- Email: [email protected]
Behavior to complete failure of steel beams subjected to cyclic loading
Engineering Structures 25 (2003) 525–535 www.elsevier.com/locate/engstruct
Behavior to complete failure of steel beams subjected to cyclic loading D. Liu a, M. Nakashima b,∗, I. Kanao b a
Graduate School of Engineering, Kyoto University, Yoshida-Honmachi, Sakyo, Kyoto 606-5801, Japan b Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan
Abstract This paper presents an experimental study on steel beams subjected to cyclic loading to extremely large deformations. The study aims to collect information on beam hysteretic behavior up to complete failure, in the belief that such information is needed for the establishment of performance-based design. Test beams were about 1/10-scaled models, and effects of the RBS detail and lateral braces arranged at beam top flanges were examined. Behavior up to the cyclic loading amplitude of 0.06 rad was commensurate with behavior observed in many previous studies. Behavior in extremely large deformations from 0.1 to 0.5 rad amplitudes was significantly different from the behavior in large deformations (to the 0.06 rad amplitude). The RBS beam failed earlier in the reduced cross-section primarily due to strain concentrations at the section. Lateral braces also caused strain concentrations, leading to earlier fractures. Significant increase in the maximum resistance was observed in extremely large deformations for beams not braced laterally. Tensile axial forces induced in the beam according to the geometry change were responsible for the increase. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Steel beam; Earthquake; Cyclic loading; Complete failure; Lateral braces; Pinching
1. Introduction Performance-based design is an issue of great interest in the earthquake engineering community. In this design format, various limit states should be determined, and structures be designed to satisfy these limit states, each of which is checked against a different level of seismic hazard. Among the limit states, the collapse limit is most important, because this limit is associated directly with human loss. Here, collapse means the condition in which the structure cannot sustain gravity load any longer. Characterization of this limit, however, is still premature. Numerous studies, both experimental and analytical, have been carried out to investigate the damage and failure of structural members and systems, but the vast majority of them did not explore the final collapse for many reasons. On the experimental side, tests to duplicate the collapse are difficult to achieve because of limitations of loading facilities and also because of safety. On the analytical side, numerical analysis procedures
∗
Corresponding author. Fax: 81-774-38-4086. E-mail address: [email protected] (M. Nakashima).
that can allow for strong geometrical and material nonlinearities as well as for drastic changes in structural topology are very limited to begin with. Furthermore, experimental or observational data on collapse are so scarce that means to calibrate the reliability of numerical results are nearly nonexistent. Although it is a great challenge, quantification of the collapse limit is crucial for the establishment of performance-based design. Seismic design of steel building structures has greatly evolved in the past few years in response to serious damage to steel buildings observed in the 1994 Northridge and 1995 Hyogoken–Nanbu (Kobe) earthquakes [1–4]. Design guidelines [5–6] recently released, consider significantly larger deformation demands to structural members and frames than those considered before those earthquakes, and various means to ensure their deformation capacity have also been proposed. Even with the latest design guidelines, the true collapse limit has not been clearly stated, primarily because of the scarcity of data on this limit as well as lack of reliable analytical procedures to predict this limit state. In view of these circumstances, the writers and their group have been conducting a series of experiments for collecting quantitative data on collapses of steel struc-
0141-0296/03/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0141-0296(02)00164-5
526
D. Liu et al. / Engineering Structures 25 (2003) 525–535
tural members and frames. Here, collapse of structural members means complete fracture (separation) of members at which they cannot provide any resistance. Such information becomes an important ingredient for the evaluation of complete collapse of structural frames and also provides background data for future numerical analysis that will aim at collapse simulation. The tests on steel beam members were completed, and this paper presents details of the test program and the test results. Numerical analysis using the large-deformation theory was also conducted to supplement and interpret the experimental data as well as to examine the applicability and limitation of the numerical analysis.
2. Test program 2.1. Specimen and test setup Fig. 1 shows the test specimen used in this study. The specimen was fabricated by machine-grinding a steel bar. The length was 810 mm, and the cross-section was 70 mm (in depth), 30 mm (in width), 3 mm (in flange thickness), and 2 mm (in web thickness). These dimensions indicate that the specimen was approximately a 1/10-scaled model. The specimen had a slenderness ratio of 121 about the weak axis, and its flange and web width-to-thickness ratios were 5.0 and 35, respectively. Such scaling violates some important similitude, and therefore effects of some critical parameters such as local buckling and brittle fracture would not be duplicated. The objective of this test was to examine the behavior to complete failure involving extremely large deformations, which forced the use of reduced-scaled specimens. The size of the beam specimens was chosen as a compromise. The end of the beam was attached to a steel plate, which in turn was fastened to the test setup shown in Fig. 2. The beam end was not welded directly to the end plate to avoid possible fracture from this location. Instead, the end plate was holed, and the specimen was squeezed into the hole and welded from the other end of the plate. The test setup is shown in Fig. 2. It consists of a pair
Fig. 1.
Beam test specimen.
Fig. 2.
Test setup: (a) elevation; (b) plan.
of frames, each of which has three rigid columns that are pin-supported at the reaction beam (at the bottom) and loading beam (at the top). Two beam specimens are inserted in between the three rigid columns and fastened to the columns by high-tension bolts. A pair of these frames is arranged in parallel at a distance of 1.0 m, and a transverse plate is attached to bridge each loading beam. These transverse plates are arranged to restrict deformations of the pair of frames against the direction perpendicular to the loading direction. With this setup, four beam specimens (Fig. 1) were tested at one time. Two quasi-static jacks were used for loading, with each one attached at the end of each loading beam. The two jacks imposed an identical horizontal displacement so that the two frames were to move in parallel, and therefore the four beam specimens would take the same forced deformation. According to these loading and support conditions, the beam specimens were to sustain bending in double curvature. The jacks were operated by a computer-control loading system, which consisted of a PC that sent displacement signals, a controller that controlled the hydraulic pump, and a hydraulic pump whose oil flow was adjusted by an inverter motor. A digital displacement transducer and an analog load cell measured the displacement and force of the jack, respectively. Signals of those measuring devices were continuously fed back into the PC. Although a predetermined displacement history was applied in the tests presented in this paper, the loading system has a capacity of conducting a quasi-static pseudo dynamic loading test. Details of this system are presented elsewhere [7]. In some tests, the reduced beam section (RBS) detail was adopted, and portions of the beam flanges were trimmed. The trimming procedures followed those pro-
D. Liu et al. / Engineering Structures 25 (2003) 525–535
posed by Engelhardt [8], setting the full-plastic moment of the most reduced cross-section at 0.85 times that of the unreduced cross-section. In some tests, members having a channel section were placed on top of the top flanges of the specimens and in the direction perpendicular to the specimens’ longitudinal direction (Fig. 2). These members, named lateral braces, were arranged to mimic floor slabs that would provide restraint against the specimens’ torsion and out-of-plane deformations. Note, however, that the braces did provide restraint only at beam top flanges and beam bottom flanges were unbraced laterally. Three lateral braces were placed on each beam specimen pair and fastened to the specimen’s top flange by mechanical clamps. Four tests were conducted in this study, so that a total of 16 beam specimens were loaded. One test featured four specimens of the standard cross-section and without lateral braces, designated as ‘SS-1’. One test featured four specimens with the RBS detail and without lateral braces, designated as ‘SR-1’. One test featured four specimens of the standard cross-section and with lateral braces, designated as ‘SS-2’. One test featured four specimens with the RBS detail and with lateral braces, designated as ‘SR-2’. 2.2. Measurement and loading history To measure the strains of beam specimens in the longitudinal direction, strain gauges were glued on the beam top and bottom flanges. The deformations of a cross-section were measured by three wire-type displacement transducers, placed in three different directions, i.e. one oriented in the horizontal direction, another in the direction inclined by +45° from the horizontal, and the last in the direction inclined by ⫺45° from the horizontal. Those measurements were converted to the in-plane displacement (v), out-of-plane displacement (u), and torsion (qz). The deformations were measured at the three locations in each of the four beam specimens. Fig. 3 shows the loading history adopted in the test. The deformation was expressed as the horizontal displacement of the loading beam divided by the height of the rigid-columns (664 mm, measured between the two pins). This value is identical with the end rotation (qx) of the beam specimen because the columns were rigid. The amplitudes adopted (Fig. 3) were 0.02, 0.04, 0.06, 0.08, 0.10, 0.15, and 0.20 rad in terms of qx. For each amplitude, loading was repeated for two cycles. After the amplitude of 0.20 rad, amplitudes were increased to 0.30 and 0.50 rad, with each amplitude for one cycle. All specimens were completely fractured and separated by the end of loading with the 0.50 rad amplitude. The yield stress of the specimens was 300 MPa, and the calculated yield end rotation was 0.007 rad. The amplitude of 0.10 rad corresponds to about 15 in terms of ductility.
