for greater impact velocities it is too large to be ascribed to such an error. On the other hand, for the relatively long durations of impact chosen for these experiments, the discrepancy might well be related to the influence of the supports. h.15> Variation of distance x0 with impact velocity. Equation (2) gives the distance x0 as a function of the dimensions of the beams and the time of impact t and shows that xQ is independent of the impact velocity. This formula, however, is valid only for impact velocities smaller than VQ. For higher velocities, the distance xQ cannot be expressed by a simple analytic formula, but a few computations have shovm that the variation of xQ is negligible within the range of impact velocities used in the experiments.As shown in Table IV,.the distance xQ found experimentally does not seem td' exhibit any systematic variation -with -the-impact velocity and can be considered as essentially constant. A"‘comparison’of the experimental and theoretical values of xQ has been given under Sec. h.12 for an impact velocity of 100 ft/sec.'- h. 16 Shape of deflection curve after the beam has come to rest. The discussion so far has centered on the deflection characteristics of the beam at the instant the motion of the hammer is stopped. That is, at the end of impact. The beam continues in motion after this instant by virtue of its inertia and comes to rest at some later time, assuming a deflection .curve that differs from the shape it had at the end of impact. The final shape of the deflection curve is measured, and its qualitative comparison vdth the curve indicated by the .theory is of considerable interest. For purposes of this conparison, a series of tests was made on beams 3/8 in. wide and 3/h in. deep at impact velocities ranging from . 23 to 160 ft/s ec. The distance through wrhich the hammer acted while In contact with the beam was approximately the same for all tests. It was chosen so that at the highest impact velocity the center of the-beam would not strike the base of the machine. Consequently, the duration of impact decreased with increasing velocity. The final deflection curve was obtained by the method described in Sec. 2.20 of this report. The conditions'of each test, the measured angle 0 and the total permanent deflection at the center of the beam are given in Table V. Some of the deflection curves are shown in Fig. 12. While these curves have been drawn vdth a sharp angle at the point of impact, there is actually an upward curvature extending.along the beam for about 1 in. from the center. This curvature could not’be measured accurately. From Eq. (1), evaluated at the true elastic limit ko = 0.36 x 10”a in.-1, the minimum velocity Vg at .which the first measurable permanent deflection angle occurs is 20 ft/sec. The minimum.velocity at which the first permanent deflection is observed in the test results is 23 ft/ sec or slightly below, which agrees favorably vdth the theoretical value. The deflection curves obtained f u? different impact velocitieso 5 10 15 I I I I I I I I I I I 20 25 30 I I I I I I I I I I I 35 40 I I I I I I Fig. 11. Deflection curves for cold-rolled steel beams 10 ft long for a duration of impact of 1.22 millisec at various impact velocities as indicated on each curve. Doth horizontal and vertical scales are in inches.-Pe dZecZ/or) (tnl) d^edZecZton (tn.J J)isZance X aZong /he hearn (tn.') o /o zo 30 4-0 ^o eo £oft/J*C nsfdjec .nfu’z' -f. \r coZd- ro/Zed s/ee/ heams 2 8.0 h-3 ■ 100 • 9h .79 9.7 6.0 and are presented in Fig. 13. The angle 9 for the final state of the beam is much smaller than the angle at the end of impact. The difference is large enough so that it cannot be attributed to experimental error. The reaction of the supports plays an important part during the motion of the beam after impacts and the variation of the angle 9 is related to this reaction. h.20 Annealed copper beams, 3/8 in. x 3/h in. x 10 ft. — h.21 Bending moment-curva.ture curve. Two static stress-strain curves were obtained for specimens 0.3 in. in diameter and 8 in. long. The difference between the two curves was smaller than 300 Ib/in? A compression curve was made following the technic described in Sec. 1 .h0. The tension and compression curves are given in Fig. 1li. From these curves, the bending moment-curvature curve was computed and is given in curve 1_> of Fig. 1£. The proportional limit, ultimate strength, percentage strain in tension and compression, and the bending moment and curvature at the elastic limit are given in Table VII. For the theoretical computations, the bending moment-curvature curve was approximated by two straight lines as shown in curve 2 of Fig. 1^. This approximation differs'.from the one taken for cold-rolled steel in that the straight line in the plastic range of the curve makes an angLe with the horizontal axis instead of being horizontal. The bending