DISTORTIONAL BUCKLING OF C AND Z MEMBERS IN BENDING

Transcription

1 DISTORTIONAL BUCKLING OF C AND Z MEMBERS IN BENDING Progress Report Submitted to: The American Iron and Steel Institute (AISI) February 2004 Cheng Yu, Ben Schafer Department of Civil Engineering The Johns Hopkins University Baltimore, Maryland 1

2 Table of Contents Abstract 1. Introduction 2. Distortional Buckling Tests 2.1 Selection of Specimen 2.2 Testing Setup 2.3 Panel-to-Purlin Fastener Configuration 2.4 Tension Tests 2.5 Test Results 2.6 Examination of Several Tests of Note Re-Test of D8C Test of D8.5Z059-6E5W Test of D3.62C054-3E4W Test of D8C033-1E2W 2.7 Comparison with Local Buckling Tests 2.8 Comparison with Current design Specifications 2.9 Comparison with Direct Strength Method 3. Finite Element Analysis 3.1 Finite Element Modeling 3.2 Geometry Imperfection 3.3 Comparison with Experimental Data 4. Conclusions Acknowledgement References 2

3 Abstract Laterally braced cold-formed steel beams generally fail due to local and/or distortional buckling in combination with yielding. For many cold-formed steel studs, joists, purlins, or girts, distortional buckling may be the predominant buckling mode, unless the compression flange is partially restrained by attachment to sheathing or paneling. Distortional buckling of cold-formed steel C and Z members in bending remains a largely unaddressed problem in the current North American Specification for the Design of Col- Formed Steel Structural Members (NAS 2001). Further, adequate experimental data on unrestricted distortional buckling in bending is unavailable. Therefore, a series of distortional buckling tests on C and Z members in bending has been completed at the structural testing lab of Johns Hopkins University. These tests are essentially the extended research (phase 2) of a recently completed experimental project on local buckling of C and Z beams (phase 1, as summarized in Yu and Schafer 2003). This second phase of four point bending tests use nominally identical specimens to the first phase and a similar testing setup. However, the corrugated panel attached to the compression flange is removed in the constant moment region so that distortional buckling may occur. The experimental data is being used to help develop and validate an efficient and reliable design procedure for C and Z members in bending. Further, the experiments are being combined with detailed non-linear finite element modeling in order to make it possible to extend the findings into a broader range of section dimensions, material properties, restraint, and loading conditions. 1. Introduction Determination of the ultimate bending capacity of cold-formed steel C and Z members is complicated by yielding and the potential for local, distortional, and lateral-torsional buckling of the section, as shown in the finite strip analysis of Figure AISI96 Ex M y =12.75kN m M cr / M y Lateral torsional 0.5 Local M cr /M y =0.86 Dist. M cr /M y = half wavelength (mm) Figure 1 Buckling modes of a cold-formed steel beam 3

4 Existing experimental and analytical work indicates that the current North American Specification provisions (NAS 2001) are inadequate for predicting bending capacity of C and Z members when distortional buckling occurs (e.g., Hancock et al. 1996, Rogers and Schuster 1995, Schafer and Peköz 1999, Yu and Schafer 2002). To investigate this problem, a joint MBMA-AISI project was recently completed at Johns Hopkins University entitled: Test Verification of Effect of Stress Gradient on Webs of Cee and Zee Sections. This Phase 1 testing focused on the role of web slenderness in local buckling failures. A panel was through-fastened to the compression flange at a close fastener spacing to insure that distortional buckling and lateral-torsional buckling were restricted. The Phase 1 testing provided the upper-bound capacity for a bending member failing in the local mode. This progress report details Phase 2 work on distortional buckling of the same C and Z members examined in Phase 1. Although many C and Z members in bending have attachments (panel or otherwise) which stabilize the compression flange and help restrict distortional buckling, many do not. Negative bending of continuous members (joists, purlins, etc.) and wind suction on walls and panels (without interior sheathing) are common examples where no such beneficial attachments exist these members are prone to distortional failures. Even when attachment to the compression flange exists it may not fully restrict distortional buckling, as may be the case for thin panels with large center to center fastener spacing and/or thick insulation between the purlin and panel. Flexural members are typically more prone to distortional failures than compression members, due to the dominance of local web buckling in standard compression members. Geometry, unique to flexural members, such as the sloping lip stiffener used in Zs is inefficient in retarding distortional buckling. For example, a typical 8 in. deep Z with t = in. has a distortional buckling stress that is ½ the local buckling stress. The advent of higher strength steels also increases the potential for distortional failures (Schafer and Peköz 1999, Schafer 2002a). In many flexural members, left unrestricted, distortional buckling is the expected failure mode. 2. Distortional Buckling Tests 2.1 Selection of Specimen The distortional buckling tests reported here employ nominally the same geometry as the previously conducted local buckling tests. Specimens were selected to provide systematic variation in web slenderness (h/t) while also varying the other non-dimensional parameters that govern the problem such as flange slenderness (b/t), edge stiffener slenderness (d/t) and relevant interactions, such as the web height to flange width (h/b) ratio. However, as commercially available sections were used, the manner in which the h/t variation could be completed was restricted by the availability of sections. A selection of the cross-sections selected for testing is summarized in Figure 2 and Table 1. 4