Fig. 3.
527
Loading history adopted.
3. Behavior in the large deformation range 3.1. Overall behavior Fig. 4 shows the beam end moment (M / Mp) versus beam end rotation (qx) relationship obtained from the four tests. The beam moment was estimated as the total reactional force (measured by the two load cells) times the height of the rigid-columns (664 mm), then divided by 8. (The resistance was provided by the four identical beam specimens.) The four beam specimens in each test exhibited nearly identical behavior up to the loading cycles with the 0.08 rad amplitude. The moment was normalized by the calculated beam full-plastic moment (Mp ⫽ 2.37 kNm). Fig. 4(a)–(d) are the plots of loading to the amplitudes of 0.06 rad, whereas Fig. 4(e)–(h) are those plotting the data to complete failure. In conventional seismic design, a rotation of 0.05 rad is regarded as a large deformation [5]. To assist the interpretation of the test results and also to highlight the difference of the behavior under extremely large deformations relative to the behavior under deformations commonly considered, here the deformation up to the amplitude of 0.06 rad is named as ‘large deformation’ and deformation beyond that amplitude is named as ‘extremely large deformation.’ First, test results are examined in detail for ‘large deformation’. As evidenced from Fig. 4(a)–(d), all beam specimens show stable behavior, with some strain hardening beyond Mp. Fig. 5(a) (up to 0.06 rad) summarizes the transition of in-plane resistance of the specimens, given as the maximum moment obtained during prescribed amplitudes. The moment was smaller for RBS specimens than for the corresponding standard specimens (SR-1 relative to SS-1 and SR-2 relative to SS-2). This is natural because of the reduced in-plane capacity at the RBS portion. The difference between SR-1 and SS-1 (those without lateral braces), however, is rather minimal,
528
D. Liu et al. / Engineering Structures 25 (2003) 525–535
Fig. 4. End moment versus end rotation relationships: (a) SS-1 (to 0.06 rad); (b) SR-1 (to 0.06 rad); (c) SS-2 (to 0.06 rad); (d) SR-2 (to 0.06 rad); (e) SS-1 (to failure); (f) SR-1 (to failure); (g) SS-2 (to failure); (h) SR-2 (to failure).
which will be reasoned later. Some degradation in resistance due to load reversal is evident in the specimens without lateral braces (Fig. 4(a), (b)), and this is primarily because of lateral–torsional buckling and succeeding instability observed in those specimens (Fig. 6). No such buckling was observed, and no degradation was present in specimens with lateral braces (Fig. 4(c), (d)). Fig. 5(b) shows the accumulation of dissipated energy (E / Ey) with the progress of loading. E is the area enclosed by the hysteresis loops, and Ey is the strain energy at the calculated elastic limit. The dissipated energy is very similar between SS-1 and SR-1 (those without lateral braces) and also between SS-2 and SR2 (those with lateral braces). This is rather puzzling because the strength is larger in SS-1 than in SR-1 and particularly in SS-2 than in SR-2. The reason is examined in the next section. In SS-1, SS-2, and SR-2, local buckling was observed in beam flanges during the first cycle of the 0.06 rad amplitude but did not grow until the end of loading with this amplitude. In SR-1, no local buckling was observed up to the end of the 0.06 rad amplitude. It appeared during the first cycle of the 0.08 rad amplitude. 3.2. Lateral instability and energy dissipation in RBS Fig. 4(a)–(b) and Fig. 5(a) indicate that reduction in resistance due to lateral instability is somewhat less significant for the RBS beam than for the standard beam, although the out-of-plane stiffness is smaller in the RBS
beam because of the reduced section. The writers conducted an analytical study to look into the lateral instability of RBS beams. To this end, a computer program code using the larger deformation theory was employed. The main feature of this program is as follows: (1) finite element formulation along the length; (2) discretization of the cross-section into many fiber segments and adoption of a hypothesis that a plane section should remain plane; (3) consideration of large deformation consisting of large rigid-body motion and small strains; (4) adoption of coordinate systems that consider finite rotations; and (5) incorporation of the perturbation method for tracing correct deformation paths particularly in the instability range. Details of the program code are given elsewhere [9]. Note, however, that distortion of crosssections (local buckling of plate elements) was not included in the analysis. To calibrate the validity of this program in tracing lateral–torsional buckling and post-buckling instability of steel beams, behavior of SS-1 up to the 0.06 rad amplitude was analyzed. The results are shown in Fig. 7, in which the bold and dotted lines show the analytical and experimental hysteresis loops, respectively. The figure includes the relationship between the beam end moment (M / Mp) and the beam end rotation (qx) (Fig. 7(a)), the relationship between M / Mp and out-of-plane displacement (u) at the beam quarter-span (normalized by the beam length of L) (Fig. 7(b)), and the relationship between M / Mp and the torsional rotation (qz) at the beam mid-length (Fig. 7(c)). The figures demonstrate that the
D. Liu et al. / Engineering Structures 25 (2003) 525–535
Fig. 5. Summary of results: (a) transition of maximum moment attained in each amplitude; (b) accumulation of dissipated energy.
Fig. 6. Out-of-plane deformations in cycles with 0.06 rad amplitude (SS-1).
529
analysis predicts the initiation of lateral torsional buckling accurately (that occurred during the first cycle of the 0.04 rad amplitude) and duplicates closely the deterioration in resistance after buckling and the growth of out-of-plane deformations in the post-buckling region. Using the program, three beam sections: W24 × 76, W30 × 99, and W36 × 194, were analyzed for the double curvature moment condition. The loading sequence adopted in the analysis was: two cycles with the 0.015 rad amplitude, followed by two cycles with the 0.030 rad amplitude, and then followed by two more cycles with the 0.045 rad amplitude in the beam end rotation. Fig. 8 shows the results for the slenderness ratio about the weak axis (L / ry) between 40 and 480 and for the yield stress of 250 MPa. In this figure, three strength measures were adopted: i.e. the maximum strength (the largest resistance obtained during the entire cycles), the last strength (the maximum strength obtained during the last half-cycle of the 0.045 rad amplitude), and the monotonic strength (the maximum strength obtained for monotonic loading up to the beam end rotation of 0.045 rad). The last strength was adopted as a measure of the strength that can be sustained by the beam under cyclic loading. Fig. 8(a) is for standard beams. The maximum and monotonic strengths are nearly the same, but the last strength is reduced significantly relative to the maximum strength for L / ry greater than 100 due to lateral–torsional buckling and succeeding progresses of out-of-plane deformations. Fig. 8(b) is for RBS beams, in which the strength shown in the ordinate is normalized by the fullplastic moment of the original (unreduced) cross-section. The general trend is similar, but significant reduction in last strength occurs for a larger L / rx than for standard beams. Fig. 9(a) shows the yielding region at the instant when lateral–torsional buckling occurs (during the first cycle of the 0.03 rad amplitude) for the standard beam having the cross-section of W36 × 194 and L / ry ⫽ 150. Fig. 9(b) shows the yielding region at the same loading instant for the corresponding RBS beam, where no buckling occurs. The yielding region is longer for the standard beam than for the RBS beam, and the applied end moment is also greater for the standard beam (M / Mp ⫽ 1.11) than for the RBS beam (M / Mp ⫽ 0.83). In view of a smaller length of the plastified region and smaller compressive stresses induced in the flanges and web, it is possible that the RBS beam, although having a smaller out-of-plane stiffness, could be less susceptible to lateral instability than the corresponding standard beam. More detailed discussion on this topic is presented elsewhere [10]. The tested beams (SS-1 and SR-1) had L / rx of 121. In reference to Fig. 8(a) and (b), the RBS beam has a larger last strength relative to the maximum strength than the corresponding standard beam. Although the test and analysis presented in this study
530
D. Liu et al. / Engineering Structures 25 (2003) 525–535
Fig. 7. Comparison between test and analysis: (a) end moment versus end rotation; (b) end moment versus out-of-plane displacement (at quarter span); (c) end moment versus torsional rotation (at mid-span).
Fig. 8. Beam strength with respect to slenderness ratio about weak axis: (a) standard beam; (b) RBS beam.
Fig. 9. Comparison between standard and RBS beams: (a) plastic region at instant of buckling of standard beam; (b) plastic region of RBS beam at loading corresponding to buckling of standard beam; (c) experimental hysteresis loop obtained in second cycle of 0.06 rad amplitude.
