CYCLIC BEHAVIOR OF DEEP SLENDER WIDE-FLANGE STEEL BEAM- UNDER COMBINED LATERAL DRIFT AND AXIAL LOAD

Transcription

1 CYCLIC BEHAVIOR OF DEEP SLENDER WIDE-FLANGE STEEL BEAM- UNDER COMBINED LATERAL DRIFT AND AXIAL LOAD A. Elkady 1 and D. G. Lignos 2 ABSTRACT During a seismic event, first-story ( drift demands coupled with - selected behavior of wide-flange beam-columns i FEA simulations are used to evaluate: (a) the flexural strength of the wide-flange beam-columns; (b) their - a - web and flange slenderness ratios near the compactness limits of current seismic design provisions. It is found that a deep slender wide flange column experiences more than two times the axial shortening of a stocky wide flange column. - wide-flange sections undergo lateral loading - current modeling recommendations 1 PhD Candidate, Dept. of Civil Eng. and Applied Mechanics, McGill University, Montreal, Canada H3A C3 2 Assistant Professor, Dept. of Civil Eng. and Applied Mechanics, McGill University, Montreal, Canada H3A C3 Elkady A, Lignos DG. Cyclic behavior of deep - - axial load. Proceedings of the 1 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 214.

2 1NCEE Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21-25, 214 Anchorage, Alaska - - Drift And Axial Load A. Elkady 1 and D. G. Lignos 2 ABSTRACT During a seismic event, first-story columns in steel Special - In this paper, - - flexural strength of the wide-flange beam-column - - sections with web and flange slenderness ratios - capping plastic rota - - Introduction Deep wide-flange columns are typically used as part of steel buildings with perimeter special moment frames (SMFs) in North America. These members are often selected over stocky sections due to their high inertia-to-weight ratio. Deep slender wide-flange sections are characterized by: 1) depths larger than 46 mm (i.e., 16 inches); and 2) compact webs and λ hd )based on [1]. The high slenderness ratios of the webs and flanges imply weak resistance to twisting and out-of-plane buckling in addition to susceptibility to local buckling failure modes. Experimental data on the cyclic behavior of deep slender wide-flange beam-columns is not presently available in the literature. Limited beam-column tests were conducted on either small and/or stocky sections [2-4]. These tests demonstrated that stocky sections (i.e., h/t w <3 and b f /2t f <6) perform well even at high compressive axial load ratios of P/P y =.8, where P y is the axial yield strength of a steel column (P y is the product of the cross section gross area A gross 1 PhD Candidate, Dept. of Civil Eng. and Applied Mechanics, McGill University, Montreal, Canada H3A C3 2 Assistant Professor, Dept. of Civil Eng. and Applied Mechanics, McGill University, Montreal, Canada H3A C3 Elkady A, Lignos DG. - b - axial load. Proceedings of the 1 th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 214.

3 and the yield stress F y ). Experimental studies on deep slender wide-flange beams [5, 6] have demonstrated that these sections deteriorate fast in flexural stiffness due to web and flange local buckling under cyclic loading. This paper investigates the cyclic behavior of a range of wide-flange beam-columns through detailed finite element (FE) analysis. These beam-columns are subjected to combined lateral drift and axial load ratios. A typical first-story interior column is considered for this investigation. The FE model simulates the rotational flexibility expected at the column's top end. The FE results are used to evaluate and assess the following issues: (a) the cyclic deterioration in flexural strength of beam-columns; (b) the pre-capping plastic rotation of beam-columns; and (c) the axial shortening of a steel column when it is subjected to cyclic loading. Member Sizes and Loading Protocols A set of 4 wide-flange steel sections is selected for the FE study. The selected sections represent a wide spectrum of web and flange slenderness ratios. The range of sections covers those typically used as first-story columns of SMFs in low- and mid-rise buildings in North America. The selected sections along with some of their main geometric properties are summarized in Table 1. The web and flange slenderness ratios of 35 sections comply with the compactness limits for highly ductile members (λ hd ) as specified by [1] even at high levels of axial load ratio (i.e., P/P y =.5). The remaining 5 sections (marked with asterisk), are compact sections with web and flange slenderness ratios that comply with the compactness limits specified for moderately ductile members (λ md ). It should be noted that for highly ductile members, the compactness limits for the flange slenderness is 7 and the compactness limit for the web slenderness is 43, at P/P y =.5. Table 1.Summary of wide-flange steel sections used in the FE study. W36 W33 W3 W27 Section b f /2t f h/t w r y [mm] C w [mm 6 ] J [mm 4 ] W36x E E+8 W36x E E+8 W36x E+14 8.E+7 W36x E E+7 W36x E E+7 W36x E E+7 W36x135* E E+6 W33x E E+7 W33x E E+7 W33x E+13 2.E+7 W33x E E+7 W33x E E+6 W33x118* E E+6 W3x E E+7 W3x E E+7 W3x E E+7 W3x E E+7 W3x E+13 6.E+6 W3x9* E E+6 W27x E E+8 W27x E E+7 W27x E E+7