5 Table 1 Summary of specimens selected for testing h/t b/t d/t h/b d/b Performed Tests number min max min max min max min max min max Z Study 1: h, b ~d fixed, t varied Z Study 2: h, b ~d fixed, t varied C Study 1: h, b d fixed, t varied C Study 2: b, d, t fixed, t varied Total Figure 2 Typical geometry of tested C and Z members Geometry of the C and Z members used in the distortional buckling tests is summarized in Figure 3. The dimensions of the specimens were recorded at mid-length and middistance between the center and loading points, for a total of three measurement locations for each specimen. The mean dimensions, as determined from the three sets of measurements, are given in Table 2. b c d c b c r hc rdc θ c r hcrdc d c θ c h t h t θ t d t r ht r dt b t r ht r dt bt θ t d t Figure 3 Definitions of specimen dimensions for Z and C The variables used for the dimensions and the metal properties are defined as follows: h b c d c θ c b t d t out-to-out web depth out-to-out compression flange width out-to-out compression flange lip stiffener length compression flange stiffener angle from horizontal out-to-out tension flange width out-to-out compression flange lip stiffener length 5

6 θ t r hc r dc r ht r dt t f y E tension flange stiffener angle from horizontal outer radius between web and compression flange outer radius between compression flange and lip outer radius between web and tension flange outer radius between tension flange and lip base metal thickness yield stress modulus of elasticity Group 1 Group 2 Group 3 Group 4 Table 2 Measured geometry of beams for distortional buckling tests h b Test label Specimen c d c θ c b t d t θ t r hc r dc r ht r dt t f y (in.) (in.) (in.) (deg) (in.) (in.) (deg) (in.) (in.) (in.) (in.) (in.) (ksi) D8.5Z120-4E1W D8.5Z D8.5Z D8.5Z115-1E2W D8.5Z D8.5Z D8.5Z092-3E1W D8.5Z D8.5Z D8.5Z082-4E3W D8.5Z D8.5Z D8.5Z065-7E6W D8.5Z D8.5Z D8.5Z065-4E5W D8.5Z D8.5Z D8.5Z059-6E5W D8.5Z D8.5Z D11.5Z092-3E4W D11.5Z D11.5Z D11.5Z082-3E4W D11.5Z D11.5Z D8C097-7E6W D8C D8C D8C097-5E4W D8C D8C D8C068-6E7W D8C D8C D8C054-7E6W D8C D8C D8C043-4E2W D8C D8C D8C033-1E2W D8C D8C D12C068-10E11W D12C D12C D12C068-1E2W D12C D12C D10C068-4E3W D10C D10C D3.62C054-3E4W D3.62C D3.62C Note: Typical specimen label is DxZ(or C)xxx-x. For example, D8.5Z120-1 means the specimen is 216 mm (8.5 in.) high for the web, Z- section, 3.05 mm (0.12in.) thick and the beam number is 1. Typical test label is DxZ(or C)xxx-xExW. For example, test D8.5Z120-4E1W means the two-paired specimens are D8.5Z120-4 at the east side and D8.5Z120-1 at the west side. 6

7 2.2 Testing Setup The basic testing setup is illustrated in Figure 4 through Figure 6. The 16 ft span length, four-point bending test, consists of a pair of 18 ft long C or Z beams in parallel loaded at the 1/3 points. The members are oriented in an opposed fashion, such that in-plane rotation of the C or Z leads to tension in the panel, and thus provides additional restriction against lateral-torsional buckling. Small angles, in., are attached to the tension flanges every 12 in. Hot-rolled tube sections, in., bolt the pair of C or Z beams together at the load points and the supports, and insure shear and web crippling problems are avoided at these locations. Figure 4 Elevation view of distortional buckling tests Figure 5 Panel setup for distortional buckling tests Figure 6 Loading point configuration 7

8 No panel is placed inside the constant moment region (Figure 5). Instead, the throughfastened panel, t = in., 1.25 in. high rib, is attached to the compression flanges in the shear spans only to restrict both the distortional and the lateral-torsional buckling in these regions, but leaving distortional buckling free to form. The loading system employs a 20 kip MTS actuator, which has a maximum 6 in. stroke. The test was performed in displacement control at a rate of in./sec An MTS 407 controller and load cell monitored the force and insured the desired displacement control was met. Meanwhile, specimen deflections were measured at the 1/3 points with position transducers. 2.3 Panel-to-Purlin Fastener Configuration The panel-to-purlin fastener configuration employed in the distortional buckling tests is the same as that used in the earlier local buckling tests, except the through-fastened panel in the constant moment region is removed. This setup is expected to restrict latertorsional buckling while allowing distortional and local buckling to occur. Examination of the ratio of the elastic distortional buckling moment (M crd ) to elastic local buckling moment (M crl ) indicates that a large number of members, particularly the Zs, are anticipated to fail in a mechanism dominated by distortional buckling (i.e., M crd /M crl 1). Even when M crd /M crl >1 distortional buckling may govern because of reduced postbuckling strength in distortional failures (Schafer and Peköz 1999). Table 3 shows the elastic buckling loads of all performed tests. In Table 3, M crl -CUSFM and M crd -CUSFM respectively is the summation of local and distortional buckling moments for the two specimens in each test (calculated by the finite strip software CUSFM). M crl -ABAQUS, M crd -ABAQUS and M crltb -ABAQUS respectively is elastic local, distortional and lateral-torsional buckling loads calculated by ABAQUS considering the complete testing setup. M y is the summation of yield moments of both specimens in each test. As shown in Table 3, all tests except D8C097-5E4W have either local (M crl -ABAQUS) or distortional buckling (M crd -ABAQUS) to be the first buckling mode and later-torsional buckling (M crltb -ABAQUS) is successfully restricted. Further, the test setup does not change the local and distortional buckling moments significantly (compare CUFSM vs. ABAQUS results to observe this). Distortional buckling is expected to be the failure mechanism for all Z beams and local or distortional for the C beams. Figure 7, 8 and 9 show typical elastic buckling results by ABAQUS. 8