moment and the curvature at the elastic limit given by the approximate curve are listed in Table VII. h.22 Comparison between the theoretical and experimental deflec-tion curves. The theoretical deflection curve has been computed for an impact velocity of 97.1 ft/sec and a deflection of U.O in. at the- 27 - Table VII. Results of static tests of annealed copper. Proportional limit in tension (lb/in?) 2b00 2h00 Ultimate strength in tension (lb/in?) 29600 29^0 Elongation in 3 in. (percent) 38 ho Reduction of area (percent) 68 69 Rockwell F hardness h3 Proportional limit in compression (lb/in?) 2200 Apparent stress for 10-percent compression (lb/in?) 29000 Proportional limit in bending (in. lb) 69.2 Proportional limit for the approximate moment-curvature curve (in. lb) 208 Curvature at proportional limit in bending (in.-1) 0.032 x 1O“2 Curvature at proportional limit for approximate moment-curvature curve (in.-1) 0.09 x 10“2 point x — 0. The duration of impact in this case is 3»hh millisec. The curve computed from the theory and the curve obtained fron the experiment are plotted in Fig. 16, in which it will be noted that the shapes of the two curves are similar. An upward curvature is observed in the vicinity of the impact, wrhich occurs over an appreciable distance, and the angle A at x = 0 is negligible in contrast with the angle found in the case of cold-rolled steel. In the latter case, the angle may be attributed to a very localized curvature. The curvature in the experimental curve is greater than in the theoretical curve, probably as a result of the approximation taken for the bending moment curvature curve. The value of xQ ccmputed from the theory is smaller than the observed value (9 in. instead of ih in.). This difference may also be attributed to the choice of the approximate bending moment curvature curve and the influence of the supports.0t— o — Fig. 17. Deflection curves for copper beams 10 ft long at an impact velocity of 100 ft/sec for a duration of impact as indicated on each curve. Both horizontal and vertical scales are in inches.- 30 - A deflection curve vra.s also computed on the assumption that the beam remains perfectly elastic; this is also given in Fig. 16. In the case of the cold-rolled beam discussed in Sec. h. 12, the elastic deflection curve agrees rather well with the experimental curve (Fig. 6) except for the angle © at the point of impact. In the case of annealed copper, the elastic deflection curve is not in good agreement with the measured deflection curve. This fact shows that for copper beams the influence of the plasticity cannot be overlooked. Plastic strains are found a,t some distance from the point of impact, whereas in cold-rolled steel the plastic deformation was localized in a very small region near the point of impact. h.23 Variation of distance x0 with time. It has been shown in the theory that for a given impact velocity, the distance x0 between the point of impact and the point at which the deflection curve cuts the horizontal axis varies directly with This relation is general and independent of the shape of the bending moment-curvature curve. This fact has been verified by a series of experiments made with an impact velocity of 100 ft/sec and different durations of impact. The results are given in Table VIII, and the deflection curves are presented in Fig. 1 7. Table VIII. Relation between distance x0 and duration of impact for annealed copper beams at an impact velocity of 100 ft/sec. Distance D (in.) Duration of Impact, t (millisec) VT (secs) Distance / xo, (m. ) 0.75 0.62 0.02f> 1. hh 1 .2 .035 7.3 2.^0 2.1 .0h6 10.£ h.00 3.3 .0^7 1h.O f>.25> )4.h .066 1^.2 5.9b .070 17.h 7.00 $.9 •077 18.5In Fig. 18 it is 'shown that xQ varies linearly vdth \/t. The theoretical straight line drawn. on the same' diagram is far frcm the experi-mental points. Another approximation of the bending moment-curvature Fig.18. Distance x0 versus ’/E for annealed copper beam at an impact velocity of 100 ft/sec> turves; —theoretical: - experimental. curve could have given a closer agreement. This could be accomplished by taking a straight line in the plastic range of the curve making a larger angle with the horizontal axis. However, for such an.approximation, the shape of the theoretical deflection curve (Fig. 16) would have been far from the experimental curve in the region of the point of impact. In fact, the approximate bending moment-curvature curve cannot lead to a theoretical deflection curve that will be in perfect agreement with the experimental curve. In order to accomplish this, 'it ’would be necessary to'choose- different approximations for eich of the different portions of the beam since the different portions of the beam are subjected to curvatures of markedly different values. h.2h Variation of distance x0 with impact velocity. Deflection curves have been measured for beams tested at