4 Table 1. (continue) Summary of wide-flange steel sections used in the FE study. Section b f /2t f h/t w r y [mm] C w [mm 6 ] J [mm 4 ] W27x E E+7 W27 W27x E E+6 W27x84* E E+6 W24x E E+7 W24x E E+7 W24 W24x E E+7 W24x E E+7 W24x E E+6 W24x E+13 4.E+6 W24x68* E E+5 W21x E E+7 W21 W21x E E+6 W21x E E+6 W21x E E+6 W16x E E+6 W16 W16x E E+6 W16x E E+5 W16x E E+5 * Compact λ md ) sections In order to thoroughly evaluate the FE results, the selected 4 wide-flange sections are divided into 8 sets based on their flange and web slenderness (see Fig. 1). From this figure, sets numbered 1, 2, and 3 represent compact sections with low, moderate, and high web and flange slenderness ratios, respectively. Set 4 represents the 5 compact () sections (see Fig. 1). Set F1 Set F2 Set F3 Set F compact sections compact sections compact sections compact sections hd md hd md Web Slenderness, h/t w Flange Slenderness, b f /2t f (a) (b) Figure 1. Wide flange sections categorized by (a) web and (b) flange slenderness ratios. The set of 4 sections is subjected to cyclic lateral loading combined with different levels of constant compressive axial load ratio. The lateral loading protocol implemented in the FE study is the symmetric SAC protocol [7]. This protocol is combined with different levels of constant compressive axial loads: %, 2%, 35%, and 5% P y. The axial load (2% and higher) is representative of that applied to a first-story interior column of a steel SMF. These levels of compressive axial loads are based on observations from nonlinear response history analyses of perimeter SMFs of archetype buildings with heights ranging from 2 to 2 stories [8, 9]. q Y -X SYM refers to the symmetric SAC protocol, and X is the level of axial load as a percentage of P y.

5 Additionally, a monotonic lateral loading case with no axial load, MON-, is also conducted. Finite Element Modeling A typical first-story column is modeled ABAQUS-FEA/CAE [1] as shown in Fig. 2a. The column has a length, L, of 4.6 meters (15 feet) with a fully-fixed support at one end (i.e. fixed support) and partially-fixed support at the other end (i.e. flexible support). These boundary conditions simulate the variation of the moment gradient observed in first-story columns during an earthquake once plastification occurs at the column base. The simulated boundary conditions are a better representation of the boundary conditions of first-story columns than the ones used in previous exploratory FE studies for the same purpose [3, 11, 12]. The flexible support is required rotational stiffness such that the inflection point of the column before flexural yielding is at 75% of the column height, measured from the column base. Typically, this preset location of the inflection point is expected in first-story columns while they remain elastic. Once yielding occurs at the column base, the location of the inflection point changes and hence its movement needs to be monitored. This is achieved by first monitoring the normal stresses at several sections along the column length. These stresses are then used to calculate and construct the moment gradient along the column in order to trace the position of the inflection point. Flexible Support Ux=Restrain Uy=Guide Ry=Restrain Rz=Restrain Flexible Beam M Flexible δ P Lateral Support Ux=Restrain Ux=Restrain Rx=Restrain Ry=Restrain Rz=Restrain δ L i L Inflection Point 18" L Fixed Support Ux=Restrain Uy=Restrain Uz=Restrain Rx=Restrain Ry=Restrain Rz=Restrain (a) L brace L i P θ (b) V base M base Figure 2. (a) Boundary conditions of the FE model; and (b) external forces acting on the deformed first-story column. As illustrated in Fig. 2b, the moment at the column base can be calculated using Eqn. 1, where V base is the first order shear force at the base, L i is the distance between the column base and the inflection point, P is the applied axial load, and δ is the displacement at the column's top end. All the variables in Eqn. 1 are known during the analysis except L i (i.e. location of inflection

