Welded Continuous Frames and Their Components THE POST-BUCKLING STRENGTH OF WIDE-FLANGE BEAMS. G. C. Lee and---t. V. Galambos

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.. Welded Continuous Frames and Their Components '.. ' THE POST-BUCKLING STRENGTH OF WIDE-FLANGE BEAMS by v G. C. Lee and---t. V. Galambos ThiS work has been carried out as part of an investigation sponsored
.. Welded Continuous Frames and Their Components '.. ' THE POST-BUCKLING STRENGTH OF WIDE-FLANGE BEAMS by v G. C. Lee and---t. V. Galambos ThiS work has been carried out as part of an investigation sponsored jointly by the Welding Research Council and the Department of the Navy with funds furnished by the following:. American Institute of Steel Construction American Iron and Steel Institute Office of Naval Research (Contract No.6l003) Bureau of Ships Bureau of Yards and Docks Reproduction of this report in whole or in part is permitted for any purpose of the United States Government. Fritz Engineering Laboratory Lehigh University Bethlehem, Pennsylvania June, Fritz Engineering Laboratory Report No. 205E.12 20SE.12 TABLE OF CONTENTS Page SYNOPSIS i 1. INTRODUCTION 1 2 PREVIOUS RESEARCH 4 3. DESCRIPTION OF THE EXPERIMENTAL PROGRAM 7 4. DESCRIPTION OF THE EXPERIMENTAL RESULTS 11 S. DISCUSSION OF TEST RESULTS SUMMARY 20 7 ACKNOWLEDGEMENT TABLES AND FIGURES REFERENCES 39 205E.12 i,, SYNOPSIS This paper presents a discussion on a series of beam experiments which were performed to study the behaviour of short, laterally braced wide-flange beams of as-rolled ASTM A-7 structural steel. Special reasons for this research were, (1) the determination of the maximum permissible spacing of lateral supports for beams subjected to constant plastic moment, and (2) the study of the post-buckling strength and the cause of failure of,relatively short beams~, It was found that for beams with L/r y ~ 45 failure is triggered by local buckling of the compression flange. All the beams showed,considerable post~buckling strength and in each case a plastic hinge of sufficient rotation capacity was formed. Beams longer than 45 r y failed due to lateral buckling. ~,' '.. .. 205E INTRODUCTION This report contains a discussion on a series of beam experiments which were performed to study the behaviour of short, laterally braced wide-flange beams in the post-buckling range. The test specimens were as-rolled ASTM A-7 structural steel members. A special reason for this research was the determination of the maximum permissible spacing of lateral supports for beams subjected to constant moment, and thus to check previously. (1) (2) developed theoret~cal solutions. Information on the required spacing of lateral bracing is of considerable practical importance in the plastic design of steel structures. One of the most fundamental assumptions in plastic theory is that plastic hinges will develop at certain parts of the structure, such that when a sufficient number of these hinges have formed the structure will fail as a kinematic mechanism. This assumption presupposes that the members of a structure are capable of delivering the required rotations for the formation of a mechanism... Because of the severe plastification of the member in the vicinity of the hinges, the internal stiffness of the member is reduced, and thus the possibility of the premature termination of the rotation capacity due to instability effects must be considered. Instability phenomena 205E.12-2 are here defined as those effects which cause a decrease of the applied loads when the deformation of the member is increased. For example, instability effects would cause a termination of the flat hinge-plateau in the moment-end rotation curve of a beam bent by equal terminal moments. For beams subjected to bending moment only, instability may be initiated by lateral buckling of the member between lateral supports, and by local buckling of the compressive plate elements of the cross section. Local buckling can be postponed until the material has reached the onset of strain-hardening by proper proportioning of the width-thickness ratios of each of the plate elements~3)(4) Instability caused by lateral buckling can be prevented by properly spaced lateral bracing~1)(2)(3) It is the purpose of this study to investigate the in-. fluence of the above mentioned instability effects on the rotation capacity of beams proportioned to develop plastic hinges. The selection of the length of the test beams was based on the results of a series of theoretical studies presented in Refs. 1 and 2. Since a simply supported beam subjected to uniform moment is the most critical loading condition with respect to 205E.12-3 lateral buckling, the testprogrirri was limited to this loading case. Another aim of this research is to investigate the postbuckling strength of s~ch beams. An explanation of this is.. shown in Fig. 2 where the' moment-versus-end rotation curve is illustrated by a schematic diagram. This curve consists of' four parts:' (1) the :~lastic.range (portion OA), in which the 205$.12-4 M-e relationship is linear, (2) the