velocities ranging from 2$ to 1^0 ft/s ec for which the duration of impact was approximately~ 32 - constant (about 3.33 millisec). These curves are shown in Fig. 19} and the conditions of each test are given in Table IX. The principle Table IX. Relation between velocity of impact and the distance x^ for annealed copper beams tested at a duration of impact of 3.3S~~*nilli- sec. Impact Distance Duration Distance Velocity D of Impact, t X (in?) (ft/sec) (in.) (millisec) 29 1.00 3.3b 1-2.2 : .90 2.00 3.3b 1;2.0 79 3.00 3.3b T'3-0 ' 100 h.00 3.3b 13.0 129 9.00 3.3b 13.3 190 6.00 3.3b 1,3.8 result of these tests is the evidence they provide that the distance xQ docs not present any systematic variation with the impact velocity and that it can be' considered as independent of the velocity of impact. ■ •29 Shape of deflection curve after the beam has come to rest. Do’flection measurements made on annealed copper beams after the beams had come to rest gave deflection curves of the shape indicated by curve 8 of Fig.3> in which there is a reversal of curvature. It was difficult to check experimentally the minimum velocity of impact V for which a plastic strain takes place in the beam. This minimum velocity ccmputed by Eq. (1) is equal to 1.1 ft/sec. Such a small.velocity would have required a change in equipment without revealing any now important experimental result. h.30 Hot-rolled low-carbon steel beams, 3/8 in. x 1 in.__x 10 ft. h.31. Bending moment-curvature curve. Two static tests were made on specimens of this material 0.3 In. in diameter and 8 in. long. The dif ference between the two curves was not more than 1000 Ib/in? 'One of these curves appears as curve 1_ of Fig. 20. ■ A compression test was made, and curve 2, Fig. 20, is the resulting stress-strain curve. Both0 5 10 15 20 25 30 35 40 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I | I | Fig. 19. Deflection curves for copper beams 10 ft long for a duration of impact of 3.3U millisec at various impact velocities. Both horizontal and vertical scales are in inches•-J4- » § U 0 ’O 1 • \ 0 <* < - ^%L wi o . * A'* k ?■ 0 u M\ 1 \ \ L 2 «0 \9 X> N (J"/"?/*0') M'S-W' Ss! 3^$ r •> < S 3 ^£> to to tovn >x? K’S < ? 6 2 3'$ n u h -4§>> s^etension and compression curves have a very definite yield point. The proportional limit, ultimate strength and percentage strain are listed in Table X. The bending moment-curvature curve was computed from these Table X. Results of static tests on hot-rolled low-carbon steel. Proportional limit in tension (ib/in?) b3 300 hhooo Ultimate strength in tension (lb/in?) 67 300 68000 Elongation in 8 in. (percent) 23 20 Reduction of area (percent) 73 66 Rockwell B hardness 7h Proportional limit in compression (lb/in?) h2 300 • Apparent stress for 10-percent compression (lb/in?) 83/00 Proportional limit in bending (in. lb) 26I4O Proportional limit for the approximate moment- curvature curve (in. lb) 7O7O Curvature at proportional limit in bending (in."1) 0.28 x 10"2 Curvature at proportional limit for the approximate moment-curvature curve (in."1) 0.73 x 10"2 two curves and appears as curve 1_ °f Big. 21. This curve presents a change in curvature which is due to the presence of a yield point in both the tension and compression curves. The bending moment and the curvature at the elastic limit are also given in Table X. Curve _2, Fig. 21, is the approximate bending mement-curvature curve taken for the computations. The curve in the plastic range is a horizontal line as in the case of cold-rolled steel. The bending, moment and the curvature at the elastic limit for the approximate curve are given in Table X. The approximate bending moment-curvature curve was chosen in order to obtain a reasonable agreement between the experimental and the theoretical deflection curves as explained in the next sub-section.- 36 - h.32 Comparison between theoretical and experimental deflection curves. The theoretical deflection 'curve based on the approximate bending moment-curvature curve has been computed for an-impact velocity of 73 ft/sec and a deflection of 1.13 in. at the point xQ. The duration of impact for this case is 1.28 millisec. The theoretical and experimental deflection curves are compared in Fig.22. The angle 0 and the distance xQ obtained from theory and experiment are as follows: Theoretical Experimental Angle Q_ (percent) 6*9 Distance xQ (in.) 13.8 17*3 The agreement between the theoretical and experimental angles 0 and distances xQ is as good as was found for cold-rolled steel. This shows that the choice of the approximate bending -moment-curvature is justified, and so it can be inferred that under dynamic conditions the yield point is higher than under static - conditions- (see end of Sec. b.36) Another deflection curve, obtained by assigning that the beam is perfectly elastic, is also plotted in Fig. 22. As in the- case of the cold-rolled beam, the elastic deflection curve is not so far from the experimental curve, but at the point 'of impact the deflection angles 0 are riot in agreement. h.33 Variation of angle 0 and distance x0 with time. A series of experiments was made with an impact velocity of 100 ft/sec and different durations .of impact. The results are given in Table XI, and Table XI. The angle 0 and the distance xo for hot-rolled steel beams at an Impact"velocity of"100 ft/sec and for different durations of impact. Distance D (in.) Duration of 1mpa ct, t (millisec) ' r- 1 (secb) Distance xo (in.) Angle 0 ' (percent) 0.62 0.32 0.023 11.0 6. 