D. Liu et al. / Engineering Structures 25 (2003) 525–535
imply that the RBS beam may not necessarily be more susceptible to lateral instability, lateral instability effects of RBS beams involve more complexity than that presented. The beam cross-sections used in the test were stocky, and local buckling was not included in the analysis. According to a statistical study conducted by Uang and Fan [11], however, web local buckling can seriously deteriorate the resistance of RBS beams. Quantification of lateral instability effects of RBS beams is a subject of continuing research by the writers. Fig. 5(b) shows that the dissipated energy is about the same between SS-2 and SR-2 (both with lateral braces), although the maximum resistance is about 15% larger in SS-2 than in SR-2. Fig. 9(c) shows the normalized moment versus end rotation relationships obtained for SS-2 and SR-2 during the second cycle of the 0.06 rad amplitude. Here the ordinate is normalized by the largest moment obtained for respective specimens. SR-2 (the RBS detail) is fatter in hysteresis loops than SS-2, and this fatness was the reason why the SR-2 had about the same dissipated energy despite the smaller strength. The fatness was the result of a combined effect of: (1) concentration of the yielding region within the RBS portion; and (2) relatively uniform strains distributed along the yielding region, which retards strain hardening for a given beam plastic rotation. Such behavior observed in the RBS beam looks advantageous in terms of beam energy dissipation. This, however, involves relatively larger strains induced in the RBS portion, and becomes the source of detrimental behavior in the extremely large deformation range, as shown in the next section.
4. Behavior in the extremely large deformation range 4.1. Cracks and fractures In SS-1 and SR-1 (those without lateral braces), outof-plane deformations grew (and accelerated by the progress of local buckling) during the cycles of 0.08 and 0.1 rad amplitudes, accompanied by some pinching in hysteresis loops. Cracks initiated near the beam ends (for SS-1) and at the reduced cross-section (for SR-1) during the cycles with the 0.2 rad amplitude; then, the in-plane resistance dropped less than half the maximum resistance. Four beam specimens in each test were fractured (complete separation) one after another during the cycles of 0.2 to 0.5 rad amplitudes for SS-1 (Fig. 10) and of the 0.2 to 0.3 rad amplitudes for SR-1. Although very small, the beams possessed some resistance and dissipated energy until they were separated completely. In SS-2 and SR-2, stable hysteresis was maintained up to the end of the 0.08 rad amplitude thanks to the lateral braces. Cracks initiated near the beam ends (for SS-2) and at the reduced cross-sections (for SR-2) during the
Fig. 10.
531
Near separation at beam end (SS-1).
first cycle of the 0.1 rad amplitude, causing drops in inplane resistance. The restraint provided by the lateral braces caused strain concentrations in some parts of the beam, which triggered early cracks in SS-2 and SR-2 as compared to SS-1 and SR-1. In SS-2, four beam specimens were fractured one after another during the cycles with 0.1 to 0.3 rad amplitudes, with slight pinching observed in the hysteresis loops. In SR-2, fractures occurred earlier than other three tests, and all beam specimens were separated completely during the cycles with the 0.15 rad amplitude. The progress of cracks was more rapid for SS-2 and SR-2, in which the entire tension flange and large portion of the web were cracked during the cycles of the amplitude (0.1 rad) when cracks initiated. In SS-1 and SR-1, cracks started at a larger amplitude (0.2 rad) and extended only half the flange width at the end of the cracking amplitude. Among the four tests, SR-2 (laterally braced beams with the RBS detail) fractured earliest. Because of this, the dissipated energy to complete fracture was the smallest in SR-2 (Fig. 5(b)). SS2 (with lateral braces) showed consistently larger energy dissipation than SS-1 (without lateral braces) to the end of the 0.15 rad amplitude (Fig. 5(b)), but the difference becomes smaller after the 0.2 rad amplitude, and the total dissipated energy to complete fracture was about the same for SS-1, SR-1, and SS-2. In summary, behavior in extremely large deformations is significantly different from behavior in large deformations. The RBS detail and lateral braces, both beneficial in maintaining stable hysteresis in large deformations, do not necessarily promise larger deformation and energy dissipation capacity when load is advanced in extremely large deformations that lead to complete failure. Also note that the lateral braces adopted in the test were arranged at three discrete locations rather than
532
D. Liu et al. / Engineering Structures 25 (2003) 525–535
continuously along the beam, which might have enhanced the strain concentrations. Finally, caution should be made when interpreting the range of deformations presented in this study. The tests were conducted for reduced-scaled specimens with a scale-ratio of about 1/10. As a result, the cross-sections were made of thin plates (2–3 mm thickness), which are commonly more ductile than thicker plates. Welds, which are the most critical source that triggers earlier fractures, were also avoided in the fabrication of the specimens. Although it is difficult to quantify the size and weld effects on the behavior, the writers’ comments on them are as follows. Tests of great many full-scale beams having the post-Northrdige and post-Kobe connection details exhibit stable behavior up to the cyclic amplitude of 0.04 rad. [5,12]. Stable behavior up to the cyclic amplitude of 0.06 rad (the amplitude defined as the large deformation range in this study) is not necessarily easy to achieve in full-scale welded beams, while all beams tested in this study survived without notable strength deterioration up to the amplitude of 0.08 rad. This difference is most likely attributed to the welds and thicker plates adopted in the full-scale beams; hence the present discussion on the behavior in the large deformation range remains valid within the assumption of ‘no fracture’. In some of the previous full-scale tests with welded connections [13], the beams did not fail completely (no separation) with the cyclic amplitude of about 0.08 rad. In this amplitude of loading, one flange and a large portion of the web were broken, but the beam was still continuous in the other flange. (Because of the stroke capacity of the loading device, the tests had to be terminated before leading the beams to complete failure.) This behavior in the full-scale tests was very similar to the behavior in the extremely large deformation range (the cyclic amplitudes beyond 0.1 rad) presented in this study. It remains a question, however, whether or not full-scale welded beams can reach a deformation of 0.2 to 0.5 rad before complete failure as did the small-scale beams of this study. Thus, the experimental results in the extremely large deformation range should best be regarded as the background data for future numerical simulation of complete failure of steel members and frames.
0.15 and 0.2 rad amplitudes, however, show visible increases in resistance beyond the maximum resistance attained in previous cycles (up to the 0.1 rad amplitude). Such pinching involving the increase in maximum resistance was found to be a result of tensile axial forces applied in the beam specimens. Fig. 11(a) shows the growth of a beam axial strain, estimated from a pair of strain gauges glued on the beam top and bottom flanges (for SS-1). The axial strain remains nearly zero up to the amplitude of 0.04 rad, but started growing (in tension) precipitously for amplitudes larger than 0.08 rad. This means that large tensile strains (and stresses) were applied to the beam cross-sections. Fig. 11(b) shows the transition of the out-of-plane displacement (u) at a crosssection located at a beam quarter length. In amplitudes not greater than 0.06 rad, the displacement increases with the increase of beam end rotation. For larger amplitudes, it is reversed, with the out-of-plane displacement decreasing for larger in-plane rotations. This indicates that the beam specimens were stretched for larger rotations. The stretching generated tensile strains in the beam cross-section. Geometrical deformation and presence of column rigid-zones were found to be attributable to beam stretching. Fig. 12 shows the geometrical deformation of the loading system adopted in this study, in which a rigid-zone has a length (a) that equals the sum of the half column width (75 mm) and the beam end-plate thickness (20 mm). When the rigid columns rotate by qx, the beam is inclined by an angle of f (measured from the horizontal). Assuming that the columns are rigid, the length of the beam has to be elongated to L1 from the original length of L0 due to the geometry change. The kinematic condition leads to the following relationship: L1 ⫽ 冑(L⫺2acosqx)2 ⫹ (2asinqx)2
(1)
Here, L is the length between column supports. If the
4.2. Pinching effect Notable in the extremely large deformation range is a significant ‘pinching’ observed in the specimens without lateral braces (SS-1 and SR-1). Hysteresis loops in the 0.08 and 0.1 rad amplitudes exhibit pinching, but the maximum resistance never exceeded the resistance attained in previous cycles (Fig. 4(e) and (f)). Such pinching is typical of cyclic loading of buckled (and deformed in the out-of-plane direction) members. Hysteresis loops (with significant pinching) observed for the
Fig. 11. Stretching of beam under extremely large deformations: (a) accumulation of axial tensile strain; (b) behavior of out-of-plane displacement.
D. Liu et al. / Engineering Structures 25 (2003) 525–535
Fig. 13.
533
Increase in resistance due to geometry change.
