Buckling control of cast modular ductile bracing system for seismic-resistant steel frames

Journal of Constructional Steel Research 71 (2012) 74–82

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Journal of Constructional Steel Research

Buckling control of cast modular ductile bracing system for seismic-resistant steel frames G. Federico, R.B. Fleischman ⁎, K.M. Ward Dept. of Civil Engineering and Engineering Mechanics, Univ. of Arizona, Tucson, AZ, United States

a r t i c l e

i n f o

Article history: Received 16 June 2011 Accepted 21 November 2011 Available online 20 December 2011 Keywords: Steel castings Special concentric braced frames Steel lateral resisting system Earthquake resistant design Structural stability Modular construction

a b s t r a c t The buckling behavior of a new ductile bracing concept for steel structures is examined. The system makes use of cast components introduced at the ends and the center of the brace to produce a special bracing detail with reliable strength, stiffness and deformation capacity. The system takes advantage of the versatility in geometry offered by the casting process to create configurations that eliminate non-ductile failure modes in favor of stable inelastic deformation capacity. This paper presents analytical research performed to determine the buckling strength and buckling direction of the bracing element based on the geometries of the cast components. Limiting geometries are determined for the cast components to control the buckling direction. Design formulas for buckling strength are proposed. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction A cast modular ductile bracing system is under development as an alternative to steel special concentric braced frames (SCBFs). The system, termed a cast modular ductile bracing system (CMDB), produces a controlled energy-dissipation mechanism through the use of specially detailed cast steel components introduced at the ends and the center of the brace. These components are designed to produce controlled stable and ductile plastic hinge regions when the bracing element undergoes buckling and straightening cycles during seismic loading. The steel SCBF is a popular seismic resistant system. However recent research has indicated a number of seismic performance issues [1] related to: member low-cycle fatigue life; fracture at connections; induced distortion in the surrounding members, and unbalanced shear load in the beam [2]. Improvements to SCBFs have been proposed [2–4] and new innovative systems have been proposed as alternatives to SCBFs [5–8]. The CMDB system falls into the latter group. Ward et al. [9] have presented the CMDB component geometries that produce ductile mechanisms leading to greater low-cycle fatigue life and greater post-buckling strength. These designs were shown to have the potential for improved seismic response in comparison to a SBCF. A necessary further step in developing the CMDB prototype design is the ability to: (1) reliably control the buckling direction to ensure the desired post-buckling mechanism; and, (2) adequately predict the CMDB critical load. This paper presents analytical research to establish

⁎ Corresponding author. E-mail address: rfl[email protected] (R.B. Fleischman). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.11.010

the relationship between CMDB geometry and these behaviors. Design expressions for buckling strength and required geometry for buckling control are proposed.

2. Casting modular ductile brace concept Fig. 1 shows a schematic of the CMDB system in the evaluation frame used for this paper. The CMDB bracing element is constructed by inserting cast components at the ends and center of HSS (Hollow Structural Section) members. The components, termed the end and center (cast ductile) components, or EC and CC, are shown in detail in the next section. For simplicity, the CMDB design is developed using a single diagonal configuration, as is commonly done in research. However, the system is anticipated to be used in chevron or X-brace configuration, which represents a more accommodating case in terms of tolerance. Ward et al. [9] demonstrated analytically that the CMDB system can develop a stable and ductile plastic mechanism in the postbuckling region. Fig. 2(a) shows the controlled plastic mechanism developed in the CMDB analytical model, in which dark regions of the contour plot are elastic, while the lighter regions indicate the plastic hinge regions contained in the specially designed cast components (EC and CC). Ward et al. [9] compared the performance of the CMDB system to a SCBF of similar strength and bay geometry under cycling loading protocols. The accumulated plastic strain comparison is shown in Fig. 2(b), with the hatched area representing the predicted fracture range. The lower strain demand in the CMDB is due to the spreading of the inelastic demands in the special cast component, and the elimination of strain concentrations and local buckling in the HSS.

G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

75

properties of the cast components are determined based on the reduced cross-section geometry.