3 1 .12 - 0.93 .030 1L|.2 8.0 1 .26 1 .03 .032 16.0 9-3 1 .U6 1 .22 .033 16.3 9.6 1.7h 1.U3 .038 18.0 10.3 2.10 1-73 .0h2 20.2 11 .0■ o I 2P Fig# 23* Deflection curves for hot-rolled steel beams 10 ft long at an Impact velocity of 100 ft/sec for various durations of Impact* Both horizontal and vertical scales are in inches*-J8- * V s 5 u -5 o this is a justification for the choice of the approximate bending moment-curvature curve. U.39 Variation of distance xQ with impact velocity. From the results of the tests discussed in Sec. Lj.. 13.? it is apparent from - h-1 - Table XII that the distance xQ does not vary with impact velocity. This coincides with.the results obtained from tests on cold-rolled steel and annealed copper. h.36 Shape of the deflection curve after the beam has come to rest. A series of tests was made for the purpose of verifying the computation of the minimum impact, velocity for which a permanent angle occurs at the point of impact by Eq. (1). It was shown in Sec. U.16 that this formula was approximately verified by tests on cold-rolled low-carbon, steel. As In the previous case, the permanent deflections of beams tested at different velocities were measured in order to determine the lowest velocity.at which a permanent angle was produced. Rectangular beams 3/8 in. wide by 3/h in. deep were used for these experiments. The theoretical elastic velocity given by Eq. (1) based on the static bending moment-curvature curve is 13-5 ft/sec. .The results of experiments made with impact velocities ranging frcm 21 to 7.5 ft/sec (Table XIII) show that the.minimum velocity at which, a measured permanent angle appears in the beam is about 26 ft/sec. This Table XIII. Measured deflection after impact and angle Q for hot-rolled steel beams at various impact velocities with the distance through which the hammer ns in contact with the beam approximately constant. Measurements were made after impact. Impact Velocity (ft/sec) Distance D • (in.) Deflection after Impact (in.) ■ Duration of impact, t . (millisec.) Angle © (percent) 21 0.520 . . 0.065 2.06 0 . 25 0 515 .185 1.70- .0 . 26 .520 .107 1 .60 O.O7 38 .520 • 5b5 1 .1 h. • 53 hh .. .510 .666 0.97 • 78 55 . .538 1 =255 .80. . . . 1.80 75 .500 1.910 .60 3.20 noticeable discrepancy between the experimental and the theoretical elastic velocity might be due to the rise of the yield point in hot-rolled low-carbon steel under dynamic conditions (Sec. h.32).- h2 - 5-.00 Summary and conclusions ■ ‘ The deflection of a beam subjected to an impact is influenced by the bending due to the bending moment, the displacement due to the .shear force and the tension in the beam produced by the localized deflection. The results of this investigation verify the relations established on a theory based on the bending alone. Therefore, it can be concluded that shear and tension are of secondary importance. The comparison of experimental and theoretical results is based on the deflection curves at the end of the impact and after the beam, has come to rest. In the former curve, the distance x.Q along the beam at which the deflection curve crosses the horizontal axis was measured, and for low-carbon steel also the angle 9 formed at the point of impact. The latter curve Tas used to determine the elastic velocity, that is, the lowest velocity at which a plastic deformation occurs. The angle 9 was also measured. It was noticed that 9 is twice as large at the end of the impact as it is after the beam has come to rest. At the end of the impact, the beam is thence under much higher strain than would be implied from the final deflection curve. ■ 'The fundamental result of the theory is the fact that the strain is not propagated along the beam at constant velocity as is the case in a longitudinal impact, on a rod. The strain depends hero on the ratio x2/t and, therefore, the distance xQ and the angle 9 must vary as for a given Impact velocity. This holds true independently of the shape of the bending moment-curvature curve end was found strictly verified within experimental errors. The computation of the factor of proportionality had to be based on simple approximations to the bending moment-curvature curve, and no perfect agreement can be expected. However, for cold-rolled low-carbon steel