6 point), which is calculated as discussed in the previous paragraph. base base i i M V L P L / L (1) The column is laterally supported at a distance L brace measured from the column base. The length L brace is taken as the minimum of L b,hd and L p as given by Eqns. 2 and 3, where L b,hd is the minimum length between points that are braced against lateral displacement for h members [1] and L p is the maximum laterally unbraced length required to avoid lateral torsional buckling for the yielding limit state in steel beams [13]. This is done in order to investigate the effect of plastic hinging on the overall column performance without considering the possibility of lateral torsional buckling throughout the member. The latter is currently investigated by a separate FE study. In all cases, the length L brace was controlled by L p. L. 86 r E / F (2) b,hd y y L r E / F (3) p y y The column is meshed using 1"x1" 4- q 4 global geometric imperfections are introduced to the model by scaling and superimposing the G with ASTM [14] limits. The approach to introduce such imperfections has been discussed in detail in a companion paper [12]. Residual stresses was shown to be insignificant based on the same study [12]; hence they are not considered as part of the FE model discussed herein. The nonlinear cyclic behavior of the column is simulated using the nonlinear isotropic/kinematic hardening material model in ABAQUS. A modulus of elasticity E=28 MPa (3165 ksi) and an expected yield stress of F y =383 MPa (55.5 ksi) is considered to represent typical ASTM A992 Gr.5 steel (F y =345MPa). The additional four parameters, used to define the cyclic hardening properties of the material model, are as follows: C=6895 MPa (1 ksi), γ=25, Q =172 MPa (25 ksi), and b=2. These tests conducted by Krawinkler et al. [15] and MacRae et al. [2]. These parameters are independent of the loading history and axial load demand that a steel column is subjected to [12]. The aforementioned FE modelling approach has been validated against past experimental data - tests [2] as discussed in [12]. Results and Discussion This section summarizes the results based on the FE analysis of the 4 sections shown in Table 1. Emphasis is placed on the following issues: (a) the cyclic deterioration in flexural strength of the steel columns; (b) the pre-capping plastic rotation of steel beam-columns; and (c) the column axial shortening. Cyclic Deterioration in Flexural Strength Fig. 3 shows the simulated peak strength response at the column base (normalized by the plastic

7 strength, M p ) measured at a range of chord rotations, θ: 1, 2, 3, 4, and 5% rads. From this figure, it is evident that the applied axial load and the section cross sectional slenderness affect both the level of cyclic strain hardening and the rate of cyclic deterioration in flexural strength of a steel column. Fig. 3a shows that, in the absence of axial load, cyclic strain hardening (expressed by the ratio of the maximum moment to plastic moment, M max /M p ) can reach a factor of 2. for highly compact sections (i.e. Sets W1 and F1). This is primarily attributed to: 1) the large number of small inelastic cycles within the symmetric SAC protocol; and 2) the absence of axial load. Generally, the level of cyclic strain hardening decreases when the applied axial load increases. The same observation holds true when the web and flange slenderness ratios increase (see Table 2). In the absence of axial load (see Fig. 3a), all the sections are able to maintain their plastic bending strength at 4% chord rotation. When the applied axial load increases (see Figs. 4b to 4d), the rate of cyclic deterioration in flexural strength of steel columns increases. For an applied axial load larger than 2% P y, seismically compact sections, with high web and flange slenderness ratios (i.e., 32.5<h/t w <43 and 5.5<b f /2t f <7), approach zero flexural strength before reaching a 4% chord rotation. At 5% P y (see Fig. 4d), almost all the cross sections reach a zero flexural strength before reaching a 4% chord rotation. M/M p M/M p Chord Rotation, [rad] (a) SYM Chord Rotation, [rad] (b) SYM-2 M/M p M/M p Chord Rotation, [rad] (c) SYM Chord Rotation, [rad] (d) SYM-5 Figure 3. Normalized peak strength response of steel columns at different chord rotations.