9 Test label Table 3 Elastic buckling loads of performed tests M crl - M crd - M cr-64in. - CUFSM CUFSM CUFSM (kips-in.) (kips-in.) (kips-in.) M y (kips-in.) M crl - ABAQUS (kips-in.) M crd - ABAQUS (kips-in.) M crltb - ABAQUS (kips-in.) D8.5Z120-4E1W D8.5Z115-1E2W > D8.5Z092-3E1W D8.5Z082-4E3W >566 D8.5Z065-7E6W >302 D8.5Z059-6E5W >255 D11.5Z092-3E4W >760 D11.5Z082-3E4W >608 D8C097-5E4W D8C068-6E7W >338 D8C054-7E6W >144 D8C043-4E2W >107 >107 D8C033-1E2W >42 >42 D10C068-4E3W >227 D12C068-1E2W >256 D3.62C054-3E4W > Note: lower bounds are given to those modes which are not included in the first 30 eigenmodes calculated in the ABAQUS analysis. Figure 7 Lateral-torsional buckling mode of beam D8C097-5E4W Figure 8 Distortional buckling mode of beam D8C097-5E4W 9

10 2.4 Tension Tests Figure 9 Local buckling mode of beam D8C097-5E4W Tension tests were carried out following ASTM E8 00 Standard Test Methods for Tension Testing of Metallic Material (ASTM, 2000). Three tensile coupons were taken from the end of each specimen: one from the web flat, one from the compression flange flat, and one from the tension flange flat. A screw-driven ATS 900; with a maximum capacity of 10 kips was used for the loading. An MTS E-54 extensometer was employed to monitor the deformation. Two methods for yield strength determination were employed: 1) 0.2% Offset Method for the continuous yielding materials; and 2) Auto Graphic Diagram Method for the materials exhibiting discontinuous yielding, Figure 10. Test results are summarized in Table 4. All the Z beams have similar material properties; the yield stresses are between 60 to 70 ksi and for most Z beams the f u /f y ratios are around 130%. On the contrary, the C beams have greatly varying material properties, the yield stresses are measured from 20 to 85 ksi, and the range of f u /f y ratios is from the lowest 101% (for a high strength material) to the highest 207% (for a low strength material). In all cases the tested yield stresses are employed to calculate the beam strength. The elastic moduli E is assumed to be ksi in all of the members. This E value is supported by tension test results during Phase 1 experiments. (a) Continuous yielding curve (b) Discontinuous yielding curve Figure 10 Typical stress-strain curve of a tensile test 10

11 Table 4 Summary of tension test results Specimens t (in.) f y (ksi) f u (ksi) f u / f y D12C % D12C % D12C % D12C % D10C % D10C % D8C % D8C % D8C % D8C % D8C % D8C % D8C % D8C % D8C % D8C % D8C % D8C % D3.62C % D3.62C % D11.5Z % D11.5Z % D11.5Z % D11.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % D8.5Z % 2.5 Test Results Summaries of the distortional buckling test results are given in Table 5. Included for each test are the elastic buckling moments (M crl, M crd ) as determined by the finite strip method using CUFSM (Schafer 2001) and the ratios of test-to-predicted capacities for several design methods including the existing American Specification, M AISI (AISI 1996), the 11