Fig. 12. Geometry change of loading system: (a) upright position; (b) inclined position.
beam is elastic, the resultant axial force (N) exerted into the beam is: N ⫽ EA / L0(冑(L⫺2acosqx)2 ⫹ (2asinqx)2⫺L0)
(2)
The axial force is in self-equilibrium within the loading system. Because of the beam inclination (f), however, the arm-length measured between one beam end (A in Fig. 12) and the support-column’s bottom (where a mechanical pin is arranged) is no longer the same as the arm-length measured between the other end (B in Fig. 12) and the corresponding support-column’s bottom. The difference produces a moment equal to the axial force (N) times a length of Lsinf. This moment is applied to the direction to resist the horizontal movement (in the anti-clockwise direction in Fig. 12), becoming a source to increase the resistance in larger deformations. The dotted lines of Fig. 13 show the hysteresis during the first cycle of the 0.2 rad amplitude (SS-1). The solid lines of Fig. 13 show the increase due to the geometry change, estimated by applying Eq. (1) and (2), with N capped to the yield strength to allow for beam tensile yielding. Here, the origin is shifted in the ordinate to the experimental moment at qx ⫽ 0 for comparison purpose. Although the estimate is rather crude, it still captures the degree of increase in resistance due to the geometry change. Fig. 14 shows comparisons between the test and
Fig. 14. Comparison between test and analysis up to complete failure: (a) end moment versus end rotation; (b) end moment versus outof-plane displacement (at quarter span).
analysis (introduced in the previous section) for the entire loading range (SS-1). The analysis could simulate the behavior very reasonably in the large deformation range (up to the 0.06 rad amplitude) as evidenced in Fig. 7. The analysis explicitly allowed for geometrical nonlinearity in large deformations but did not consider distortion and fracture of beam cross-sections. Because of this limitation, the analysis fails to simulate the degradation in resistance (primarily due to cracks and fractures) for extremely large deformations. The analysis traces pinching (due to the accumulation of out-of-plane deformations and resultant relaxation around the neutral deformations) and significant increase in resistance due
534
D. Liu et al. / Engineering Structures 25 (2003) 525–535
to beam’s geometry change in the extremely large deformation range. The latter appears because of the beam stretching and inclination, as described before. The rate of increase in pinching resistance is relatively similar between the test and the analysis. In the test, contraction of beams was not permitted, because the columns were rigid and the column ends were connected to rigid horizontal frames. That became the source of beam stretching. Fig. 15(a) shows the effect of beam contraction on the in-plane resistance, obtained analytically for monotonic loading. The solid line shows when the beam axial contraction is not permitted, as in Fig. 12, and the dotted line shows when such contraction is allowed. For smaller to medium deformations, no difference is observed. For larger deformations (those beyond 0.06 rad in this example), the resistance increases when axial contraction is not allowed, but the resistance keeps decreasing when such contraction is permitted. The length of the rigid-zone also affects the degree of increase associated with geometry change. For a longer rigid-zone, inclination of the beam (f) is more conspicuous relative to the beam end rotation (qx) (Fig. 12). Fig. 15(b) shows this effect, comparing analytical moment versus end rotation relationships for the rigid-zone length equal to 95 mm (adopted in the test and shown by solid lines) and for the length of zero (dotted lines). The difference becomes more conspicuous for larger deformations, and the strength is larger for a longer rigid-zone. Increased resistance associated with geometry change may not occur in real moment frames, because columns are not rigid and restraining members that would prevent relative movement between adjacent columns are not present at the mid-height of columns. Floor slabs are
Fig. 15. Effect of support condition and rigid-zone on hysteretic behavior: (a) effect of beam contraction; (b) effect of rigid-zone length.
neglected in this study, but they may provide some resistance against beam contraction, although floor slabs also should suffer great damage for large deformations. Refinement of numerical analysis so that it can allow for distortion, cracks, and fracture of cross-sections, and further investigation into the effects of floor slabs on beam contraction are the continuing subjects of research for the writers.
5. Conclusions This paper presented a series of tests for steel beams subjected to cyclic loading to extremely large deformations, and investigated their behavior to complete failure. Major findings obtained from this study are summarized as follows. 1. Four tests having a combination of standard and RBS beams, with and without lateral braces, were conducted. The lateral braces were adopted to mimic floor slabs and arranged on beam top flanges at discrete locations. The behavior in the large deformation range was commensurate with the behavior observed in many previous studies. That is, both the standard and RBS beams exhibited stable hysteresis, and the lateral braces prevented growth of lateral–torsional buckling. 2. Analysis using the large deformation theory was performed to strengthen the experimental findings observed in the large deformation range. The analysis was able to accurately duplicate the experimental behavior involving lateral–torsional buckling and succeeding accumulation of out-of-plane deformations. Additional analysis was conducted to examine the lateral instability effects of RBS beams relative to standard beams. 3. The behavior in extremely large deformations, i.e. the behavior beyond the cyclic loading amplitude of 0.06 rad, was significantly different from the behavior in large deformations. The RBS beam ensured stable behavior up to large deformations, but failed earlier from the reduced cross-section, primarily due to strain concentrations at the RBS portion. Lateral braces that provided restraint against out-of-plane deformations caused strain concentrations in extremely large deformations, leading to earlier fractures for laterally braced beams than for those without such braces. 4. In the extremely large deformation range, significant pinching and increase in the maximum resistance was observed for beams without lateral braces. This increase was given as a result of tensile axial forces induced in the beam, which in turn was generated due to its geometry change. The degree of pinching was affected by the length of beam rigid-zones as well as by the degree of restraint against the beam contraction in its longitudinal direction.
D. Liu et al. / Engineering Structures 25 (2003) 525–535
5. Caution should be made when interpreting the range of deformations presented in this study. The tests were carried out for reduced-scaled specimens with a scale-ratio of about 1/10. As a result, the cross-sections were made of thin plates, which are commonly more ductile than thick plates. Welds, which often trigger earlier fractures, were also avoided in the fabrication of specimens. Deformations that correspond to the limit of stable behavior and the instant of complete failure are likely to be smaller in full-scale beams with welded connections than those presented in this study.
[5]
[6] [7]
[8]
[9]
References [10] [1] Bertero VV, Anderson JC, Krawinkler H. Performance of steel building structures during the Northridge Earthquake. Report no. UCB/EERC-94-09, University of California at Berkeley. The Earthquake Engineering Research Center; 1994. [2] Reconnaissance report on damage to steel building structures observed from the 1995 Hyogoken–Nanbu Earthquake. Kinki Branch of the Architectural Institute of Japan, Steel Committee, Osaka; 1995 [in Japanese with attached abridged English version]. [3] Interim guidelines: evaluation, repair, modification and design of welded steel moment frame structures (August 1995), plus Interim Guidelines Advisory No. 1. Federal Emergency Management Agency, Washington, DC; 1997. [4] Nakashima M, Inoue K, Tada M. Classification of damage to
[11]
[12]
[13]
535
steel buildings observed in the 1995 Hyogoken–Nanbu earthquake. Eng Struct 1998;20(4–6):271–81. Recommended seismic design criteria for new steel moment– frame buildings. Federal Emergency Management Agency, FEMA350, Washington, DC; 2000. Recommendation for design of connections in steel structures. Tokyo: The Architectural Institute of Japan; 2001. Nakashima M, Akazawa T, Igarashi H. Pseudo dynamic testing using conventional testing devices. J Earthquake Eng Struc Dyn 1995;24(10):1409–22. Engelhardt MD. Design of reduced beam section moment connections. Proceedings of 1999 North American Steel Construction Conference. American Institute of Steel Construction, Ill; 1999. Kanao I, Morisako K, Nakamura T. Numerical analysis of elastoplastic lateral buckling behavior of H-shaped beams by beamcolumn finite element model. J Struc Constr Eng Architect Inst Jpn Tokyo 2000;527:95–101 [in Japanese]. Nakashima M, Kanao I, Liu D. Lateral instability and lateral bracing of steel beams subjected to cyclic loading. J Struct Eng ASCE 2002;128(10):1308–16. Uang CM, Fan CC. Cyclic stability criteria for steel moment connections with reduced beam section. J Struct Eng ASCE 2001;127(9):1021–7. Nakashima M, Teteyama E, Morisako K, Suita K. Full-scale test on beam-column subassemblages having connection details of shop-welding type. Proceedings of the Structural Engineers World Conference, San Francisco, USA; 1998 [Paper#:T158–7]. Suita K, Nakashima M, Engelhardt MD. Comparison of seismic capacity between post-Northridge and post-Kobe beam-to-column connections. Proceedings of the Second International Conference on Behavior of Steel Structures in Seismic Areas, Montreal, Canada; 2000. p. 271–8.