8 Another way to assess the cyclic performance of the steel beam-columns, analyzed as part of this paper, is to examine if they can sustain 8% of their full plastic flexural strength at 4% chord rotation. This assessment is consistent with the way that fully restrained beam-tocolumn connections are evaluated in terms of flexural resistance according to [1]. The average values of the chord rotation at 8% M p for each section set are summarized in Table 2. This table indicates that deep slender wide-flange beam-columns close to the compact limits for highly ductile members as per [1] (i.e. Sets F3 and W3) reached 8% of their plastic bending moment, M p, at chord rotations less than 4% rads even at low axial load ratios (P/P y =.2). Table 2. Summary of average chord rotations at 8% M p and average levels of cyclic strain hardening, M max /M p. Chord Rotation at 8% M p [% rad] Cyclic Strain Hardening, M max /M p P/P y [%] Sets based on flange slenderness Sets based on web slenderness Pre-Capping Plastic Rotation F1 b f /2t f <3.9 >5 > F2 3.9<b f /2t f <5.5 > F3 5.5<b f /2t f <7 > < F4 b f /2t f >7 >5 1.5 <1 < NA W1 h/t w <22 >5 > W2 22<h/t w <32.5 >5 > W3 32.5<h/t w <43 > < W4 h/t w >43 >5 1.5 <1 < NA The pre-capping plastic rotation is commonly used by structural engineers to model structural components as part of steel frame buildings in order to conduct a nonlinear static and/or dynamic analysis to evaluate their seismic performance. The PEER/ATC [16] modeling recommendations for steel columns (modeling option 1) use the pre-capping plastic rotation θ p, in addition to other strength and deformation parameters, to construct a monotonic "backbone" curve that bounds the behavior of a steel component as shown in Fig. 4a. For the PEER/ATC [16] modeling option 1, θ p is computed based on the multivariate regression equations developed by [6]. These equations were obtained from extensive calibrations in which the parameters of the backbone curve were matched to cyclic experimental data from steel beams with 2<h/t w <55 and 4<b f /2t f <8. The precapping plastic rotation calculated using PEER/ATC [16] θ p, ATC-Option1 compared to the θ p, values as predicted by the FEA. For both the cyclic and monotonic loading cases, θ p is defined as the difference between the capping rotation (at maximum flexural strength) and the yield rotation (at plastic flexural strength) as shown in Figs. 4b and 4c. It should be noted that in the case of cyclic loading, the capping rotation is deduced based on the cyclic envelope curve as shown in Fig. 4b.

9 Moment [kn.m] Moment [kn.m] Moment [kn.m] 5 M max 4 3 M p M max 2 1 Cyclic data Cyclic data -5 Mono. Backbone Envelope curve Chord Rotation, [rad] Chord Rotation, [rad] Chord Rotation, [rad] (a) (b) (c) Figure 4. Definition of the pre-capping plastic rotation; (a) PEER/ATC [16], modeling option 1; (b) FEA cyclic loading; (c) FEA monotonic loading. Figs. 5a and 5b show the θ p, ATC-Option1 versus the pre-capping plastic rotation as predicted by the FEA when the column is subjected to the loading cases MON- (noted as θ p, MON-) and SYM- (noted as θ p, SYM-), respectively. For monotonic loading, the θ p, ATC- Option1 values match the θ p, MON- values for low compactness sections (i.e., 32.5<h/t w <43 and 5.5<b f /2t f <7). However, for highly and moderately compact sections (i.e., h/t w <32.5 and b f /2t f <5.5), the values of θ p, ATC-Option1 are lower than the corresponding ones from θ p, MON-. This is primarily attributed to the fact that θ p, ATC-Option1 is calculated from equations that are based on calibrations against cyclic and not monotonic data. Normally, the capping rotation moves towards the yield rotation in such case compared to the monotonic one [6, 16]. In the case of cyclic loading, values of θ p, ATC-Option1, by definition, are expected to be larger than θ p, SYM- values. However, this is not the case for set W1 as shown in Fig. 5b. The same observation holds true for set F1. This is attributed to the cyclic strain hardening observed in highly compact sections due to the delayed formation of flange and web local buckling. Also, sets W1 and F1, represent sections with h/t w <22 and b f /2t f <3.9; this range of web and flange slenderness is outside the valid range of the Lignos and Krawinkler [6] regression equations. ATC-Option1 [rad] ATC-Option1 [rad] MON- [rad] (a).5.1 SYM- [rad] (b) Figure 5. Plastic rotation as calculated per PEER/ATC [16] versus the plastic rotation predicted by the FE analysis: (a) MON-; and (b) SYM-.