12 Group 1 Group 2 Group 3 Group 4 existing Canadian Standard, M S136 (S ), the newly adopted North American Specification, M NAS (NAS 2001), the existing Australia/New Zealand Standard, M AS/NZS (AS/NZS 1996), the existing European Standard EN1993, M EN1993 (EN ) and the recently proposed Direct Strength Method (DSM, Schafer and Peköz, 1998; Schafer, 2002a,b M DSl for local failure M DSd for distortional failures). The actuator load-displacement response of all distortional buckling tests are given in Figure 11 to Figure 14. Compared with the phase 1 local buckling tests, more non-linear response is observed prior to formation of the failure mechanism. The specimens which have a capacity at or near the yield moment (M test /M y ~ 1, see Table 5) exhibit the most nonlinear deformation prior to failure; while the more slender specimens have essentially elastic response prior to formation of a sudden failure mechanism. Table 5 Distortional buckling test results M test (kips-in.) Test label Specimen M y M crl M crd M test / M test / M test / M test / M test / M test / M test / M test / (kips-in.) (kips-in.) (kips-in.) M y M AISI M S136 M NAS M AS/NZS M EN1993 M DSl M DSd D8.5Z120-4E1W D8.5Z120-4* D8.5Z D8.5Z115-1E2W D8.5Z D8.5Z115-1* D8.5Z092-3E1W D8.5Z092-3* D8.5Z D8.5Z082-4E3W D8.5Z D8.5Z082-3* D8.5Z065-7E6W D8.5Z065-7* D8.5Z D8.5Z065-4E5W D8.5Z D8.5Z065-4* D8.5Z059-6E5W D8.5Z059-6* D8.5Z D11.5Z092-3E4W D11.5Z D11.5Z092-3* D11.5Z082-3E4W D11.5Z082-4* D11.5Z D8C097-7E6W D8C D8C097-6* D8C097-5E4W D8C097-5* D8C D8C068-6E7W D8C D8C068-7* D8C054-7E6W D8C D8C054-6* D8C043-4E2W D8C043-4* D8C D8C033-1E2W D8C D8C033-1* D12C068-10E11W D12C068-11* D12C D12C068-1E2W D12C D12C068-1* D10C068-4E3W D10C068-4* D10C D3.62C054-3E4W D3.62C D3.62C054-3* Note *: Controlling specimen, weaker capacity by AISI (1996) 12

13 D8.5Z120-4E1W D8.5Z115-1E2W D8.5Z092-3E1W D8.5Z082-4E3W D8.5Z065-4E5W D8.5Z065-1E1W 7 D8.5Z059-2E1W 7 Figure 11 Actuator force-displacement responses of distortional buckling tests - Group 1 D8C097-7E6W D8C097-5E4W D8C054-7E6W D8C043-4E2W D8C033-1E2W Figure 12 Actuator force-displacement responses of distortional buckling tests - Group 2 13

14 D11.5Z092-3E4W D11.5Z082-5E4W Figure 13 Actuator force-displacement responses of distortional buckling tests - Group 3 D12C068-1E2W D12C068-10E11W D10C068-4E3W D3.62C054-3E4W Figure 14 Actuator force-displacement responses of distortional buckling tests - Group 4 14

15 2.6 Examination of Several Tests of Note Re-Test of D8C097 As shown in Table 3, D8C097-5E4W is the only test which has a lower lateral-torsional buckling moment than that of local or distortional buckling (Figure 7, 8 and 9). In fact, lateral-torsional buckling failure was observed in the testing, as shown in Figure 15. Therefore, a re-test of D8C097-7E6W with proper modifications was performed. In the new test, an additional angle was used to connect the two purlins at midpsan, in order to restrict later-torsional buckling (Figure 16) while at the same time not to boost the distortional buckling moment. Figure 15 Test D8C097-5E4W with standard setup Figure 16 Test D8C097-7E6W with angle added Elastic finite element analysis was performed to examine the new testing setup. After the angle is added to the compression flange the lateral-torsional buckling moment is increased from 600 kips-in., to 618 kips-in. Distortional buckling remains at almost the same value of 631 kips-in. Without the benefit of a nonlinear finite element analysis it was still decided to examine the experimental results with the angle attached. Figure 16 shows the failure mode of the new test (D8C097-7E6W) and Figure 7 shows the comparison of the two tests on nominally identical beams. The new test provided an increased strength and featured a distortional buckling failure mechanism. Further, the rotation and lateral movement observed in the previous test did not occur. Comparison of the test-to-predicted results for the distortional buckling prediction of DSM improve from 84% for D8C097-5E4W to 99% for D8C097-7E6W. Test D8C097-7E6W is therefore believed to fail in distortional buckling, and its data is included in subsequent analyses. D8C097-5E4W which provides an examination of the role of lateral-torsional buckling is not considered a distortional buckling test for the purposes of this report. 15

16 D8C097-7E6W D8C097-5E4W Test of D8.5Z059-6E5W Figure 17 Actuator load-displacement curves As shown in Figure 18, the first failure in test D8.5Z059-6E5E occurred outside of the constant moment region. Initial geometric imperfections, uneven specimen setup and shear-bending interaction are possible reasons for the unexpected failure mode. As a result, the test provided a very low bending capacity, about 17% off the Direct Strength distortional mode prediction and 35% off the section strength by NAS (2001). The test data for this specimen is not included in the subsequent analyses of distortional buckling tests and indicates a need for further analysis of these thinnest of members. (a) local failure begins Test of D3.62C054-3E4W Figure 18 Local failure of test D8.5Z059-6E5W (b) deformation continues Test D3.62C054-3E4W is believed to fail primarily by material yielding, based on the experimental data and observation during the test. The beam s calculated yield moment is 33 kips-in. which is much lower than the first elastic buckling (distortional buckling) moment of 66 kips-in. The test result is 34.2 kips-in. and this beam showed significant non-linearity but no sharp strength loss during the loading process. 16