5.39 m

W14x48

2.2. Design parameters

bolted interfaces

The CMDB element is described by a number of design parameters, typically geometry and/or material strength ratios. These design parameters control the characteristics of the response, as will be shown in this paper. The parameters include:

CC (see Fig. (3b))

EC (see Fig. (3a))

7.62 m (L)

W14x132

W14x132

Brace main member (HSS typ.)

5.39 m

W14x48

Fig. 1. CMDB element: evaluation frame schematic.

These analytical results suggest that the CMDB system may produce a significantly improved low-cycle fatigue life. The CMDB design may also permit relaxing of seismic local slenderness limits for the HSS since it remains essentially elastic. Likewise, the cruciform shaped interface decreases the reduction required for shear lag associated with net section fracture. Ward et al. [9] also showed that postbuckling strength can be improved in cases. 2.1. CMDB component Each CMDB component is composed of (see Fig. 3): (1) an interface region of length, Lint, in which the CMDB element is connected to the main member (typically shop-welded to the brace main member and bolted to surrounding main members, producing replaceable brace elements); (2) a special ductile region of length, Lsd, possessing a reduced cross-section in order to control the inelastic mechanisms; and (3) a filleted radius transition section of length, Ltr. A cruciform cross-section is chosen for the CMDB component due to its ease of castability, facilitation of fabrication, reduction of shear lag effects, and straightforward modulation of in-plane and out-ofplane component properties [9]. Fig. 3(c) shows a schematic of the CMDB cross-section indicating the main geometric parameters in the special ductile region, the in-plane (of the braced frame) depth, d; and corresponding thickness t; the out-of-plane width, b; and corresponding thickness w. It is noted that while this study treats the cruciform as an assembly of rectangular elements, for castability the cruciform section will include fillet radii at each juncture. Section

(1) a cast component stiffness ratio, γ, which is important in determining the critical buckling direction: γ ¼ Iy =Ix

ð1Þ

where I is the moment of inertia of the CMDB component reduced cross-section and the x–x and y–y axes are shown in Fig. 3(c). A value of γ greater than unity indicates a propensity to buckle in-plane. Since the end castings and center casting need not be identical, parameters γe and γc are defined for these components, respectively. The term γm is used to describe this same property in the HSS section. (2) a bending stiffness factor, κ, which is an indicator of the modification of flexural stiffness in the brace due to the presence of the CMDB component: κ ¼ Im =Ic

ð2Þ

where Im and Ic are the moment of inertia of the HSS member and the casting, respectively. Once buckling direction is determined through the parameter γ, the κ value in the primary buckling direction is used in the determination of buckling strength. Different values, κe, κc, can exist for the EC and CC. (3) a length ratio, Λ, which affects the influence of the CMDB component properties on the brace element response: Λ ¼ Lsd =L

ð3Þ

where L and Lsd are defined in Figs. 1 and 3. Two capacity design factors (casting relative to HSS), a flexural overstrength factor, Ωb, and an axial overstrength factor, Ωa, control the plastic mechanism as described in [9]. 2.3. Controlled buckling response In order to develop the intended ductile mechanism, a controlled buckling direction is desired. Current SCBF systems essentially follow this approach since the gusset plate is significantly more flexible in the out-of-plane direction [10]. For controlled CMDB response, in-plane buckling is chosen because of less non-structural damage to partitions, or to glazing and other façade elements, for instance at the perimeter of structures, where bracing elements are currently discouraged. A “balanced” condition, in which the CMDB element buckles in a bi-axial flexural mode, is used to delimit the transition between buckling directions for CMDB system. 2.4. Analytical modeling

Fig. 2. CMDB vs. SCBF comparison [9]: (a) plastic mechanism; (b) accumulated plastic strain under cyclic load.

The CMDB system is evaluated in this paper through threedimensional nonlinear finite element modeling, using the general purpose commercial software ANSYS [11]. All models employ geometric nonlinearity. Depending on the study, models are created from twonode beam, eight-node solid, or four-node shell elements. Other than the initial elastic study, elements constitutive relationships possess material inelasticity utilizing multi-linear kinematic hardening principles, the Von Mises yield function, and the Prandtl–Reuss flow equation [11]. Large deformation formulations are used with the true-stress

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G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

Fig. 3. Cast components: (a) end casting; (b) center casting; (c) cruciform dimensions (after [9]).

logarithmic strain curves represented in detailed piecewise-linear fashion in order to accurately capture stress under high strain gradients within the plastic zones.