Behavior to complete failure of steel beams subjected to cyclic loading D. Liu a, M. Nakashima b,∗, I. Kanao b a
Graduate School of Engineering, Kyoto University, Yoshida-Honmachi, Sakyo, Kyoto 606-5801, Japan b Disaster Prevention Research Institute, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan
Abstract This paper presents an experimental study on steel beams subjected to cyclic loading to extremely large deformations. The study aims to collect information on beam hysteretic behavior up to complete failure, in the belief that such information is needed for the establishment of performance-based design. Test beams were about 1/10-scaled models, and effects of the RBS detail and lateral braces arranged at beam top flanges were examined. Behavior up to the cyclic loading amplitude of 0.06 rad was commensurate with behavior observed in many previous studies. Behavior in extremely large deformations from 0.1 to 0.5 rad amplitudes was significantly different from the behavior in large deformations (to the 0.06 rad amplitude). The RBS beam failed earlier in the reduced cross-section primarily due to strain concentrations at the section. Lateral braces also caused strain concentrations, leading to earlier fractures. Significant increase in the maximum resistance was observed in extremely large deformations for beams not braced laterally. Tensile axial forces induced in the beam according to the geometry change were responsible for the increase. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Steel beam; Earthquake; Cyclic loading; Complete failure; Lateral braces; Pinching
1. Introduction Performance-based design is an issue of great interest in the earthquake engineering community. In this design format, various limit states should be determined, and structures be designed to satisfy these limit states, each of which is checked against a different level of seismic hazard. Among the limit states, the collapse limit is most important, because this limit is associated directly with human loss. Here, collapse means the condition in which the structure cannot sustain gravity load any longer. Characterization of this limit, however, is still premature. Numerous studies, both experimental and analytical, have been carried out to investigate the damage and failure of structural members and systems, but the vast majority of them did not explore the final collapse for many reasons. On the experimental side, tests to duplicate the collapse are difficult to achieve because of limitations of loading facilities and also because of safety. On the analytical side, numerical analysis procedures
∗
Corresponding author. Fax: 81-774-38-4086. E-mail address: [email protected] (M. Nakashima).
that can allow for strong geometrical and material nonlinearities as well as for drastic changes in structural topology are very limited to begin with. Furthermore, experimental or observational data on collapse are so scarce that means to calibrate the reliability of numerical results are nearly nonexistent. Although it is a great challenge, quantification of the collapse limit is crucial for the establishment of performance-based design. Seismic design of steel building structures has greatly evolved in the past few years in response to serious damage to steel buildings observed in the 1994 Northridge and 1995 Hyogoken–Nanbu (Kobe) earthquakes [1–4]. Design guidelines [5–6] recently released, consider significantly larger deformation demands to structural members and frames than those considered before those earthquakes, and various means to ensure their deformation capacity have also been proposed. Even with the latest design guidelines, the true collapse limit has not been clearly stated, primarily because of the scarcity of data on this limit as well as lack of reliable analytical procedures to predict this limit state. In view of these circumstances, the writers and their group have been conducting a series of experiments for collecting quantitative data on collapses of steel struc-
0141-0296/03/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0141-0296(02)00164-5
526
D. Liu et al. / Engineering Structures 25 (2003) 525–535
tural members and frames. Here, collapse of structural members means complete fracture (separation) of members at which they cannot provide any resistance. Such information becomes an important ingredient for the evaluation of complete collapse of structural frames and also provides background data for future numerical analysis that will aim at collapse simulation. The tests on steel beam members were completed, and this paper presents details of the test program and the test results. Numerical analysis using the large-deformation theory was also conducted to supplement and interpret the experimental data as well as to examine the applicability and limitation of the numerical analysis.
2. Test program 2.1. Specimen and test setup Fig. 1 shows the test specimen used in this study. The specimen was fabricated by machine-grinding a steel bar. The length was 810 mm, and the cross-section was 70 mm (in depth), 30 mm (in width), 3 mm (in flange thickness), and 2 mm (in web thickness). These dimensions indicate that the specimen was approximately a 1/10-scaled model. The specimen had a slenderness ratio of 121 about the weak axis, and its flange and web width-to-thickness ratios were 5.0 and 35, respectively. Such scaling violates some important similitude, and therefore effects of some critical parameters such as local buckling and brittle fracture would not be duplicated. The objective of this test was to examine the behavior to complete failure involving extremely large deformations, which forced the use of reduced-scaled specimens. The size of the beam specimens was chosen as a compromise. The end of the beam was attached to a steel plate, which in turn was fastened to the test setup shown in Fig. 2. The beam end was not welded directly to the end plate to avoid possible fracture from this location. Instead, the end plate was holed, and the specimen was squeezed into the hole and welded from the other end of the plate. The test setup is shown in Fig. 2. It consists of a pair
Fig. 1.
Beam test specimen.
Fig. 2.
Test setup: (a) elevation; (b) plan.
of frames, each of which has three rigid columns that are pin-supported at the reaction beam (at the bottom) and loading beam (at the top). Two beam specimens are inserted in between the three rigid columns and fastened to the columns by high-tension bolts. A pair of these frames is arranged in parallel at a distance of 1.0 m, and a transverse plate is attached to bridge each loading beam. These transverse plates are arranged to restrict deformations of the pair of frames against the direction perpendicular to the loading direction. With this setup, four beam specimens (Fig. 1) were tested at one time. Two quasi-static jacks were used for loading, with each one attached at the end of each loading beam. The two jacks imposed an identical horizontal displacement so that the two frames were to move in parallel, and therefore the four beam specimens would take the same forced deformation. According to these loading and support conditions, the beam specimens were to sustain bending in double curvature. The jacks were operated by a computer-control loading system, which consisted of a PC that sent displacement signals, a controller that controlled the hydraulic pump, and a hydraulic pump whose oil flow was adjusted by an inverter motor. A digital displacement transducer and an analog load cell measured the displacement and force of the jack, respectively. Signals of those measuring devices were continuously fed back into the PC. Although a predetermined displacement history was applied in the tests presented in this paper, the loading system has a capacity of conducting a quasi-static pseudo dynamic loading test. Details of this system are presented elsewhere [7]. In some tests, the reduced beam section (RBS) detail was adopted, and portions of the beam flanges were trimmed. The trimming procedures followed those pro-
D. Liu et al. / Engineering Structures 25 (2003) 525–535
posed by Engelhardt [8], setting the full-plastic moment of the most reduced cross-section at 0.85 times that of the unreduced cross-section. In some tests, members having a channel section were placed on top of the top flanges of the specimens and in the direction perpendicular to the specimens’ longitudinal direction (Fig. 2). These members, named lateral braces, were arranged to mimic floor slabs that would provide restraint against the specimens’ torsion and out-of-plane deformations. Note, however, that the braces did provide restraint only at beam top flanges and beam bottom flanges were unbraced laterally. Three lateral braces were placed on each beam specimen pair and fastened to the specimen’s top flange by mechanical clamps. Four tests were conducted in this study, so that a total of 16 beam specimens were loaded. One test featured four specimens of the standard cross-section and without lateral braces, designated as ‘SS-1’. One test featured four specimens with the RBS detail and without lateral braces, designated as ‘SR-1’. One test featured four specimens of the standard cross-section and with lateral braces, designated as ‘SS-2’. One test featured four specimens with the RBS detail and with lateral braces, designated as ‘SR-2’. 2.2. Measurement and loading history To measure the strains of beam specimens in the longitudinal direction, strain gauges were glued on the beam top and bottom flanges. The deformations of a cross-section were measured by three wire-type displacement transducers, placed in three different directions, i.e. one oriented in the horizontal direction, another in the direction inclined by +45° from the horizontal, and the last in the direction inclined by ⫺45° from the horizontal. Those measurements were converted to the in-plane displacement (v), out-of-plane displacement (u), and torsion (qz). The deformations were measured at the three locations in each of the four beam specimens. Fig. 3 shows the loading history adopted in the test. The deformation was expressed as the horizontal displacement of the loading beam divided by the height of the rigid-columns (664 mm, measured between the two pins). This value is identical with the end rotation (qx) of the beam specimen because the columns were rigid. The amplitudes adopted (Fig. 3) were 0.02, 0.04, 0.06, 0.08, 0.10, 0.15, and 0.20 rad in terms of qx. For each amplitude, loading was repeated for two cycles. After the amplitude of 0.20 rad, amplitudes were increased to 0.30 and 0.50 rad, with each amplitude for one cycle. All specimens were completely fractured and separated by the end of loading with the 0.50 rad amplitude. The yield stress of the specimens was 300 MPa, and the calculated yield end rotation was 0.007 rad. The amplitude of 0.10 rad corresponds to about 15 in terms of ductility.