and copper beams, the agreement is satisfactory, particularly in the measurement of 9. The discrepancy, especially in xQ, may also be attributed to the influence of the supports. The dependence of xQ and 9 on the Impact velocity is also in satisfactory agreement with the theory. The higher the velocity of impact the more localized is the plastic deformation in theregion of the blow. Hence, the higher the velocity the less important are the supports. For copper, the elastic velocity is. too small to be of any significance, but for cold-rolled steel beams it was found in agreement with the theory. ■ To obtain satisfactory agreement between the experimental results and the theory for hot-rolled low-carbon steel beams, it was found necessary to replace the static bending moment-curvature curve by a dynamic one, in which the yield point is doubled. Indications that the yield point is higher under dynamic conditions than under static 21/ conditions have been discussed in previous reports.— It should be noted that for cold- and hot-rolled low-carbon steel, the deflection curve may be predicted on the basis of elastic behavior outside a neighborhood of the point of impact. This Is not the case for copper beams. The maximum velocity for which no plastic deformation occurs in low carbon steel is of the order of 23 ft/sec. For any material, the elastic impact velocity is equal to the longitudinal elastic impact velocity multiplied by the ratio of the radius of gyration of the cross section to half the depth of the beam. Thus for a structural I beam the elastic velocity is about 30 percent greater than for a solid beam of the same material. So far, these results cannot bo applied to the design of beams in structures to withstand impact loading. However, the work presented in this report is a first step and, therefore, the investigation is being extended to the study of the energy absorbed during impact for very long and for short beams. This study will not be limited to impact at constant velocity, but will include impact in which all the impact energy of the blow is absorbed by the beam. The ultimate object of this investigation is to bo able to specify the characteristics of a structure designed to withstand an impact produced by a given mass striking the structure with a given velocity. -21/ Particularly by P. E. Duwez, D. S. Wood and D. S. Clark, Factors influencing the propagation of plastic strain in long tension spe ciraens, hLfrtO" Heport A-1hy (udKU h o. 1 JuU).PART II. A THEORY OF THE PLASTIC DEFORMATION OF BEAMS OF INFINITE LENGTH UNDER IMPACT 6.00 Introduction The theory of the elastic deformation of a beam subjected to a 22/ transverse impact -was developed by Boussinesq—' and others several-decades ago. This theory will now be extended to the case of plastic deformation. The. success of this extension, rests on the fact that .Boussinesq ’ s solution gives a clue to the form of the solution for the plastic case. Boussinesq's theory is not an exact.study'of the •threer-dimensional stress-strain distribution in the beam, but is ' based on the simplified equations commonly used in this type of problem. The present theory uses the same simplifications, but the linear equation between bending moment and curvature is replaced by a more general relation. A summary and discussion of the results obtained was presented in Sec. 3.00 of the first part of this report. 6.10 The bending moment-curvature curve. — The differential equation for the elastic motion of a beam, is obtained from the equi- librium conditions, . '• : ... . 0 he- C 1 = DM/3x, (1) 3Q + U = 0 (2) lx and the relation, M = Elk = El 32y/ax2, (3) between the bending moment M and the curvature k. In these equa- tions the symbols have the following meanings. x Distance along the beam, y Deflection. k S2,y/t)x2, the curvature of the deflection curve. p Density of the beam. 22/ Boussinesq, Application dos potentials (Paris, 18£5). See also PPAB restricted reports Nos. T3 and lit. - hh - A Cross-sectional area of the beam E Young’s modulus. E Young’s modulus. M Q t cQ Velocity of propagation of elastic wave equal to VjS/^ . I Moment of inertia of the cross section. M Bending moment. Q Shearing force, t Time. If the beam is deformed plastically3 Eqs. (1) and (2) remain valid, but Eq. (3) is replaced by a nonlinear relation M-M(k) between the bend-23/ ing moment and the curvature. This function M(k) can be computed—7 from the stress-strain relation .- U6 - .y Fig.- 28. bending moment-curvature curve, ancy hysteresis- effect. The description just given corresponds' exactly to the assumption commonly made about the hysteresis effect for the stress-strain relationship. The two effects are not independent of each other; indeed, it is shown in the Appendix that the hysteresis behavior for the bending moment-curvature curve can be derived from the hysteresis behavior of the stress-strain curve. 