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1、PMFSEL REPORT NO. 82-5 AUGUST 1982 THE BEHAVIOR OF BEAMS SUBJECTED TO CONCENTRATED LOADS By Paul B. Summers Joseph A. Yura Sponsored by American Iron and Steel Institute PHIL M. FERGUSON STRUCTURAL ENGINEERING LABORATORY Department of Civil Engineering / Bureau of Engineering Research The University
2、 of Texas at Austin THE BEHAVIOR OF BEAMS SUBJECTED TO CONCENTRATED LOADS by Paul B. Summers Joseph A. Yura Sponsored by American Iron and Steel Institute Phil M. Ferguson Structural Engineering Laboratory Department of Civil Engineering University of Texas at Austin PMFSEL Report No. 82-5 August 19
3、82 A C K N O W L E D G M E N T S The report herein, which is the M.S. thesis of Paul Summers, was sponsored by the American Iron and Steel Institute. Their finan- cial support is greatly appreciated. The facilities of both the Phil M. Ferguson Structural Engineering Laboratory and the Department of
4、Civil Engineering at The University of Texas at Austin were used. The computer analysis was done using the University computer system. The authors also acknowledge the guidance of Dr. C. Phillip Johnson, who was the other member of Paul Summers supervisory commit- tee, Dr. Karl H. Frank and Dr. Rich
5、ard E. Klingner. Special thanks are extended to graduate students John M. Joehnk and to David R. Wright who helped in this research. The help of Hank Franklin and James Stewart in the prepara- tion of the model beams, and the assistance of Dave Whitney and Bob Gedies in the photography is sincerely
6、appreciated. Also acknowledged is the continual assistance of Laurie Golding and Maxine DeButts, and the friendly cooperation of both Tina Robinson who typed the complete manuscript and Deanna Thomas who so excellently drafted all of the figures. ii A B S T R A C T Tests on composite steel and concr
7、ete beams, continuous steel beams and simply supported plexiglass model beams, subjected to a concentrated load at midspan have demonstrated a buckling phenomenon where the bottom flange, which is primarily in tension, moves laterally. A complete parametric study was undertaken using a linear elasti
8、c finite element buckling analysis program to gain insight into this behavior. Results are presented for beams under a single midspan concentrated load with varying end restraint in-plane. The phenom- enon was found to be a combination of local web buckling and lateral- torsional buckling, the forme
9、r effect being dominant in simply supported beams and the latter in fixed ended beams. For beams with end restraint in between these cases (including most continuous beams), an interaction of the two effects occurs. It was found that bending stresses significantly influence the load at which there i
10、s a local web buckle between the top and bottom flange braces at the load point. Also observed was the fact that when the bottom flange was unbraced, the addition of a brace there did little to increase the buckling load if the beam was simply supported, but dramatically increased it when the beam w
11、as fixed ended. Design recommendations for beams subjected to concentrated loads are provided along with a design example. iii T A B L E O F C O N T E N T S Chapter Page 1 INTRODUCTION 1 1.1 Structural Behavior 1 1.2 Specification Provisions 6 1.3 Yura Theory 12 1.4 Purpose and Scope 12 2 PRELIMINAR
12、Y STUDIES 17 2.1 Analysis Program 17 2.2 Orals Plexiglass Model 23 2.2.1 Test Set-Up 23 2.2.2 Testing 23 2.2.3 Computer Analysis 29 2.3 Extensions of Yuras Theory 33 2.3.1 Stress Analysis 33 2.3.2 Generation of Theory 37 3 PARAMETRIC COMPUTER ANALYSIS 39 3.1 Background 39 3.2 Analyses 42 3.2.1 Varia
13、tion in Length 43 3.2.2 Variation in Tension Flange Width 45 3.2.3 Variation of Other Parameters 47 3.2.4 Rotational End Restraint 49 3.3 Interpretation 55 4 DESIGN RECOMMENDATIONS-SIMPLY SUPPORTED BEAMS 58 4.1 Tension Flange Braced 58 4.2 Tension Flange Unbraced 65 4.3 New Plexiglass Models 67 4.4
14、Other Considerations 69 4.4.1 Inelastic Effects 69 4.4.2 Post-Buckling Strength 74 iv Chapter Page 5 DESIGN RECOMMENDATIONS-FIXED ENDED BEAMS 76 5.1 Bottom Flange Braced 76 5.2 Bottom Flange Unbraced 77 5.2.1 Bottom Flange Models 78 5.2.2 Single Spring Model 83 5.2.3 Limitations 87 6 DESIGN RECOMMEN
15、DATIONS-BEAMS WITH VARYING END MOMENTS . 89 6.1 Bottom Flange Braced 89 6.2 Bottom Flange Unbraced 89 6.3 Continuous Beams 91 6.4 Design Example 95 7 SUMMARY AND CONCLUSIONS 98 APPENDIX 1 Basler Theory 101 APPENDIX 2 Yura Theory . . . 104 APPENDIX 3 Lateral-Torsional Buckling Examples 107 APPENDIX 4
16、 Results of Parametric Study 112 BIBLIOGRAPHY 116 v N O T A T I O N Af Area of flange a b Width of flange beff Effective length of web used in calculation of spring stiffness bs Width of stiffener Cb Distance from neutral axis of beam cross section to mid- thickness plane of bottom flange Cw Torsion
17、al warping constant C1 Coefficient for lateral-torsional buckling of a beam d Distance between flange centerlines dwc Distance from web-compression flange junction to neutral axis E Modulus of elasticity G Elastic shear modulus G,Gc,GD Joint bending stiffness ratio (subscripts apply to respective en