Fig. 3.
527
Loading history adopted.
3. Behavior in the large deformation range 3.1. Overall behavior Fig. 4 shows the beam end moment (M / Mp) versus beam end rotation (qx) relationship obtained from the four tests. The beam moment was estimated as the total reactional force (measured by the two load cells) times the height of the rigid-columns (664 mm), then divided by 8. (The resistance was provided by the four identical beam specimens.) The four beam specimens in each test exhibited nearly identical behavior up to the loading cycles with the 0.08 rad amplitude. The moment was normalized by the calculated beam full-plastic moment (Mp ⫽ 2.37 kNm). Fig. 4(a)–(d) are the plots of loading to the amplitudes of 0.06 rad, whereas Fig. 4(e)–(h) are those plotting the data to complete failure. In conventional seismic design, a rotation of 0.05 rad is regarded as a large deformation [5]. To assist the interpretation of the test results and also to highlight the difference of the behavior under extremely large deformations relative to the behavior under deformations commonly considered, here the deformation up to the amplitude of 0.06 rad is named as ‘large deformation’ and deformation beyond that amplitude is named as ‘extremely large deformation.’ First, test results are examined in detail for ‘large deformation’. As evidenced from Fig. 4(a)–(d), all beam specimens show stable behavior, with some strain hardening beyond Mp. Fig. 5(a) (up to 0.06 rad) summarizes the transition of in-plane resistance of the specimens, given as the maximum moment obtained during prescribed amplitudes. The moment was smaller for RBS specimens than for the corresponding standard specimens (SR-1 relative to SS-1 and SR-2 relative to SS-2). This is natural because of the reduced in-plane capacity at the RBS portion. The difference between SR-1 and SS-1 (those without lateral braces), however, is rather minimal,
528
D. Liu et al. / Engineering Structures 25 (2003) 525–535
Fig. 4. End moment versus end rotation relationships: (a) SS-1 (to 0.06 rad); (b) SR-1 (to 0.06 rad); (c) SS-2 (to 0.06 rad); (d) SR-2 (to 0.06 rad); (e) SS-1 (to failure); (f) SR-1 (to failure); (g) SS-2 (to failure); (h) SR-2 (to failure).
which will be reasoned later. Some degradation in resistance due to load reversal is evident in the specimens without lateral braces (Fig. 4(a), (b)), and this is primarily because of lateral–torsional buckling and succeeding instability observed in those specimens (Fig. 6). No such buckling was observed, and no degradation was present in specimens with lateral braces (Fig. 4(c), (d)). Fig. 5(b) shows the accumulation of dissipated energy (E / Ey) with the progress of loading. E is the area enclosed by the hysteresis loops, and Ey is the strain energy at the calculated elastic limit. The dissipated energy is very similar between SS-1 and SR-1 (those without lateral braces) and also between SS-2 and SR2 (those with lateral braces). This is rather puzzling because the strength is larger in SS-1 than in SR-1 and particularly in SS-2 than in SR-2. The reason is examined in the next section. In SS-1, SS-2, and SR-2, local buckling was observed in beam flanges during the first cycle of the 0.06 rad amplitude but did not grow until the end of loading with this amplitude. In SR-1, no local buckling was observed up to the end of the 0.06 rad amplitude. It appeared during the first cycle of the 0.08 rad amplitude. 3.2. Lateral instability and energy dissipation in RBS Fig. 4(a)–(b) and Fig. 5(a) indicate that reduction in resistance due to lateral instability is somewhat less significant for the RBS beam than for the standard beam, although the out-of-plane stiffness is smaller in the RBS
beam because of the reduced section. The writers conducted an analytical study to look into the lateral instability of RBS beams. To this end, a computer program code using the larger deformation theory was employed. The main feature of this program is as follows: (1) finite element formulation along the length; (2) discretization of the cross-section into many fiber segments and adoption of a hypothesis that a plane section should remain plane; (3) consideration of large deformation consisting of large rigid-body motion and small strains; (4) adoption of coordinate systems that consider finite rotations; and (5) incorporation of the perturbation method for tracing correct deformation paths particularly in the instability range. Details of the program code are given elsewhere [9]. Note, however, that distortion of crosssections (local buckling of plate elements) was not included in the analysis. To calibrate the validity of this program in tracing lateral–torsional buckling and post-buckling instability of steel beams, behavior of SS-1 up to the 0.06 rad amplitude was analyzed. The results are shown in Fig. 7, in which the bold and dotted lines show the analytical and experimental hysteresis loops, respectively. The figure includes the relationship between the beam end moment (M / Mp) and the beam end rotation (qx) (Fig. 7(a)), the relationship between M / Mp and out-of-plane displacement (u) at the beam quarter-span (normalized by the beam length of L) (Fig. 7(b)), and the relationship between M / Mp and the torsional rotation (qz) at the beam mid-length (Fig. 7(c)). The figures demonstrate that the
D. Liu et al. / Engineering Structures 25 (2003) 525–535
Fig. 5. Summary of results: (a) transition of maximum moment attained in each amplitude; (b) accumulation of dissipated energy.
Fig. 6. Out-of-plane deformations in cycles with 0.06 rad amplitude (SS-1).
529
analysis predicts the initiation of lateral torsional buckling accurately (that occurred during the first cycle of the 0.04 rad amplitude) and duplicates closely the deterioration in resistance after buckling and the growth of out-of-plane deformations in the post-buckling region. Using the program, three beam sections: W24 × 76, W30 × 99, and W36 × 194, were analyzed for the double curvature moment condition. The loading sequence adopted in the analysis was: two cycles with the 0.015 rad amplitude, followed by two cycles with the 0.030 rad amplitude, and then followed by two more cycles with the 0.045 rad amplitude in the beam end rotation. Fig. 8 shows the results for the slenderness ratio about the weak axis (L / ry) between 40 and 480 and for the yield stress of 250 MPa. In this figure, three strength measures were adopted: i.e. the maximum strength (the largest resistance obtained during the entire cycles), the last strength (the maximum strength obtained during the last half-cycle of the 0.045 rad amplitude), and the monotonic strength (the maximum strength obtained for monotonic loading up to the beam end rotation of 0.045 rad). The last strength was adopted as a measure of the strength that can be sustained by the beam under cyclic loading. Fig. 8(a) is for standard beams. The maximum and monotonic strengths are nearly the same, but the last strength is reduced significantly relative to the maximum strength for L / ry greater than 100 due to lateral–torsional buckling and succeeding progresses of out-of-plane deformations. Fig. 8(b) is for RBS beams, in which the strength shown in the ordinate is normalized by the fullplastic moment of the original (unreduced) cross-section. The general trend is similar, but significant reduction in last strength occurs for a larger L / rx than for standard beams. Fig. 9(a) shows the yielding region at the instant when lateral–torsional buckling occurs (during the first cycle of the 0.03 rad amplitude) for the standard beam having the cross-section of W36 × 194 and L / ry ⫽ 150. Fig. 9(b) shows the yielding region at the same loading instant for the corresponding RBS beam, where no buckling occurs. The yielding region is longer for the standard beam than for the RBS beam, and the applied end moment is also greater for the standard beam (M / Mp ⫽ 1.11) than for the RBS beam (M / Mp ⫽ 0.83). In view of a smaller length of the plastified region and smaller compressive stresses induced in the flanges and web, it is possible that the RBS beam, although having a smaller out-of-plane stiffness, could be less susceptible to lateral instability than the corresponding standard beam. More detailed discussion on this topic is presented elsewhere [10]. The tested beams (SS-1 and SR-1) had L / rx of 121. In reference to Fig. 8(a) and (b), the RBS beam has a larger last strength relative to the maximum strength than the corresponding standard beam. Although the test and analysis presented in this study
530
D. Liu et al. / Engineering Structures 25 (2003) 525–535
Fig. 7. Comparison between test and analysis: (a) end moment versus end rotation; (b) end moment versus out-of-plane displacement (at quarter span); (c) end moment versus torsional rotation (at mid-span).
Fig. 8. Beam strength with respect to slenderness ratio about weak axis: (a) standard beam; (b) RBS beam.
Fig. 9. Comparison between standard and RBS beams: (a) plastic region at instant of buckling of standard beam; (b) plastic region of RBS beam at loading corresponding to buckling of standard beam; (c) experimental hysteresis loop obtained in second cycle of 0.06 rad amplitude.