inelastic range (portion AB) where the curve becomes nonlinear because of partial yielding, (3) the plastic hinge plateau (portion BC) where the beam is fully yielded and it is not capable of sustaining additional moment, and (4) the unloading range (portion CD) where the beam is in an unstable equilibrium. The experiments have shown that lateral buckling usually starts at point B. If the length of the beam is higher than the critical length (L cr in Fig. 1), then point Band C coincide and no plastic hinge is formed (that is, the curve unloads at the start of lateral buckling). If, on the other hand, the beam is short enough, the start of lateral buckling has no effect on the formation of a hinge and failure is triggered by local buckling at point C, after sufficient hinge rotation has taken place. The flat portion of the curve between the slopes e B and e C represents the post-buckling strength of the beam..the length of this range is of utmost importance in plastic design, since it is in this region that a true plastic hinge exists PREVIOUS RESEARCH A theoretical solution for the det~rmination of the critical length and thus the required spacing of lateral bracing is 205E.12-5 pres~nted in Ref. 1. In this theory it is assumed that lateral buckling starts when the material has reached the onset of strainhardening. Thus the beam material is either strain-hardened or \ elastic; for a beam subjected to uniform moment, lateral buckling sets in when the entire beam has the stiffnesses prescribed by the strain-hardening modul~ Est (strain-hardening modulus) and G st (~hear modulus of strain-hardening material). Since the beam has already undergone considerable inelastic deformation when strain-hardening is reached, this the~ry assures sufficient rotation capacity before lateral buckling. In the analysis in Ref. 1 no elastic unloading of already yielded fibers is considered, and therefore this approach furnishes a lower bound solution similar to the tangent modulus solution of axially loaded columns. The critical spacing for all rolled wide-flange shapes was found to be about 18 r y (where r y is the weak axis radius of gyration) for simply supported beams subjected to a uniform plastic moment M P An extension of the work in Ref. 1 is presented in Ref. 2. This study considers unloading at buckling and thereby gives an upper bound of about 45 r for the critical spacing for rolled y beams. This solution is analogous to the reduced modulus load of axially loaded columns. 205E.12-6 The above mentioned solutions are Eigenvalue solutions in which the rotation of a member could not be directly taken into account. They imply that lateral buckling at the start of strainhardening is the termination of the usefulness of relatively short beams. A limited number of experiments have been carried out to verify the theoretical results. In Ref. 1 four tests are reported. These tests were conducted on simply supported beams using the third point loading. The distances between loading points (that is, the critical spans) were 22.4 r y, 44.6 r y and 71.6 r y respectively. It was found that the experimental results were much closer to the upper bound solution than to the lower bound solution. The experiments of Ref. 1 were not sufficient to lead to concrete conclusions because of the following two factors: (1) the variation of the unbraced span was limited and therefore the optimum unsupported length was not found. (2) The end conditions of the critical span (that is, the loading points) were not well defined. The loading device supplied unknown restraints into the system, thus prohibiting an exact evaluation of the results. However, they did indicate that lateral buckling could not be the reason of failure of relatively short beams. 205E.12-7 In summary, the purpose of the present experimental program is to conduct tests with a better control of test conditions and with a greater variation of the length in order to determine experimentally the optimum unbraced length for beams in plastically designed structures. 3. DESCRIPTION OF THE EXPERIMENTAL PROGRAM 3.1 Scope of the Experiments Five beam tests on lowf25 rolled beams were performed in this research program. A summary of these experiments is given in Table 1. In this table the test numbers, the variable length and the support conditions for each of the tests are giv~n. All beam specimens were divided by lateral supports into three equal spans (see the schematic view of the test set-up in Table 1). In each test beam the center span was subjected to constant moment and this was the critical span. Lateral supports at A, B, C and D were such that, at these sections, the beam section could neither twist nor deflect laterally, but rotations of the.. beam. in both principal directions were free. All beam specimens were 10WF25 as-rolled sections of ASTM A-7 structural carbon steel. They were processed from the same ingot and the same rolling, and each piece was subjected to 205E.12-8 identical cooling and straightening procedures. A summary of the material properties, based on laborat~ry coupon tests, is given in Table 2. In this table 0yf is the average yield stress of the flanges, ~ yw is the same property for the web material and a ya is the weighted average yield stress level of the average values of 0- for the flanges and of the web. y The full plastic moment values for each test are equal to the product of cr and the plastic modulus Z of each specimen section based. ya _..I,,' on measured dimensions. The strain at the start of strain-hardening f st was 14 y' the strain at initiation of yielding. Also included in this table are results of a stub column test for comparison with the co~~bn to 50 r, y i ~ i, i re~~~ts.