17 2.6.4 Test of D8C033-1E2W According to the Direct Strength Method predictions, test D8C033-1E2W has lower local buckling strength than that of distortional buckling. Local failures in the compression flanges were observed during the testing (Figure 19). Therefore this test is believed to fail in local buckling mode and the data is sorted in the local buckling tests. This test is indicative of a trend throughout the data and one observed elsewhere before (Schafer 2002a) thinner members are more prone to local, not distortional, failures. Figure 19 Failure of test D8C033-1E2W 2.7 Comparison with Local Buckling Tests For the Z beams Figure 20 provides a typical comparison between the local buckling and distortional buckling tests. The buckling wavelength is visibly longer in the distortional buckling test and the compression flange rotates about the web/compression flange junction. This is expected as the Z beams have an elastic distortional buckling moment (M crd ) which is lower than local buckling for all the tests. Some of the C beams exhibited similar behavior, but in general the response of the C beams is more complicated. All of the C beams were observed to buckle at longer wavelengths than in the local buckling tests. Typically the compression flanges of the C beams did not exhibit the same large rotations as observed in the Z beams. In the post-buckling range the majority of C beams include some rotation of compression flange, but in many cases translation and rotation of the cross-section as well. This observation indicates a more complicated collapse response and the possible interaction of distortional buckling with local/lateraltorsional buckling in the C beams. (a) Local buckling test of 11.5Z092-1E2W Figure 20 Comparison of tests on 11.5Z092 (b) Distortional buckling test of D11.5Z092-3E4W 17

18 Among 25 local buckling and 19 distortional buckling tests, 9 pairs of tests use beams with nominally identical geometry and material. The test comparison for these specimens is summarized in Table 6. The notations of P y, P crl, P crd are respectively the actuator load, P, that causes yielding, elastic local buckling, or elastic distortional buckling in the beam. On average, the beam bending strength will lose 17% when the through-fastened panel is removed from the compression flanges. The actuator load-displacement curves and failure modes for these 9 pairs of tests are provided in Figure 21 through 38. It is shown that the two tests in each pair have the same elastic stiffness, but different peak loads and buckling and post-buckling behavior. The distortional buckling tests present more nonlinear behavior before failure than the local buckling tests. More significant deformations in the web were observed in the local buckling tests. For the distortional buckling tests, the failure in the compression flange is dominant. At the same time, lateral-torsional buckling is also involved in post-buckling region for some distortional buckling tests. It can be seen that with the through-fastened panel removed, the cold-formed steel C and Z beams have lower bending strength and less flexibility (less out-of-plane deflection before buckling) as expected. Pair No. Table 6 Comparison of 9 pairs of tests having the same nominal geometry and material property Local buckling test label Distortional buckling test label Actuator peak load of local test P L (klb) Actuator peak load of distortional test P D (klb) P D /P L 1 8.5Z120-3E2W D8.5Z120-4E1W % 2 8.5Z092-4E2W D8.5Z092-3E1W % 3 8.5Z082-1E2W D8.5Z082-4E3W % 4 8.5Z059-2E1W D8.5Z059-4E3W % 5 8C054-1E8W D8C054-7E6W % 6 8C043-5E6W D8C043-4E2W % 7 12C068-3E4W D12C068-1E2W % 8 12C068-9E5W D12C068-10E11W % C054-1E2W D3.62C054-3E4W % Average 83% 18

19 Figure 21 Comparison of tests on 8.5Z120 (a) Local buckling test Figure 22 Comparison of tests on 8.5Z120 (b) Distortional buckling test 19

20 Figure 23 Comparison of tests on 8.5Z092 (a) Local buckling test (b) Distortional buckling test Figure 24 Comparison of tests on 8.5Z092 20

21 Figure 25 Comparison of tests on 8.5Z082 (a) Local buckling test (b) Distortional buckling test Figure 26 Comparison of tests on 8.5Z082 21

22 Figure 27 Comparison of tests on 8.5Z059 (a) Local buckling test (b) Distortional buckling test Figure 28 Comparison of tests on 8.5Z059 22

23 P y Figure 29 Comparison of tests on 8.5Z054 (a) Local buckling test (b) Distortional buckling test Figure 30 Comparison of tests on 8.5Z054 23

24 Figure 31 Comparison of tests on 8C043 (a) Local buckling test (b) Distortional buckling test Figure 32 Comparison of tests on 8C043 24

25 Figure 33 Comparison of tests on 12C068 (a) Local buckling test (b) Distortional buckling test Figure 34 Comparison of tests on 12C068 25

26 Figure 35 Comparison of tests on 12C068 (a) local buckling test (b)distortional buckling test Figure 36 Comparison of tests on 12C068 26

27 Figure 37 Comparison of tests on 3.62C054 (a) Local buckling test (b) Distortional buckling test Figure 38 Comparison of tests on 3.62C Comparison with Current Design Specifications Six design methods are considered for comparison: AISI (1996), S136 (1994), AS/NZS (1996), NAS (2001), EN1993 (2002) and DSM (2003). Specific specification predictions of the tested beams are listed in Table 5. Table 7 provides a summary of the test-topredicted ratios. In average, all six methods give good strength predictions to the local buckling tests. DSM uses a single strength curve, while the other five methods apply effective width concepts in the calculation of bending strength. For the distortional buckling, only AS/NZS, EN1993 and DSM have specific equations for this mode. Table 27