10 Axial Displacment [%] Axial Displacment [%] Axial Displacment [%] Axial Displacment [%] Axial Shortening Steel columns subjected to inelastic lateral cyclic drift ratios experience axial shortening primarily due to web local buckling. The level of axial shortening increases in the presence of high compressive axial loads. The issue of axial shortening has been reported by MacRae et al. [2] after conducting tests on 1.1m tall cantilever columns with 25UC73 sections (equivalent to W1x49). Based on this data, MacRae et al. [17] developed empirical equations to predict the column axial shortening as a function of the applied axial load level, P/P y and the cumulative inelastic drift, Σθ H. Fig. 6 shows the axial shortening a, as a percentage of the column length, versus the cumulative inelastic drift Σθ H for all the column sections that were analyzed as part of this paper. The axial shortening, a, is deduced by monitoring the axial displacement of the column's top end. At any loading stage, the cumulative inelastic drift, Σθ H, is defined as the sum of absolute peak drift values for all cycles following the yield drift (i.e. inelastic cycles). It should be noted that at the second 4% drift cycle of the SYM protocol, the average value of the cumulative inelastic drift, for all sections, is.7. This value for cumulative inelastic drift, Σθ H =.7, is used next as reference value for assessing the column axial shortening Cummulative Inelastic Drift [rad] (a) SYM Cummulative Inelastic Drift [rad] (b) SYM Cummulative Inelastic Drift [rad] (c) SYM Cummulative Inelastic Drift [rad] (d) SYM-5 Figure 6. Axial shortening versus cumulative inelastic drift ratio.

11 Fig. 6a shows that, at % P y and Σθ H =.7, the level of axial shortening is below 2% of the total column length for all the seismically compact sections. This is attributed to the occurrence of web/flange local buckling at chord rotations larger than 4% when subjected to zero axial load (i.e., SYM-). When an axial load of 2% P y is applied, a significant increase in column axial shortening is observed (see Fig. 6b). At 2% P y and Σθ H =.7, highly compact sections (i.e., h/t w <22 and b f /2t f <3.9) experience an average a equal to 3% compared to 7% for low compactness sections (i.e., 32.5<h/t w <43 and 5.5<b f /2t f <7). The column axial shortening increases while the applied column axial load increases (see Figs. 6c and 6d). However, for axial load ratios P/P y >35%, a further increase in axial load level has a minor effect on the column axial shortening. This agrees with MacRae et al. [17] findings that axial shortening does not increase after a critical axial load level. Summary and Conclusions The cyclic behavior of a range of wide-flange steel beam-columns was investigated through detailed finite element (FE) analysis. The beam-columns are subjected to the SAC symmetric lateral loading protocol, as well as, different levels of constant compressive axial load. The main findings of the FE analysis are summarized as follows: At % P y axial load, wide-flange sections with h/t w <22 and b f /2t f <3.9 experience high levels of cyclic strain hardening. The maximum moment M max reaches about 1.9M p compared to 1.4M p for sections with 32.5<h/t w <43 and 5.5<b f /2t f <7. At P/P y = 2%, slender wide-flange beam-columns with 32.5<h/t w <43 and 5.5<b f /2t f <7 reach 8% of their plastic strength at chord rotations less than 4% rads. Except for the load case SYM-5, highly compact wide-flange sections with h/t w <22 and b f /2t f <3.9 are able to maintain 8% of their plastic bending resistance at chord rotations larger than 4% rads. The pre-capping plastic rotation based on PEER/ATC [16], modeling option 1, is shown to be adequate for predicting the monotonic behavior of sections with 32.5<h/t w <43 and 5.5<b f /2t f <7. However, it is underestimated for sections with h/t w <32.5 and b f /2t f <5.5. The pre-capping rotation based on PEER/ATC [16], modeling option 1, is underestimated for sections with h/t w <22 and b f /2t f <3.9 undergoing the SYM- loading case. At 2% P y and Σθ H =.7 (corresponding to the second 4% drift cycle), highly compact wide-flange sections with h/t w <22 and b f /2t f <3.9, experience an average axial shortening a of 3% of the column length compared to 7% for slender sections with 32.5<h/t w <43 and 5.5<b f /2t f <7. At axial load levels P/P y >35%, the increase in axial shortening is minor. It should be noted that the results presented here are dependent on the applied loading protocols and the selected beam-column boundary conditions. The authors currently evaluate the behavior of the same set of beam-columns for other lateral and axial loading protocols. The findings from the extensive analytical study summarized herein will also be complemented by a series of full-scale experiments on deep wide-flange beam-columns. Acknowledgments

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