3. Buckling control: elastic response The effect of CMDB component geometry on the elastic buckling modes is first examined. Eigenvalue solutions are performed on a three-dimensional finite element (3D-FE) elastic beam model of an isolated CMDB element (see Fig. 4). This simple model is wellsuited to rapidly investigate a large parameter space. Model accuracy was evaluated via a limited number of comparisons to a 3D-FE shell model with fully-realized cross-sections used subsequently for inelastic buckling analysis (see Section 4). The comparisons [12] indicated that: (1) the beam model approximated the shell model within 5%, and exhibited similar trends; and (2) the simpler boundary conditions of the isolated element caused a slight shift (approximately 5%) in the relative values of in-plane to out-of-plane critical loads relative to models with the surrounding frame (which show an slight increased propensity toward in-plane buckling due to in-plane end-moment and shear from frame action). These differences were deemed acceptable for the large parameter initial study. Each segment of the CMDB brace element in Fig. 3 modeled with a 3D elastic beam. Rigid offsets of length Lrgd are inserted at each end to represent the projection distance from the working point to the EC special detail region. The elements representing the special detail regions include half the transition region, and thus are of length Lsd + Ltr, and incorporate a weighted average moment of inertia based on Isd and Itr. The interface region, with length Lint, is assigned a moment of inertia based on the sum of the HSS and cast component, Iint + Ihss. Different analyses in a given sequence are performed by varying γ via parametric representation of moments of inertia of the cast components, while all other dimensions are held constant: the length of the CMDB element, L, is 7.62 m (25 ft); Lsd is 22.86 cm (9 in.) (Λc = Λe = 0.03); Ltr =7.62 cm (3 in.); Lint =22.86 cm (9 in.); and, Lrgd = 38.1 cm (15 in.). The brace main member section (initially a HSS 7× 7× 5/8; and later rectangular HSS members with values of γm greater than unity) remains unchanged during a study sequence.

For each sequence, γe and γc are varied independently: γe through a range of 1 to 10; γc through a range of 0.25 to 3. The selection of γe values greater than 1.0, i.e. a larger out-of-plane moment of inertia, produces end-conditions promoting in-plane response. The extension of the γc design space to values less than unity (center cast component with strong axis oriented in-plane) is motivated by the desire to develop in-plane buckling mechanisms with significant energy dissipation [9]. Since the CMDB component moment of inertia and plastic modulus are proportional, a γc b 1.0 design will create higher energy dissipation, provided the bracing element buckles in-plane. Thus, these analyses examine if the γe > 1.0 end cast component can still produce an in-plane buckling mode for center cast components oriented strong axis in-plane. The upper limit of γe = 10.0 represents a practical limit of asymmetry in the cruciform section: for higher values of γe it will become more difficult to realize an actual casting cruciform section (due to local slenderness, i.e. b/t requirements [13] and casting integrity thinness limitations [14]). Values of γc > 3.0 are not considered since they produce too low of a CMDB element buckling strength. Two cases are considered for varying γ: (1) κ ≥ 1.0, i.e. the CMDB component strong axis moment of the inertia (larger of Ix and Iy) is held equal to Im, and, (2) κ ≤ 1.0, i.e. the CMDB component weak axis moment of inertia is held equal to Im. As an illustration, Table 1 shows each case for the design point (γe = 5, γc = 0.5). 3.1. Elastic buckling results The results of the elastic buckling study for the square HSS are shown in Fig. 5 (black lines are Case 1; gray lines are Case 2). Fig. 5 (a) plots the ratio of the in-plane to out-of-plane CMDB elastic critical P loads, ρcr ¼ Pcr;y ; Fig. 5(b) plots the ratio of the CMDB fundamental cr;x critical load normalized by the elastic buckling load corresponding to the analogous SCBF, based on the accepted effective length factor of 1.0. Focusing first on Case 1 κ ≥ 1.0 (black lines), an examination of the ρcr plot in Fig. 5(a) indicates that as γc or γe increases, ρcr decreases. As expected, the combination of high γc and γe produce ρcr below unity, denoting in-plane buckling response; and low γc and γe produce ρcr above unity, denoting out-of-plane buckling response. However, note that combinations involving γc ≤ 1.0 can produce ρcr values below unity provided γe is sufficiently large. This finding is useful because it implies that the center casting can be biased toward high inplane energy dissipation and post-buckling strength (large in-plane plastic modulus associated with a low γc [9]) with the desired inTable 1 Elastic buckling study: example γ values for castings. Case 1: κ ≥ 1.0