6.20 The impact on a bean. The beam is assumed to extend infinitely in both directions and to be at rest until time t - 0. At time t - 0, the beam is struck at, the point x = 0 with such a force P that the point x = 0 is impelled to move at a constant given velocity Vx. For small values of the deformation of the bean is elastic. Boussinesq computed—1 the deflection y - y(x,t) for this case. His 2h/ Reference 22.results show that y/t is a function of x2/t alone. It happens that this fact remains true even when the deformation of the beam is plastic To verify this statement, it is convenient to replace the variable x by the new variable} 1 x2 where = vWM- (h) Taking y = then _ 1 [ft + 2v|fl»] k = (5) o2 y/ 3t2 = h2 f l,/t. The curvature is a function of p_ alone and the same is true for the bending moment M since M is a function of k. However, the shear-ing force is given by dti 0 = J a vt> dt] (6) and, in order to deal with functions of p alone, the quantity, s = 2a.3 Q = 2a." vTJ « , (7) El El dr[ is introduced. Equation (2) becomes S’ + 2pfc3/2f” = S’ + r^1/2 (2a2k - f’) =0, (8) and, by differentiation and Eqs. (>) and (7), it follows that . Sl‘ + SlSdk/dl = 0. (?) Any solution S of Eq. (9) for which S'(0) = 0 is a solution of Eq. (3) and gives the shear distribution for an impact of a certain velocity. Every other quantity can be expressed in terms of S by integrations. The upper limit of the integrals is infinite since- h8 - for x - oo the bending, moment, the curvature, the slope and .the-deflect! on are, equal to zero. Therefore, ■ ^S(h) To = BI 2a2 diq, El S(q) 2a3 \/fc k = -i. H 2 a2 Jr] ■ ^r> ps>(q) 2 dn + -r? 2 ■ t^ f s» (n) n n3/a dv| (10) (11) (12) (13) Equation (13), evaluated at q =0, shows that Vl = _t r^odn. 1 v"l Oh) Q = y 1 Finally a short computation shows that the solution y given by Eq.'(13) satisfies all the initial and boundary conditions of the impact problem. As stated before, the.solution is of the form y = t*f(x2/t). Expressed geometrically, this fact implies that two deflection curves for the. same impact velocity, but for different values of the time, become identical when y is divided by t and x is divided by Vt. In particular,.the point xQ at which y = 0 is proportional to \/t: x0^vF . Oka) The force P = P(t) necessary tp produce the impact is twice the shear Q evaluated at v] - 0, or P - S 3(0) ± . a3 \/t (1£) The force P is seen to be inversely proportional to the square root of t. The energy W is, therefore, proportional to this square root, or w = 2 n s(o) vt, „ 36.30 Elastlc def lect ion. — The preceding discussion includes, of course, the case of the elastic beam. Equation (9) takes the simple form S” + S = 0 which, together with S’(0) = 0 and Eq. (1I4), gives the solution, S = Sv^/tT V2 cos r( . (17) Evaluation of Eq. (12) shows that the maximum curvature occurs at x = 0 and is given by k(0) = V^/oATeT. (18) The elastic velocity Vp — that is, the limiting velocity of impact which does not produce a plastic deformation — is obtained by equating k(0) to the elastic curvature ko, (19) Pi y evaluating the integrals in Eq. (13) it is seen that xQ — that is, the first value of x for which y = 0 — is obtained by the formula, (20) = 2.13.g^2,13 where c is the velocity of propagation of the elastic wave. ... 6.hO. Plastic deflection: approximation I. — The fundamental equation, Eq. (9), can be solved only by graphical methods for a bending moment-curvature curve obtained from experimental tests. To carry through the computation, it is assumed in the present section that the bending moment-curvature curve consists of two straight lines (Fig.29). The first one passes through the origin, its slope is SI, and at the clastic limit (kG,MQ) it is joined to the second straight line. The slope of this second straight line is c2EI. For such a bending moment' curvature curve, the derivative dii/dk is equal either to El or to EIc? Equation (9) shows, then, that S is composed of sections of curves of the form A cos (d - ^1) o>r A cos (1/c) (^ - Pi). At the points of junction, both S and its derivative S’ are continuous.- $0 - A typical case is illustrated in Fig, 30. In the regions I to IV, S is given by: < I. S = A cos (l/c) t], 0 < II. S = -B sin (q - Kji), 'do < < III. S = -C sin(l/c) (dx- <(i) IV. S = Dsin (r»v - tq3), The eight unknowns v]0, <|i, d3> 2b 2b C, D, are determined by the five conditions of continuity of S and S’ and the follovjing three integral conditions: TO Z'^1 s dq = / ~ dr, - hVe, '1 4 The last equation is Eq. (ih) and the first two are obtained from the following facts: (i) at rj 2 the bending moment M equals Mej (ii) the change in M between and is equal to 2Mg; this is true, first, because the curvature decreases as p varies from to and thus the bending moment decreases along the hysteresis curve, and second, because the slope of this hysteresis curve is 51 until M has decreased by the amount 2Mg. Figure 31 illustrates these facts by exhibiting .the values of k and M for the different values of n . The actual computations proceed as follows. A pair of values and rj3 is chosen. From the continuity relations of S and S’, the ratios A:B:G:D and the values of di and. d0 are determined. Then, by trial and error, a pair of values, ds, yj3? is found for which the first two integral conditions are satisfied. The impact velocity is then computed. This calculation is repeated until the given impact velocity is obtained. It vail be noticed that the integrals involved are easily reduced to the Fresnel integrals, and their values may be 23/ taken from the table of C. M. Sparrow.