* I,.' { :.. ::i: ; ~4C. The span lengths for these experiments variedf,rom 20 r y as shown in Table 1. Test LB-17 was performed to determine the full plastic moment value experimentally. oflb-20 agree closely with the M p values given in Table 2. Results During each test deflections and curvatures~ both in the loading and the late~al direction were measured. Details will be discussed later in this paper. 3.2 Description of the Test Set-Up and the Instrumentation. A scli.e~a,tic ;fr.on,~, :view of the test set-up is shown in Fig. 3a.. ~. '. '] , ,..;. -.:. -- ~.;:- ~ ~:. ;,~2;~.:.._.~..:;';,:t :.:t~~:~~_;._~,~.,~._. _ : -:. 1.. ~ ~ _~~...:.)i:. _. _,'~Results of these tl1ere notavei sged nita' rs'ya 205E.12-9 '. ... -Loads were applied downward at the ends of the test beam by two. ';' 'sok hydraulic jacks placed on a parallel pressure circuit. These jacks applied equal loads because they were controlled by a single valve of the hydraulic system. The load was transmitted to the end of the specimen by a two-inch roller in order to simulate a knife-edge loading condition (See sketch in Fig. 3c)~ The vertical supports, which correspond to the ends of the critical span, (see sketch in Fig. 3d) consisted of two high.strength steel rods. At the end of each rod, clevises connected the rod to the supporting girder and to the specimen by pins. This vertical connection permitted free rotation of the test specimen in the plane of loading and in the lateral direction. The supporting girder was bolted at its ends to the center of the beams of two parallel rectangular frames. These frames were fixed to the laboratory floor at their bases by bolts fitting into existing holes. (Fig. 3b). This loading and supporting. system provided simply supported conditions for the test beam in the plane of loading. Four sets of lateral supports were placed as follows: at., the two sections of vertical supports (third points) and at the two ends. As shown in Fig. 3, each set of lateral supports consisted of two knife-edge plates bolted to the webs of two 205E channels; these channels were welded to a base plate. The distance between the two knife-edges was adjustable by slotted holes in the channels. The test specimen was guided between these knife-edges. They permitted free lateral rotation and they prevented twisting and lateral motion of the beam at the support points. All lateral supports were fastened to the top flange of a heavy base beam which was fixed to the laboratory floor. Deflections in the plane of the web were measured by means of a surveyor's level and a travelling 1/100 in. scale which was held vertically at each of previously laid out marks. Readings along the length were taken as the scale was moved from point to point and the telescope of the level rotated. Similarily, by using a transit and a 1/100 in. scale, the lateral deflections of the tension and the compression flange along the test specimen were measured. Curvature of the test beam was obtained from strain readings and section geometry. Strain was recorded for each test by SR-4 strain gages. same for all tests. The locations of strain gages were the At the center section of the critical span five pairs of strain gages were attached (see Fig. 4). One pair 205E was located at the outer surface of the flanges in the plane of web. Four pairs of gages were attached at the four flange tips. The strain gages were warranted by the producer for accurate recording up to 2% strain, which is well above the strains expected in the experiments. An overall view of the test set-up is shown in the photograph of Fig. 11. In this picture transverse deflections are being measured and the beam is being inspected for yielding. 4. DESCRIPTION OF THE EXPERIMENTAL RESULTS In the following the results of the experiments will be described from the experimental data which were taken. That is, the behaviour of the various deformations will be discussed as the load is increased from zero to its final value. As mentioned earlier, test LB-17 (L/r y = 20) was performed for the purpose of determining the full plastic moment of the beam cross section. In this test a hydraulic universal testing machine was used and the load was applied at the third points. After completion of the test, coupons were cut from the elastic span and tested. Results show that M obtained in test LB-17 p was the same as that obtained from the tension coupons. 