Fig. 4. 3D beam representation for the elastic buckling study.

γ Ix/Im Iy/Im

EC 5.0 0.2 1.0

Case 2: κ ≤ 1.0 CC 0.5 1.0 0.5

EC 5.0 1.0 5.0

CC 0.5 2.0 1.0

G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

a

a 1.2

c = 0.25 c = 0.5 c = 1.0

1.0 0.8

r

cr

77

0.6

c = 2.0

0.4 1

2

c = 3.0 3 4

5

6

7

8

9

10

γe

b

b P CMDB/PSCBF

5.0 4.0

c = 3.0 3.0

c = 2.0

c = 0.5

c = 1.0

c

= 0.25

2.0

Pcr x controls

1.0 1

2

c

3

= 0.25 4

Fig. 6. Elastic buckling results, γm varied: (a) ρcr vs. γe; (b) PCMDB/PSCBF vs. γe.

5

6

7

8

9

10

γe Fig. 5. Elastic buckling results: (a) ρcr vs. γe; (b) PCMDB/PSCBF vs. γe.

plane buckling mode still obtained via the out-of-plane bias of the end cast components. Designs with ρcr values near unity may not reliably produce the desired direction control due to the presence of initial imperfections, secondary forces, etc. Fig. 5(a) indicates that sufficiently high values of γe produce ρcr well below unity such that reliable in-plane buckling response can be obtained. However, it is also seen in Fig. 5(b) that the controlling buckling load will decrease as ρcr decreases. Note that due to the fixity provided by the EC, these elastic buckling strengths are still well above the value associated with typical SCBFs. Thus, a design point can be obtained that maximizes buckling strength, minimizes γc for good post-buckling energy dissipation, and provides a reliable value of ρcr to assure in-plane buckling. One such design point resides at γe = 2.5, γc = 0.5 (black dot), which provides ρcr = 0.83 (Fig. 5(a)), and a PCMDB only about 20% reduced from the maximum value (Fig. 5 (b)), thus still significantly larger than that of the SCBF. The light gray lines in the background of Fig. 5 represent Case 2 (κ ≤ 1.0). As seen, these curves are more tightly spaced and rapidly converge to a stable solution (ρcr = 0.80 to 0.90 depending on γc) once γe reaches approximately 3.0, and produce a fundamental inplane critical load (PCMDB) that remains relatively constant. Thus it is seen biasing of cast components to flexural rigidities greater than the main HSS brace members has a muted effect on buckling control since the main member itself dominates the buckling response. Therefore, Case 1 κ ≥ 1.0 is only considered further for direction control. In Fig. 6, the same response indices as Fig. 5 are presented for a CMDB system with rectangular HSS sections. The HSS sections are biased toward in-plane response (γm = 3.0, γm = 2.25 and γm = 1.5 respectively), as shown in the inset of Fig. 6(a). For Fig. 6(b) PSCBF is based on the square HSS (γm = 1.0). The HSS out-of-plane dimension is selected within the typical maximum width allowed architecturally. In Fig. 6(a), each region indicated represents the range with an upper limit of γc = 0.25 and a lower limit of γc = 1.0. In Fig. 6(b), these regions “collapse” into lines since the CMDB in-plane elastic critical load, Pcr,y, is not influenced by γc b 1.0 (for Case 1). As seen, the introduction of rectangular HSS members provides an extended range of in-plane buckling for low γc values and lower γe values. For instance, selecting ρcr = 0.85 as a reliable limit for in-plane response, an acceptable design point is now γm =2.25, γe = 1.5 and γc = 0.25 (black square), which produces a ρcr = 0.8 and a PCMDB/PSCBF ratio = 3.75 (see Fig. 6(b)). This design