28 7 shows that all three methods give good predictions to the tests, though the Eurocode predictions still remain about 5% unconservative on average. The Direct Strength Method has the best statistical results for any of the distortional buckling methods. AISI, S136 and NAS provide systematically unconservative predictions for the distortional buckling strength, the average error is between 10~15%. AISI (1996), S136 (1994) and NAS (2001) are only applicable to local buckling failures. Figure 39 provides a graphical comparison of the NAS predictions for both series of tests. Local buckling tests Distortional buckling tests Table 7 Summary of test-to-predicted ratios for existing and proposed design methods Controlling specimens Second specimens Controlling specimens Second specimens M test / M AISI M test / M S136 M test / M NAS M test / M AS/NZS M test / M EN1993 M test / M DSl M test / M DSd Average Standard deviation Average Standard deviation Average Standard deviation Average Standard deviation Figure 39 Comparison of NAS predictions with experimental results Figure 40 provides a plot of the comparison focusing on distortional buckling tests only. It can be seen that DSM and AS/NAS have similar results for these beams. 28

29 Figure 40 Test-to-predicted ratios vs. web slenderness for distortional buckling tests 2.9 Direct Strength Method The design methods used in current specifications are based on the effective width concept. The effective width method examines the buckling behavior of elements forming a cross section in isolation and uses this information to form effective member properties. As sections become more complex, with, for example, additional edge and intermediate stiffeners, the computation of the effective width becomes more complex as well. Interaction between the elements also occurs in all buckling modes - so that consideration of the elements in isolation is less accurate. To overcome these problems, a new method has been developed by Schafer and Peköz 1998a, called the Direct Strength Method (DSM). DSM employs elastic buckling solutions for the entire member rather than the individual element, and strength curves for the entire member in all of the relevant buckling modes. The method has the advantage that calculations for complex sections are simple and reliable. In July 2003, the Direct Strength Method was approved by the American Iron and Steel Institute Committee on Specifications and it has been documented as Appendix 1 of the North American Specification for the Design of Coldformed Steel Structural Members, DSM (2003). DSM provides specific predictions for both local and distortional buckling of beams and is briefly summarized here. Since the beams are fully laterally braced, the maximum capacity due to long wavelength buckling M ne = M y. For local buckling the capacity is: λ l 0.776, M DS l = M y (1) λ l > 0.776, ( ( ) )( ) M 0.4 M 0.4 crl crl M DS l M M M y λ l = = (2) M y M cr (3) l y y 29

30 Where M crl = critical elastic local buckling moment and M y = moment at first yield. For distortional buckling the capacity is: λ d 0.673, M DSd = M y (4) λ d > 0.673, ( ( ) )( ) M 0.5 M 0.5 crd crd M DSd M M M y λ d = = (5) M M (6) y crd y y Where M crd = Critical elastic distortional buckling moment. As shown in Table 7, the Direct Strength Method provides good agreement with both test series. The high test-topredicted ratios for distortional buckling (M test /M DSd ) in the local buckling tests indicate that distortional buckling was successfully restricted with the testing details employed in the Phase 1 testing. The overall agreement for M DSd in the distortional buckling tests indicates that distortional buckling dominated the failure mechanism when the compression flanges were unrestrained and validates the general expression used for distortional buckling in the DSM method (which was calibrated to other data). Figure 41 provides a graphical representation of Equations 1-6 along with the results for both series tests. The average upper and lower bounds of laterally braced beams are well captured by the local and distortional buckling equations of the Direct Strength method. Local buckling test data are relatively more concentrated along the corresponding design curve than that of distortional buckling tests. The distortional buckling data show more deviation as the slenderness (M y /M cr ) increases. Figure 41 Direct Strength Method vs. test result 30

31 3. Finite element analysis 3.1 Finite Element Modeling The experimental results build a solid background for further finite element analysis that can expand the buckling study to examine further cross sections, different loading and support conditions, and other structural details used in today s industry. Elastic eigenvalue buckling analysis has already been used to determine the panel fastener configuration for local buckling tests. However, in order to simulate the real flexural bending tests, or to capture the buckling mechanism of cold-formed steel members, nonlinear inelastic finite element analysis is necessary. The commercial finite element package ABAQUS 6.2 (ABAQUS 2001) is employed in the nonlinear finite element analysis. An overall view of the finite element model is presented in Figure 42. Four-node linear shell elements with reduced integration are used for the purlin, panel and tubes (S4R). The loading beam uses 8-node linear solid elements (C3D8). The tube and purlin is connected by imposing 4 nodes (to simulate the four bolts) from purlin to the surface of tube (*Tie in ABAQUS). The same method is also used for the connection between the panel and the purlin. The loading beam is simply connected to the tubes (*pin in ABAQUS). The angle at the bottom of the purlins is simulated by keeping the distance between two nodes on the tension flanges constant (*Link in ABAQUS). Details of the various tie constraints in the model are shown in Figure 43. The tubes and the loading beam are modeled as rigid bodies by setting an artificially high elastic modulus E (E = 10E steel ). In addition, for numerical efficiency the panel is assumed to be an elastic material (note, typical yield stress for the panel is quite high, about 100 ksi, and failures were observed to initiate in the members, not the panels, for all local and distortional buckling tests). Thus, material non-linearity is considered only for the C or Z itself. Actual tension test results are used for the material stress-strain properties, after converting from engineering stress-strain ( σ e, ε e ) to true stress-strain ( σ t, ε t ) assuming the following relationship: σ = σ 1+ ε ), ε = ln(1 + ε ) t e( e t e loading point Figure 42 Finite element modeling of local buckling tests. 31