D. Liu et al. / Engineering Structures 25 (2003) 525–535
imply that the RBS beam may not necessarily be more susceptible to lateral instability, lateral instability effects of RBS beams involve more complexity than that presented. The beam cross-sections used in the test were stocky, and local buckling was not included in the analysis. According to a statistical study conducted by Uang and Fan [11], however, web local buckling can seriously deteriorate the resistance of RBS beams. Quantification of lateral instability effects of RBS beams is a subject of continuing research by the writers. Fig. 5(b) shows that the dissipated energy is about the same between SS-2 and SR-2 (both with lateral braces), although the maximum resistance is about 15% larger in SS-2 than in SR-2. Fig. 9(c) shows the normalized moment versus end rotation relationships obtained for SS-2 and SR-2 during the second cycle of the 0.06 rad amplitude. Here the ordinate is normalized by the largest moment obtained for respective specimens. SR-2 (the RBS detail) is fatter in hysteresis loops than SS-2, and this fatness was the reason why the SR-2 had about the same dissipated energy despite the smaller strength. The fatness was the result of a combined effect of: (1) concentration of the yielding region within the RBS portion; and (2) relatively uniform strains distributed along the yielding region, which retards strain hardening for a given beam plastic rotation. Such behavior observed in the RBS beam looks advantageous in terms of beam energy dissipation. This, however, involves relatively larger strains induced in the RBS portion, and becomes the source of detrimental behavior in the extremely large deformation range, as shown in the next section.
4. Behavior in the extremely large deformation range 4.1. Cracks and fractures In SS-1 and SR-1 (those without lateral braces), outof-plane deformations grew (and accelerated by the progress of local buckling) during the cycles of 0.08 and 0.1 rad amplitudes, accompanied by some pinching in hysteresis loops. Cracks initiated near the beam ends (for SS-1) and at the reduced cross-section (for SR-1) during the cycles with the 0.2 rad amplitude; then, the in-plane resistance dropped less than half the maximum resistance. Four beam specimens in each test were fractured (complete separation) one after another during the cycles of 0.2 to 0.5 rad amplitudes for SS-1 (Fig. 10) and of the 0.2 to 0.3 rad amplitudes for SR-1. Although very small, the beams possessed some resistance and dissipated energy until they were separated completely. In SS-2 and SR-2, stable hysteresis was maintained up to the end of the 0.08 rad amplitude thanks to the lateral braces. Cracks initiated near the beam ends (for SS-2) and at the reduced cross-sections (for SR-2) during the
Fig. 10.
531
Near separation at beam end (SS-1).
first cycle of the 0.1 rad amplitude, causing drops in inplane resistance. The restraint provided by the lateral braces caused strain concentrations in some parts of the beam, which triggered early cracks in SS-2 and SR-2 as compared to SS-1 and SR-1. In SS-2, four beam specimens were fractured one after another during the cycles with 0.1 to 0.3 rad amplitudes, with slight pinching observed in the hysteresis loops. In SR-2, fractures occurred earlier than other three tests, and all beam specimens were separated completely during the cycles with the 0.15 rad amplitude. The progress of cracks was more rapid for SS-2 and SR-2, in which the entire tension flange and large portion of the web were cracked during the cycles of the amplitude (0.1 rad) when cracks initiated. In SS-1 and SR-1, cracks started at a larger amplitude (0.2 rad) and extended only half the flange width at the end of the cracking amplitude. Among the four tests, SR-2 (laterally braced beams with the RBS detail) fractured earliest. Because of this, the dissipated energy to complete fracture was the smallest in SR-2 (Fig. 5(b)). SS2 (with lateral braces) showed consistently larger energy dissipation than SS-1 (without lateral braces) to the end of the 0.15 rad amplitude (Fig. 5(b)), but the difference becomes smaller after the 0.2 rad amplitude, and the total dissipated energy to complete fracture was about the same for SS-1, SR-1, and SS-2. In summary, behavior in extremely large deformations is significantly different from behavior in large deformations. The RBS detail and lateral braces, both beneficial in maintaining stable hysteresis in large deformations, do not necessarily promise larger deformation and energy dissipation capacity when load is advanced in extremely large deformations that lead to complete failure. Also note that the lateral braces adopted in the test were arranged at three discrete locations rather than
532
D. Liu et al. / Engineering Structures 25 (2003) 525–535
continuously along the beam, which might have enhanced the strain concentrations. Finally, caution should be made when interpreting the range of deformations presented in this study. The tests were conducted for reduced-scaled specimens with a scale-ratio of about 1/10. As a result, the cross-sections were made of thin plates (2–3 mm thickness), which are commonly more ductile than thicker plates. Welds, which are the most critical source that triggers earlier fractures, were also avoided in the fabrication of the specimens. Although it is difficult to quantify the size and weld effects on the behavior, the writers’ comments on them are as follows. Tests of great many full-scale beams having the post-Northrdige and post-Kobe connection details exhibit stable behavior up to the cyclic amplitude of 0.04 rad. [5,12]. Stable behavior up to the cyclic amplitude of 0.06 rad (the amplitude defined as the large deformation range in this study) is not necessarily easy to achieve in full-scale welded beams, while all beams tested in this study survived without notable strength deterioration up to the amplitude of 0.08 rad. This difference is most likely attributed to the welds and thicker plates adopted in the full-scale beams; hence the present discussion on the behavior in the large deformation range remains valid within the assumption of ‘no fracture’. In some of the previous full-scale tests with welded connections [13], the beams did not fail completely (no separation) with the cyclic amplitude of about 0.08 rad. In this amplitude of loading, one flange and a large portion of the web were broken, but the beam was still continuous in the other flange. (Because of the stroke capacity of the loading device, the tests had to be terminated before leading the beams to complete failure.) This behavior in the full-scale tests was very similar to the behavior in the extremely large deformation range (the cyclic amplitudes beyond 0.1 rad) presented in this study. It remains a question, however, whether or not full-scale welded beams can reach a deformation of 0.2 to 0.5 rad before complete failure as did the small-scale beams of this study. Thus, the experimental results in the extremely large deformation range should best be regarded as the background data for future numerical simulation of complete failure of steel members and frames.
0.15 and 0.2 rad amplitudes, however, show visible increases in resistance beyond the maximum resistance attained in previous cycles (up to the 0.1 rad amplitude). Such pinching involving the increase in maximum resistance was found to be a result of tensile axial forces applied in the beam specimens. Fig. 11(a) shows the growth of a beam axial strain, estimated from a pair of strain gauges glued on the beam top and bottom flanges (for SS-1). The axial strain remains nearly zero up to the amplitude of 0.04 rad, but started growing (in tension) precipitously for amplitudes larger than 0.08 rad. This means that large tensile strains (and stresses) were applied to the beam cross-sections. Fig. 11(b) shows the transition of the out-of-plane displacement (u) at a crosssection located at a beam quarter length. In amplitudes not greater than 0.06 rad, the displacement increases with the increase of beam end rotation. For larger amplitudes, it is reversed, with the out-of-plane displacement decreasing for larger in-plane rotations. This indicates that the beam specimens were stretched for larger rotations. The stretching generated tensile strains in the beam cross-section. Geometrical deformation and presence of column rigid-zones were found to be attributable to beam stretching. Fig. 12 shows the geometrical deformation of the loading system adopted in this study, in which a rigid-zone has a length (a) that equals the sum of the half column width (75 mm) and the beam end-plate thickness (20 mm). When the rigid columns rotate by qx, the beam is inclined by an angle of f (measured from the horizontal). Assuming that the columns are rigid, the length of the beam has to be elongated to L1 from the original length of L0 due to the geometry change. The kinematic condition leads to the following relationship: L1 ⫽ 冑(L⫺2acosqx)2 ⫹ (2asinqx)2
(1)
Here, L is the length between column supports. If the
4.2. Pinching effect Notable in the extremely large deformation range is a significant ‘pinching’ observed in the specimens without lateral braces (SS-1 and SR-1). Hysteresis loops in the 0.08 and 0.1 rad amplitudes exhibit pinching, but the maximum resistance never exceeded the resistance attained in previous cycles (Fig. 4(e) and (f)). Such pinching is typical of cyclic loading of buckled (and deformed in the out-of-plane direction) members. Hysteresis loops (with significant pinching) observed for the
Fig. 11. Stretching of beam under extremely large deformations: (a) accumulation of axial tensile strain; (b) behavior of out-of-plane displacement.
D. Liu et al. / Engineering Structures 25 (2003) 525–535
Fig. 13.
533
Increase in resistance due to geometry change.