provides higher energy dissipation than for the square HSS design case of Fig. 5, since both the EC and CC have lower γ, and hence larger inplane plastic modulus [9]. 4. Buckling control: inelastic response The controlling limit state of the CMDB element in most cases will be inelastic flexural buckling, particularly for low-slenderness braces as has been promoted for SCBFs [4]. Further, the post-buckling response of the CMDB element during seismic demand, like most SCBF systems, will involve inelastic response regardless of the initial nature of the buckling. Accordingly, the analytical evaluation must be extended to inelastic models. In these models, the material values are based on coupon tests, with yield stress for the castings, brace, and main members of 276 MPa [15], 317 MPa, and 345 MPa [2] respectively. True-stress logarithmic strain constitutive relationships [9] are used for the nonlinear material large deformation analyses [11]. The nonlinear 3D FE model (see Fig. 7) employs inelastic shell elements for the CMDB element, as shown in Fig. 7 inset. The model allows the appropriate 3D plastic zone to form in the CMDB and HSS components, thereby capturing the inelastic buckling response in each direction. The 3D model produces more accurate stress field transitions between the different elements, including shear lag and concentrations. Rigidly-connected 3D inelastic beam elements are used for the framing beams and columns that provide the back-up frame action.

Fig. 7. 3D shell model of CMDB element: (a) end view; (b) side view.

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G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

a

The onset and eventual controlling direction of inelastic buckling is examined for different combinations of γc and γe. The parameter γe is varied discretely from 0.25 (biased toward out-of-plane buckling) to 4.0 (biased toward in-plane buckling). The parameter γc is varied from 0.5 to 1.0 so as to evaluate designs with high in-plane energy-dissipation capabilities.

2

1 0

4.1. Study matrix Table 2 shows the CMDB component cross-section dimensions and design parameters examined. A square HSS section (γm = 1.0) is used initially so that the influence of the cast component geometry on buckling direction can be isolated. This section, a HSS 7×7×5/8, possesses moment of inertia, Im, of 3887.6 cm 4, cross-sectional area A = 90.3 cm 2, plastic modulus Z = 542.4 cm 3, and polar moment of inertia J = 6576.5 cm 4. Note in Table 2 that the larger moment of inertia of the cast component is set nominally to Im. This practice results in κ ≈ γ for the study (refer to Eqs. (1) and (2)). For this section in the evaluation frame (see Fig. 1), the effective slenderness of an SCBF would be 108, while for the CMDB designs in Table 2, the range is 57 to 67 based on a lower CMDB effective length factor (see Section 5). Once the relationship for the square HSS is established, CMDB designs using the rectangular HSS sections (γm > 1.0) shown in Fig. 6(a) inset are evaluated. 4.2. Inelastic buckling results Consider first the case of varying EC stiffness ratio γe for a symmetric center casting design, γc = 1.0 and square HSS (γm = 1.0). Results for this case are shown in Fig. 8. In Fig. 8(a), the midspan transverse buckling displacement is shown for selected cases 1.0 ≤ γe ≤ 4.0. The buckling displacement is decomposed into its inplane δy (solid lines) and out-of-plane δx (dashed lines) components, as indicated in Fig. 7. Transverse displacement, δ, is expressed as percentage of CMDB brace length, L. Frame drift, Δ, is expressed as

Table 2 CMDB component dimensions and section properties. γ

d (cm)

t (cm)

d/2t

b (cm)

w (cm)

b/2w

Ix (cm4)

Iy (cm4)

3.97 3.84 3.01 2.89 2.08 1.95 1.49 1.47 1.25 1.23 1.18 1.18 1.00 1.00 0.95 0.95 0.90 0.90 0.85 0.85 0.76 0.74 0.68 0.67 0.51 0.49 0.26 0.25