32 Shell element (S4R) Solid element (C3D8) Pin connection between Load beam and tube Tie connection between purlin and tube Shell element (S4R) Tie connection between purlin and panel Figure 43 Details of finite element modeling The finite element analysis is performed in a displacement control mode, mimicking the same loading method of the actual testing. The automatic Stabilization technique (*stabilize in ABAQUS) is adopted in the nonlinear static analysis, ABAQUS offers the option to stabilize this class of problems which involve local instabilities by applying artificial damping throughout the model in such a way that the viscous forces introduced are sufficiently large to prevent instantaneous buckling or collapse but small enough not to affect the behavior significantly while the problem is stable. The arc-length based Riks method was also considered in the analysis, however compared with the automatic Stabilization method, the latter has consistently provided better simulation of the actual tests and less convergence problems near the peak loads. Thus, the automatic Stabilization method is chosen for the analyses. 3.2 Geometric Imperfection Existing in every real beam, the geometric imperfections of a member include bowing, warping, twisting and local deviations. Geometric imperfections have a significant effect on the strength and post-buckling behavior of many C and Z beams. Therefore, it is necessary and important to consider geometric imperfections in the FE modeling. Previous research has measured geometric imperfections of cold-formed steel members, and sorted in two categories: type 1, maximum local imperfection in a stiffened element and type 2, maximum deviation from straightness for a lip stiffened or unstiffened flange as shown in Figure 44 (Schafer and Peköz 1998b). The histogram (Figure 45, 46) and cumulative distribution function (CDF) values (Table 8) of the maximum imperfection for the two types are available. A CDF value is written as P( <d) and indicates the probability that a randomly selected imperfection value,, is less than a deterministic imperfection, d. 32

33 1 2 Figure 44 Definition of geometric imperfections Table 8 CDF values for maximum imperfection (Schafer and Peköz 1998b) Type 1 Type 2 P( <d) d 1 /t d 2 /t mean st. dev We did not measure the geometric imperfections of the beams in both phases of the tests; therefore the imperfections used in our finite element modeling were based on the CDF values (from Schafer and Peköz 1998) summarized in Table 8. Knowing the amplitude of imperfections in the lowest eigenmodes is often sufficient to characterize the most influential imperfections. We conservatively assume that the type 1 imperfection may be applied to local buckling mode and the type 2 imperfection applied to the distortional buckling mode number of observations d/t Figure 45 Histogram of type 1 imperfection (Schafer and Peköz 1998b) number of observations d/t Figure 46 Histogram of type 2 imperfection (Schafer and Peköz 1998b) For each test two different magnitudes of geometric imperfections are generated, the first uses a 25% CDF maximum magnitude, and the second a 75% CDF imperfection, thus covering the middle 50% of anticipated imperfection magnitudes. The imperfection shape is a superposition of the local and distortional buckling mode scaled to the appropriate 33

34 CDF value. For numerical efficiency the buckling shape is obtained by using finite strip analysis in CUFSM and the resulting coordinates used to offset the geometry of the ABAQUS model appropriately. 3.3 Comparison with Experimental Data The finite element analysis results are summarized in Table 9 for the local buckling tests and Table 10 for the distortional buckling tests. On average, the failure load is bounded by the two finite element simulations with 25% and 75% CDF maximum imperfections. The pair of simulations show that the middle 50% of imperfections exhibit a range of 14% of the bending capacity, thus providing a measure of the imperfection sensitivity. Average test-to-predicted ratios for the finite element analysis of the local buckling tests are closer to 1 (100%) than the distortional buckling tests. The finite element analysis for the distortional buckling tests shows slightly greater scatter (greater imperfection sensitivity) and the mean response of the FE simulations is slightly greater than the average tested strength. Figure 47 shows the FEM accuracy vs. web slenderness, and indicates a slight tendency for the finite element analysis to over-predict the observed strength for very slender sections and under-predict for stockier sections. This may be partially driven by the choice of a constant d/t imperfection size thus leading to smaller imperfection sizes for the more slender members which typically employ thinner material. In total it is concluded that the elastic, post-buckling, and peak strength behavior of the beams are well simulated by this finite element model and its assumptions. However, the post-collapse behavior and final mechanism formation is only approximated by the model. Lack of agreement in the large deflection post-collapse range could be a function of the solution scheme (e.g., use of artificial damping via the *stabilize option) or more basic modeling assumptions, such as ignoring any plasticity in the panels. Loaddeflection response for six of the approximately 8 in. deep sections is given in Figure FEM-to-test ratio % CDF 75% CDF web slenderness = lweb = (fy/fcr_web)0.5 Figure 47 Comparison of finite element results with tests 34