Fig. 12. Geometry change of loading system: (a) upright position; (b) inclined position.
beam is elastic, the resultant axial force (N) exerted into the beam is: N ⫽ EA / L0(冑(L⫺2acosqx)2 ⫹ (2asinqx)2⫺L0)
(2)
The axial force is in self-equilibrium within the loading system. Because of the beam inclination (f), however, the arm-length measured between one beam end (A in Fig. 12) and the support-column’s bottom (where a mechanical pin is arranged) is no longer the same as the arm-length measured between the other end (B in Fig. 12) and the corresponding support-column’s bottom. The difference produces a moment equal to the axial force (N) times a length of Lsinf. This moment is applied to the direction to resist the horizontal movement (in the anti-clockwise direction in Fig. 12), becoming a source to increase the resistance in larger deformations. The dotted lines of Fig. 13 show the hysteresis during the first cycle of the 0.2 rad amplitude (SS-1). The solid lines of Fig. 13 show the increase due to the geometry change, estimated by applying Eq. (1) and (2), with N capped to the yield strength to allow for beam tensile yielding. Here, the origin is shifted in the ordinate to the experimental moment at qx ⫽ 0 for comparison purpose. Although the estimate is rather crude, it still captures the degree of increase in resistance due to the geometry change. Fig. 14 shows comparisons between the test and
Fig. 14. Comparison between test and analysis up to complete failure: (a) end moment versus end rotation; (b) end moment versus outof-plane displacement (at quarter span).
analysis (introduced in the previous section) for the entire loading range (SS-1). The analysis could simulate the behavior very reasonably in the large deformation range (up to the 0.06 rad amplitude) as evidenced in Fig. 7. The analysis explicitly allowed for geometrical nonlinearity in large deformations but did not consider distortion and fracture of beam cross-sections. Because of this limitation, the analysis fails to simulate the degradation in resistance (primarily due to cracks and fractures) for extremely large deformations. The analysis traces pinching (due to the accumulation of out-of-plane deformations and resultant relaxation around the neutral deformations) and significant increase in resistance due
534
D. Liu et al. / Engineering Structures 25 (2003) 525–535
to beam’s geometry change in the extremely large deformation range. The latter appears because of the beam stretching and inclination, as described before. The rate of increase in pinching resistance is relatively similar between the test and the analysis. In the test, contraction of beams was not permitted, because the columns were rigid and the column ends were connected to rigid horizontal frames. That became the source of beam stretching. Fig. 15(a) shows the effect of beam contraction on the in-plane resistance, obtained analytically for monotonic loading. The solid line shows when the beam axial contraction is not permitted, as in Fig. 12, and the dotted line shows when such contraction is allowed. For smaller to medium deformations, no difference is observed. For larger deformations (those beyond 0.06 rad in this example), the resistance increases when axial contraction is not allowed, but the resistance keeps decreasing when such contraction is permitted. The length of the rigid-zone also affects the degree of increase associated with geometry change. For a longer rigid-zone, inclination of the beam (f) is more conspicuous relative to the beam end rotation (qx) (Fig. 12). Fig. 15(b) shows this effect, comparing analytical moment versus end rotation relationships for the rigid-zone length equal to 95 mm (adopted in the test and shown by solid lines) and for the length of zero (dotted lines). The difference becomes more conspicuous for larger deformations, and the strength is larger for a longer rigid-zone. Increased resistance associated with geometry change may not occur in real moment frames, because columns are not rigid and restraining members that would prevent relative movement between adjacent columns are not present at the mid-height of columns. Floor slabs are
Fig. 15. Effect of support condition and rigid-zone on hysteretic behavior: (a) effect of beam contraction; (b) effect of rigid-zone length.
neglected in this study, but they may provide some resistance against beam contraction, although floor slabs also should suffer great damage for large deformations. Refinement of numerical analysis so that it can allow for distortion, cracks, and fracture of cross-sections, and further investigation into the effects of floor slabs on beam contraction are the continuing subjects of research for the writers.
5. Conclusions This paper presented a series of tests for steel beams subjected to cyclic loading to extremely large deformations, and investigated their behavior to complete failure. Major findings obtained from this study are summarized as follows. 1. Four tests having a combination of standard and RBS beams, with and without lateral braces, were conducted. The lateral braces were adopted to mimic floor slabs and arranged on beam top flanges at discrete locations. The behavior in the large deformation range was commensurate with the behavior observed in many previous studies. That is, both the standard and RBS beams exhibited stable hysteresis, and the lateral braces prevented growth of lateral–torsional buckling. 2. Analysis using the large deformation theory was performed to strengthen the experimental findings observed in the large deformation range. The analysis was able to accurately duplicate the experimental behavior involving lateral–torsional buckling and succeeding accumulation of out-of-plane deformations. Additional analysis was conducted to examine the lateral instability effects of RBS beams relative to standard beams. 3. The behavior in extremely large deformations, i.e. the behavior beyond the cyclic loading amplitude of 0.06 rad, was significantly different from the behavior in large deformations. The RBS beam ensured stable behavior up to large deformations, but failed earlier from the reduced cross-section, primarily due to strain concentrations at the RBS portion. Lateral braces that provided restraint against out-of-plane deformations caused strain concentrations in extremely large deformations, leading to earlier fractures for laterally braced beams than for those without such braces. 4. In the extremely large deformation range, significant pinching and increase in the maximum resistance was observed for beams without lateral braces. This increase was given as a result of tensile axial forces induced in the beam, which in turn was generated due to its geometry change. The degree of pinching was affected by the length of beam rigid-zones as well as by the degree of restraint against the beam contraction in its longitudinal direction.
D. Liu et al. / Engineering Structures 25 (2003) 525–535
5. Caution should be made when interpreting the range of deformations presented in this study. The tests were carried out for reduced-scaled specimens with a scale-ratio of about 1/10. As a result, the cross-sections were made of thin plates, which are commonly more ductile than thick plates. Welds, which often trigger earlier fractures, were also avoided in the fabrication of specimens. Deformations that correspond to the limit of stable behavior and the instant of complete failure are likely to be smaller in full-scale beams with welded connections than those presented in this study.
[5]
[6] [7]
[8]
[9]
References [10] [1] Bertero VV, Anderson JC, Krawinkler H. Performance of steel building structures during the Northridge Earthquake. Report no. UCB/EERC-94-09, University of California at Berkeley. The Earthquake Engineering Research Center; 1994. [2] Reconnaissance report on damage to steel building structures observed from the 1995 Hyogoken–Nanbu Earthquake. Kinki Branch of the Architectural Institute of Japan, Steel Committee, Osaka; 1995 [in Japanese with attached abridged English version]. [3] Interim guidelines: evaluation, repair, modification and design of welded steel moment frame structures (August 1995), plus Interim Guidelines Advisory No. 1. Federal Emergency Management Agency, Washington, DC; 1997. [4] Nakashima M, Inoue K, Tada M. Classification of damage to
[11]
[12]
[13]
535
steel buildings observed in the 1995 Hyogoken–Nanbu earthquake. Eng Struct 1998;20(4–6):271–81. Recommended seismic design criteria for new steel moment– frame buildings. Federal Emergency Management Agency, FEMA350, Washington, DC; 2000. Recommendation for design of connections in steel structures. Tokyo: The Architectural Institute of Japan; 2001. Nakashima M, Akazawa T, Igarashi H. Pseudo dynamic testing using conventional testing devices. J Earthquake Eng Struc Dyn 1995;24(10):1409–22. Engelhardt MD. Design of reduced beam section moment connections. Proceedings of 1999 North American Steel Construction Conference. American Institute of Steel Construction, Ill; 1999. Kanao I, Morisako K, Nakamura T. Numerical analysis of elastoplastic lateral buckling behavior of H-shaped beams by beamcolumn finite element model. J Struc Constr Eng Architect Inst Jpn Tokyo 2000;527:95–101 [in Japanese]. Nakashima M, Kanao I, Liu D. Lateral instability and lateral bracing of steel beams subjected to cyclic loading. J Struct Eng ASCE 2002;128(10):1308–16. Uang CM, Fan CC. Cyclic stability criteria for steel moment connections with reduced beam section. J Struct Eng ASCE 2001;127(9):1021–7. Nakashima M, Teteyama E, Morisako K, Suita K. Full-scale test on beam-column subassemblages having connection details of shop-welding type. Proceedings of the Structural Engineers World Conference, San Francisco, USA; 1998 [Paper#:T158–7]. Suita K, Nakashima M, Engelhardt MD. Comparison of seismic capacity between post-Northridge and post-Kobe beam-to-column connections. Proceedings of the Second International Conference on Behavior of Steel Structures in Seismic Areas, Montreal, Canada; 2000. p. 271–8.