— 23/ C. M, Sparrow, (Edwards Brothers, 123h)»- £2 - As the impact velocity decreases, region HI becomes smaller and disappears. Then only two regions need be considered, since II and IV combine into a single region. Finally, region I disappears when the impact velocity is the elastic velocity. /no For higher velocities of impact, the integral, Z S/Vq dn , '3 increases. Ihen its value equals 2VQ, a new region appears at q3 similar to the region III between hi and f|2. This complication occurs in the computation for the copper beam. 6.^0 Plastic deflection; approximation II (c = Q). — If the second straight line of the bending moment-curvature curve is taken to be horizontal, the value of c is zero and the computations of the preceding section are greatly simplified. It will be necessary to consider two cases depending on the magnitude of the impact velocity. 6.5>1 Case 15 < V! < 2.087Ve- This case corresponds to an impact velocity greater than the elastic velocity but small enough to avoid the region III of.-.the preceding section. The fact that V-,. - 2,O87Ve is the limiting value for this case will be shown presently. Starting from the results of the preceding section, let c tend to zero. Since q 0 < cff/2, the region I will shrink to the point q - 0. But the derivative S’ at q - 0 tends to a finite limit, different from zero. The function S*.is then of the form, I S = -B sin (t| - qj. The unknowns qx and B are determined from the integral conditions, (21) H PsinllLzJli) dn = 2V f - nJ Vn 15 (22) or B = 2\/(V? + Vp/n; l)x = Jn + te.n-1V.,/V1 . (23) The slope Cy/3x can be computed at x = 0 from Eo. (13). As S’(0) no longer vanishes, this slope is different from zero. It is given by tan9 = -(SJ (2h) The appearance of this angle is due to the special bending moment-curvature curve considered: as soon as the moment reaches M , the material flows and the curvature can increase indefinitely. Case I is valid as long as the moment M, for q 0, remains less than MQ in absolute valueo The integral representation of M [Eq. (10)] shows that M is an oscillating function of decreasing amplitude. The bending moment M reaches its first and largest extremum at H = Hx. The limiting velocity Vx for case 1 is determined from the equation, BEI 2? Z //’cosin(r>- qx) 7q d.q e EIV The coefficient 3 can be expressed in terms of Vjx and VqQ Substitution of this expression in the last equation gives 1 sm cos n , 1 / sinn . _ r._ , —p— dn - cos pi — / ■ —™ dq - sin Hi- cos qx for the limiting value of hi- 7ts solution is hi = 70-6°, and from Eq. (23) the corresponding value of the velocity is VT = 2.087V(1. 6.5>2 Case 2, Vx > 2.087Vo. For such velocities, region III should appear, but since v?2 - Mi < ctt/2, it must shrink to a point when c = 0. The influence of this point-region is to allow a break in the continuity of S’ . Here, The graph of the function S is given in Fig.32. S = -3 sin (q - hj.) S = -D sin (Q - Hi) < P 0 1 - u Hi 5 n The unknows 3, D and Hi are again given by the- integral c onditions, 2D / sin ~ ^i) «4 Vq ‘........... dq o sin (q - %) dq = liVf (25)- - Fig. 32. Function S for c = 0. and cos (rj - yj1) x n, /GDcos(n - *!i) V<3 dvj + D which corresponds to the value n 1 for q, the slope of the beam is continuous, but the curvature changes suddenly due to the discontinuity of S’ at y^. The change of the curvature is ke 1 . Ak = — — (B - D). 2V„ Ut The values of v, x and tanS/vtVgkg have been plotted from Bq. (25), (26) and (27) as functions of V-j/Vg in Fig. 33. For large values of impact velocity., the approximate expressions obtained are Hi = IVg/Vu and 3/2 e * 1 3 VQ * r K v' tan 9 = Vt a ~ ! \ / - / - — < v* /^•ta: / a ©//tVeke - \ / 1 1 I ! [ • 1 * 1 2.03? T T CD - 10 10 fig. 33- 'll and ten 6/’/tV k„ versus V-gV . 6.60 Numerical computations. — 6.61 Copper beam. A beam of annealed copper whose width b is 3/8 in. and whose depth h is3/h in. was considered. 'The computed bending moment-curvature curve is plotted in Fig. 15> of Part I. To apply the results of the preceding section., this curve was approximated by a curve consisting of two straight lines as indicated in Fig. 15>. In this approximation, the curvature ke at the elastic limit was taken as 0.09 * 10“2 in.