18、ds of the column) h Clear distance between flanges I Moment of inertia of cross section If Moment of inertia of flange for out-of-plane bending Iw Moment of inertia of unit width of web Iy Moment of inertia of cross section about Y-axis J Torsion constant Jtf Torsion constant of top flange vi Out-of
19、-plane geometric stiffness In-plane structural stiffness Out-of-plane structural stiffness k Stiffness of bottom flange lateral spring k* Minimum spring stiffness required to force buckling into second mode of an axially compressed flange = 4P*/L L Length of beam L1 Length between inflection points
20、of beam Length of beam between brace points Mcr Elastic buckling moment of beam MN Negative end moment Msm,M1a Smaller and larger end moments of a section of beam between braces. Ratio Msm/M1a is moment coefficient for lateral- torsional buckling. m Uniformly distributed moment P,PX,PY Applied conce
21、ntrated load; in X,Y directions, respectively. PBAS Buckling load given by Basler Theory PBFU Buckling load for fixed ended beams with unbraced bottom flange Buckling load for beams of end restraint defined by with unbraced bottom flange Pcr Elastic buckling load of beam PE Reference Euler buckling
22、load of pinned-pinned web column = PLOC Load to cause local web buckling PTFU Buckling load of simply supported beam with unbraced tension flange vii PVC Buckling load of beam when only vertical stresses are present P* Reference load at which an axially compressed flange would buckle in the second m
23、ode = R1,R2 Stiffnesses of rotational springs applied at ends of web column Nodal forces due to in-plane loading Nodal displacements due to in-plane loading Out-of-plane displacements (buckled shape) T Stiffness of lateral spring applied at end of web column t Nondimensional spring stiffness = Th/PE
24、 tf Thickness of flange ts Thickness of stiffener tw Thickness of web X,Y,Z Coordinate axes u,v,w Displacements in X, Y, and Z directions, respectively Coefficient representing negative end restraint = MN/(PL/8) Lateral deflection of bottom flange in buckled state Components of due to translation of
25、 top flange, rotation of top flange, and bending distortion of web, respectively Lateral deflection of tension flange under unit midspan lateral load = L 3/48EI f Rotation of the top flange at buckling Rotations about X,Y axes, respectively Constant used in computing lateral buckling load Poissons r
26、atio viii 3.1416 Stress Bending stress at web-flange junction under critical load to cause local web buckling Critical stress for web buckling in bending Critical buckling stress in the bottom flange Vertical compressive stress under load in web Maximum vertical compressive stress under load in web
27、Reference stress = P*/Af ix C H A P T E R 1 INTRODUCTION 1.1 Structural Behavior A beam subjected to gravity loads along a laterally supported top flange would be checked for yielding in bending and shear, local buckling of the compression flange and web behavior under any concentrated loads. The we
28、b would be checked for both buckling and yielding under each concentrated load. If the beam is simply supported, the above mentioned failure modes assume that the bottom (tension) flange does not move laterally. However, it has been observed experimentally that under certain circumstances, beams bot
29、h simply supported and those with any degree of end fixity or continuity, can reach a state of instability in a mode that previously has been rarely acknowledged. Tests at both Lehigh University and at The University of Texas at Austin have shown beams buckling with lateral movement of the bottom fl
30、ange. In a continuous beam this flange is primarily in tension (with some compression near the supports), but in a simply supported beam it is purely in tension. These beams had sufficient lateral bracing of the “compression“ flange to ensure that conventional lateral-torsional buckling had not yet
31、occurred at the loads under which they failed. Daniels and Fisher 11 conducted tests on simply supported composite beams with a span of 25 ft. between bearings. The beams consisted of a reinforced concrete slab 60 in. wide and 6 in. thick connected to a W21x62 steel beam by pairs of stud shear conne
32、ctors. The beams were supported at their ends by steel rollers that were free to move as the lower flange extended during loading. Figure 1.1 shows the beam cut at midspan after the test. At the failure load 1 Fig. 1.1 Lateral movement of tension flange in composite beam of Daniels and Fisher 11 tes
33、t 2 3 crushing of the concrete had progressed to full depth of the slab and considerable yielding of the steel beam had occurred, but astonishingly, the tension flange had deflected laterally. Bansal 4 and Yura18 have reported the results of a research program of over forty tests on three-span conti