205E Moment~Curvature Relationship The principal results obtained from th,e experiments can best be illustrated by moment-curvature curves. The curvature values are computed from strain-readings at the center of the beam. In all tests the absolute strain values at both flange surfaces were the same in the elastic range. The neutral axis moved slightly towards the tension flange as yielding commenced. This was caused by unsymmetric yielding of the tension and compression flanges due to the presence of residual stress. A typical M-0 relationship (Test LB-10, L/r y = 45) is plotted in Fig. 5. From this figure it is seen that yielding was initiated relatively early (see arrow) because of the presence of residual stress in the specimen. Lateral buckling was first observed when M p was just reached (as noted from lateral deflection readings and the strain readings at the flange tips). In spite of lateral buckling, the test beam was able to undergo large additional rotations in the post-buckling range at a nearly constant moment.. value. Eventually the test specimen began to unload as local buckling of the compression flange took place. Similar behaviour was observed for all the other tests. It should be noted in Fig. 5 that the plastic moment plateau of test LB-10 is different .._...,', E from the theqretical prediction (shown dotted). Thisl difference is due to the reason that the predicted Mp value is calculated by using the average yield stress from coupon data, and the yield stress level of an as-rolled beam can vary considerably along its length. In test LB-lO the attain~ent of the full plastic moment was observed. Figure 6 shows the mom~nt-midspancurvatr.urecurves for all four tes,ts. Specimens LB-lO, 'II and 15, with their critical lengths equal to 45 r y, 35 r y and 40 r y re!7pe~tively, showed identical behaviour in the overall load-deformation curves. The termination of their usefulness was triggered by local buckliqg of the compression flange. Specimen LB-16, having L/r y = 50, however, showed no post-buckling strength since unloading -. immediately commenced after lateral buckling., Attention was paid to the fact that during the tests the lateral supports might provide fric~ion to the s-pecimen in the loading direction. In each test he4vy grease was used between the knife edges and the specimen and it was observed that this friction effect was negligible. released to zero at least once, In every test the load was (see for example the long dashed lines in Fig. 5) and it was found that the slopes of the loading and unloading curves were the same. If there were friction the 20SE slopes of these two lines would not be the same as the slope of the elastic loading curve. During each experiment checks were made to determine whether the moment in the critical span was constant. Fig. 7. This effect was also negligible as can be seen from In this figure the curvatures at the quarter point sections are plotted for LB-ll. The even load points at one quarter section coincide well with the odd load points at the opposite quarter section to form a smooth load-deformation curve. 4.2 Transverse Deflections In Fig. 8 the moment vs. vertical deflection curves are shown. The curves in this figure indicate the midspan deflection with respect to the ends of the critical span. Since the vertical supports had elastic deformations as load increased, the deflection readings shown in Fig. 8 have been adjusted to include this effect. The load-deflection curves exhibit similar behaviour to the M- curves shown in Fig. 6. The deflected curves at various stages of loading for test LB-16 are shown in Fig. 9. Very similar situations were prevalent in other tests. As can be seen from Fig. 9, lateral buckling of the critical span started rather early before large 205E deflection took place (approximately 25% of the total deflection). This implies that the post-buckling strength, which is the ability of further deformation without unloading after buckling, is quite large and it can be utilized ~or the type of beam investigated. 4.3 Lateral Deflections The lateral deflections for both flanges at midspan are plotted in Fig. 10. The tension flanges did not show much deforma ~ tion, whereas the compression flanges deformed consideralby. In all tests lateral buckling occurred in the inelastic range, and deflection increased gradually as strain increased until the compression flange buckled locally. This was observed to be the reason whichmntrolled the termination of the usefulness of beams with L/r y ~ 45. Lateral buckling, on th~ other hand, had very little significance with respect to plastic hinge rotations. Only for beams with L/r ~ 50 is lateral buckling the cause of y failure. For all the experiments the strain readings on the tension.. edge of the compression flange showed unloading immediately after lateral buckling started The photograph in Fig. 12 shows the five test specimens after the completion of the experimental program. In each case 205E the compression flange is shown on top. The extent of lateral deformation and the
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