-1, and the ratio c2 of the two slopes as 0.0196, that is, c = O.lh. The elastic velocity Ve is then 2.33 ft/sec, and the auxiliary number a2 used in the computation is 216.9 ft2/sec. The impact velocity is 97*1 ft/sec, and the deflection D at x - 0 is h in.36 - The resulting .deflection curve is plotted in Fig. 16 of Part I together with the experimental curve' corresponding to the same deflection D and to an impact velocity of 100 ft/sec. To show the influence of the plasticity of the material, an ‘’elastic” curve is also plotted in Fig. 16. This is the deflection curve of a beam for which the bending moment-curvature curve would remain elastic 4— M - Elk. It is seen that the correspondence between the plastic curye and the experimental curve is as good as can be expected with the rough approximation used for the bending moment-curvature relation. The computed curvature at x - 0 is k(0) - 0 26h in."1, while the curvature of the experimental curve at x = 0 is greater, which, reflects the fact that for such values of k the slope of the true bending moment-curvature curve is less than that of the approximating curve used. 6.62 Steel beam. A beam of cold-rolled low-carbon steel whose width b tbs 3/8 in. and whose depth h was 1 in* was considered. The computed bending moment-curvature curve is plotted in Fig.5 of Part I and is approximated again by two straight lines; the second one is taken to have slope zero. In the approximation, the elastic limit k0 of the curvature is 0.833 x 10~2 in."1. The elastic velocity VQ is h0.9 ft/sec, and a2 is 399.2 ft2/sec. The impact velocity Vx is 100 ft/sec, and the deflection D at x = 0 is 1.0h in. The resulting curve is plotted in Fig.6 of Part I together with the experimental curve corresponding to the same dat&i As in the case of the cold-rolled steel beam, the deflection curve corresponding to the elastic bending moment-curvature curve M * Elk is presented in the same figure. The elastic curve naturally has slope 0 at x = 0, but otherwise it seems to be in better agreement with the experimental curve than with the computed plastic curve. The angle of the plastic curve at x = 0 is computed to be tan 0 =0.08 and agrees with the slope of the experimental curve at the same point.6.70- The stopping of the beam. — The motion of the beam after the hammer has been stopped is too complex to be handled by the present analysis. However, a qualitative description of the shape of the beam at rest can be obtained by assuming' that at the end of the impact the beam is frozen and then allowed to find its position of equilibrium by a static deformation. Any portion of the’ beam subjected only to an elastic bending will return to a straight line, and the plastic portion of the beam will show a residual curvature. In the case c ~ 0, discussed in Sec. 6.5>O, the following shapes should then occur: For Vx < VpJ) the beam returns to its original straight position. For lTe < V < 2.087Vg, the. beam is straight barring an angle at the origin°s thus it has the form of a ”V." For 2.087Vp < Vl5 in addition to the angle at x ~ 0, the branches of the ”Vn will be arcs of a circle up to a distance x' and will be straight from ■ there on. These qualitative considerations do not explain,, of course,, the. experimental fact that during the stopping of the beam the angle at x = 0 is reduced. Further theoretical’ work’on this question is in progress; preliminary results indicate that the reduction of the angle is due entirely to the reactions at the supports. It may be added that the influence of the supports is already felt during the impact. The distance x0 at which the deflection is zero Is increased and the angle 9 at x ~ 0 is decreased. This fact may explain part of the discrepancy between the experimental curve and the theoretical one. ■ APPENDIX Hysteresis Behavior of the Bending Moment-Curvature Curve In this appendix, the dependence of the bending moment M on the curvature k will be discussed for the case when the curvature decreases after reaching a maximum value kx. For simplicity, a rectangular beam is considered for which the material has the same stress-strain relation in tension and compression.7•QO Hysteresis assumption for the stress-strain curve If the strain a decreases after reaching the maximum value the stress r is assumed to decrease along a curve similar to the stress-strain curve in the ratio 2:1. The value of cr is thus given by the formula, “ 26- for -&!<&< The symbol c refers to the stress measured along the stress-strain curve, and r to the stress measured along the hysteresis curve. The depth of the beam is h and its width is b. ' The moment for any curvature k is rkh/2 M = ~ / £r d£ . (A-1) k J o If the curvature decreases after increasing to the. values k=k1 the same formula (A-1) can be used,, but 22 must be replaced by the value