35 With the confidence obtained in the comparison with real tests, the finite element simulation is now being extended to other practical sections which are not included in both series of tests. The maximum imperfection of 50% CDF is applied in the extended finite element simulations. 25% and 75% CDF simulation will be used to examine the imperfection sensitivity for certain sections. Based on the statistical results of finished analysis, the 50% CDF simulation, on average, will give a strength prediction within an error of 7%. Table 9 Summary of finite element analysis results for local buckling tests Note: P test (lbs) P 25% P 25% /P test P 75% (lbs) P 75% /P test 8.5Z120-3E2W % % 8.5Z105-2E1W % % 8.5Z092-4E2W % % 8.5Z082-1E2W % % 8.5Z073-4E3W % % 8.5Z065-3E1W % % 8.5Z059-2E1W % % 11.5Z073-2E1W % % 11.5Z082-2E1W % % 11.5Z092-1E2W % % 8.5Z059-4E3W % % 8C097-2E3W % % 8C068-4E5W % % 8C054-1E8W % % 8C043-5E6W % % 6C054-2E1W % % 4C054-1E2W % % 12C068-9E5W % % 3.62C054-1E2W % % 12C068-3E4W % % 10C068-2E1W % % 8C068-1E2W % % 8C043-3E1W % % mean 106% 93% standard deviation 6% 7% P test : Peak tested actuator load P 25% : Peak load of simulation with 25% CDF of maximum imperfection P 75% : Peak load of simulation with 75% CDF of maximum imperfection 35

36 Table 10 Summary of finite element analysis results for distortional buckling tests Note: Test Label P test (lbs) P 25% P 25% /P test P 75% (lbs) P 75% /P test D8.5Z120-4E1W % % D8.5Z115-1E2W % % D8.5Z092-3E1W % % D8.5Z082-4E3W % % D8.5Z065-7E6W % % D11.5Z092-3E4W % % D8.5Z065-4e5W % % D11.5Z082-4E3W % % D12C % % D8C043-4E2W % % D12C % % D8C033-1E2W % % D8C054-7E6W % % D10C068-4E3W % % D8C097-5E4W % % D3.62C054-3E4W % % mean 110% 95% standard deviation 8% 8% P test : Peak tested actuator load P 25% : Peak load of simulation with 25% CDF of maximum imperfection P 75% : Peak load of simulation with 75% CDF of maximum imperfection 36

37 Figure 48 Comparison of selected tests with finite element analysis results 37

38 4. Conclusions Experiments on a wide variety of industry standard laterally braced C and Z beams where the compression flange is unrestrained over a distance of 64 in. indicate that distortional buckling is the most likely failure mode. Distortional failures occur even when local buckling is at a lower critical elastic moment than distortional buckling. Previous testing (Phase 1 of this project) demonstrated that if additional rotational restraint can be provided to the compression flange, such as through engagement of a through-fastened deck, the distortional mode can be avoided and a local mode triggered instead. While in nearly all of the sections distortional buckling dominated the failure other limit states are possible, even at unbraced lengths of 64 inches. For example, in one test on an 8 in. deep C beam with a nominal t = in. and a 64 in. unbraced length lateral-torsional buckling (LTB) initiated the failure. The thicker specimens have high local and distortional buckling stresses and can thus be controlled by LTB. The thinnest specimens may also be controlled by other limit states, one member was observed to fail in local buckling, and another in a shear + bending failure. While these failure modes were uncommon they serve to demonstrate the variety of behavior that may occur at even modest unbraced lengths. Comparison of the experimental results with existing and proposed design specifications indicates that the previously employed American Specification (AISI) and Canadian Standard (S136) as well as the newly adopted joint North American Specification (NAS) provide a poor prediction of the strength for members failing in the distortional mode. Errors are, on average, % unconservative for these design specifications. The Eurocode which provides some measures for distortional buckling is, on average, 5% unconservative. Two methods which include explicit procedures for distortional buckling the Australian/New Zealand (AS/NZS) standard and the Direct Strength Method (recently adopted as Appendix 1 of the NAS) provide far better predictions in distortional failures with conservative errors of, on average, 2% and 1% for the respective methods. A non-linear finite model of the testing has been setup and verified by the experimental data. Using the experiments and the finite element analysis, research is now underway to explore the influence of partial restraint and moment gradient on the bending strength of laterally braced cold-formed steel members and development of code provisions to account for these phenomena. Acknowledgement The sponsorship of AISI for both phases of this work, and MBMA for phase 1 and a graduate fellowship is gratefully acknowledged. Donation of materials by Varco-Pruden, Clark Steel and the Dietrich Design Group is also gratefully acknowledged. The assistance of the AISI task group in developing the testing plan is appreciated. Don Johnson, Maury Golovin, Joe Nunnery, Joe Wellinghoff, and Steve Thomas have all been helpful with their ideas and generous with their time. The experiments would not have been possible without the help of Johns Hopkins undergraduates, Liakos Ariston, Brent Bass, Andrew Myers, Sam Phillips, and Tim Ruth, Sam s work during Phase 1 and Tim s during Phase 2 has been particularly helpful. Finally, the work has been helpfully aided by the laboratory technician Jack Spangler, and our machinist Jim Kelly. 38