34、nuous steel beams. The loading and support conditions for the test beams were so arranged as to simulate a real beam in a rigid building frame. The loads on the beams were applied to the bottom flange and upwards rather than in the gravity load orientation for ease of testing, and were only applied
35、in the central span (20 ft.). The side spans (4 ft.-6 in.) provided continuity to the beam at the interior support points where vertical stiffeners were welded to the web. Lateral bracing was provided at supports and load points. Figure 1.2 shows an Ml4xl7.2 steel beam loaded vertically upwards at m
36、idspan between the central supports, with lateral bracing on the bottom flange, which is primarily in compression, at the load point and at the 1/4-points of the central span. Furthermore, twist of the flange about the longitudinal axis is prevented under the load point by the loading mechanism itse
37、lf. The photograph shows the lateral movement of the top flange, which is primarily in tension, at the failure load. Figure 1.3, although from a different test beam, shows this movement in more detail. It is a photograph of the beam cut at midspan where the load was applied. Once again, as in the Da
38、niels and Fisher test, this movement of the “tension“ flange can be seen. Both inelastic and elastic buckling in this mode were observed in the series of tests. The phenomenon also occurred in beams with two or three equal loads, symmetrically applied within the central span. In all cases in which t
39、he lateral movement of the tension flange in the positive moment region occurred, the failure was catastrophic. There was very little post-buckling strength. Fig. 1.2 Lateral movement of “tension“ flange in three-span continuous beam 18 4 Fig. 1.3 Cross section at load point of three-span continuous
40、 beam 4 showing lateral movement of “tension“ flange 5 6 A third observation of the phenomenon was an acrylic (plexiglass) model beam of wide-flange shape developed by Oral, a graduate student of The University of Texas at Austin. The beam was simply supported over a 24 in. span and was 1-1/2 in. de
41、ep. The bracing and loading arrangements were the same as those for the continuous beam of Fig. 1.2. Figure 1.4 shows once again that the phenomenon of a failure mode, in which the flange primarily in “tension“ deflects laterally, clearly exists. A plan view of the buckled beam is seen in Fig. 1.5,
42、and a side view of it during loading is shown in Fig. 1.6. 1.2 Specification Provisions Both the 1978 AISC Specification 3 and the Structural Stability Research Council (SSRC) 17 Guide contain provisions enabling the designer to approximate the elastic buckling moment for I-beams, wide- flange secti
43、ons or doubly symmetric plate girders when such members are loaded by end couples in the plane of the web, or by transverse loads applied in the plane of the web. The load to cause lateral- torsional buckling of the beam can be obtained from the following equation: (1.1) where Mcr = elastic buckling
44、 moment C1 = buckling coefficient dependent on loading and support conditions obtained from information reported by Clark and Hill 9 and Salvadori16 = unbraced length of beam E = modulus of elasticity G = shear modulus Iy = weak-axis moment of inertia Fig. 1.4 Lateral movement of tension flange in p
45、lexiglass model beam 7 Fig. 1.5 Plan view of buckled plexiglass model beam showing lateral movement of tension flange 8 Fig. 1.6 Plexiglass model beam during loading 9 10 a 2 = EC w/GJ Cw = warping constant J = torsional constant Use of Eq. (1.1) in this form requires that both flanges are free to w
46、arp at the supports (i.e., pinned laterally). The 1978 AISC Specification also contains provisions, based on development by Basler, 5 to guard against web buckling due to concentrated transverse loading. For the case where there are no transverse stiffeners and the flange, on which the concentrated
47、load Pcr is applied, is restrained both laterally and torsionally about its longitudinal axis, (1.2) where = Poissons ratio h = clear distance between flanges tw = web thickness This check on web buckling underneath a concentrated load assumes that the other (unloaded) flange is braced laterally aga
48、inst movement. To develop the formula Basler assumed (Fig. 1.7) that the loaded area of the web underneath the concentrated load can be represented by a uniform column (of web), width h, under a triangular stress distri- bution (maximum stress under concentrated load and zero stress at other flange)
49、. Equation (1.2) then gives the buckling load of this fixed-pinned column. For complete details see Appendix 1. Since the failures presented in Sec. 1.1 are ones where the “tension“ flange moved laterally and the “compression“ flange remained relatively straight, it would appear neither Eq. (1.1) nor Eq. (1.2) is suitable to estimate the buckling load for these circumstances. 11 Fig. 1.7 Baslers Model 5 12 Furthermore, use of these equations can result in a considerable overestimate of the buckling load. 1.3 Yura Theory The previous observations led Yura 18 to develop a theory that pe
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