21.6 22.9 20.3 22.7 26.5 25.4 26.7 26.7 28.2 26.7 26.7 29.2 26.7 29.5 29.2 27.9 29.3 28.2 29.3 28.2 25.9 28.2 27.9 26.7 25.4 26.2 26.7 23.9

1.1 1.0 1.9 1.3 1.1 1.5 1.5 1.8 1.5 1.9 1.8 1.9 2.4 1.7 2.0 1.8 2.0 1.8 2.2 1.8 2.5 1.9 2.3 2.3 3.0 2.5 2.5 3.7

9.44 11.25 5.33 8.95 11.61 8.33 8.75 7.50 9.25 7.00 7.50 7.88 5.65 8.92 7.19 7.86 7.22 7.93 6.79 7.93 5.10 7.40 6.11 5.83 4.17 5.15 5.25 3.24

23.9 26.7 25.4 25.5 26.5 25.4 26.7 27.9 28.2 26.7 28.2 29.3 26.7 29.5 29.2 26.8 29.2 26.7 29.2 26.7 25.9 26.7 26.7 26.7 25.4 26.7 22.9 21.6

3.7 2.5 3.0 2.7 2.4 3.0 2.3 2.3 1.9 2.4 1.8 2.2 2.4 1.7 1.9 1.9 1.9 1.9 1.9 1.8 1.9 1.7 1.8 1.5 1.5 1.1 1.0 1.1

3.24 5.25 4.17 4.79 5.50 4.17 5.83 6.11 7.40 5.65 7.93 6.79 5.65 8.92 7.57 7.05 7.88 7.05 7.88 7.50 6.80 8.08 7.50 8.75 8.33 11.67 11.25 9.44

1053 1045 1386 1282 1811 2139 2435 2839 2864 3038 2822 3875 3763 3530 4237 3247 4291 3334 4558 3334 3696 3567 4166 3621 4171 3792 4017 4179

4179 4017 4171 3700 3763 4171 3621 4166 3567 3750 3334 4558 3763 3530 4029 3084 3871 3005 3875 2822 2793 2626 2839 2435 2139 1840 1045 1053

0

0.25

0.5

0

0.25

0.5

0.75

b 2 1 0

0.75

Fig. 8. Transverse brace displacement vs. Δ: (a) δy, δx (b) δx/δy.

percentage of the frame height, h (refer to Fig. 7). It is of interest to note in Fig. 8(a), that for controlled buckling in one direction, a component of buckling deformation exists in the other direction. As the γe value approaches unity, the magnitude of the non-controlling displacement component increases. Note that for these cases, the postbuckling displacement gradient of the non-critical direction may originally be quite steep, but at a certain point, the increase in this displacement abates and the non-critical direction displacement plateaus and then returns slightly back toward the original position. The designs with γe b 1.0 are omitted for clarity since their results are essentially the same as their inverse (e.g., γe of 2.0 and 0.5), with in-plane (δy) and out-of-plane (δx) displacements reversed. Fig. 8(b) shows the same information instead as plots of displacement ratio, δx/δy, and for all the design combinations. As seen in Fig. 8 (b), the inelastic buckling response moves from fully in-plane, δx/δy smaller than unity (e.g., γe = 4.0) to essentially out-of-plane, δx/δy greater than unity (e.g., γe = 0.25) response as the end stiffness ratio is varied, as expected. The design combination with γe = 0.95 shows the most balanced behavior (δy ≈ δx). The reason that γe is slightly offset from unity is due to added end-moment and shear in the plane of the brace due to frame action. Fig. 9 shows the effect of γc on the control of the buckling direction for different γe, again for a square HSS (γm = 1.0). In the plot, the displacement ratios (δx/δy) are taken at a frame drift Δ/h = 0.5%, i.e., beyond the point of buckling. The effect of γc, though not as great as the effect of γe, is still seen to be significant. Balanced conditions range from γe from 0.85 to 2.6 as γc changes from 1.0 to 0.5. A lower γc leads to a shift in the “balanced” condition toward greater values of γe, as expected. As was implied in the elastic analyses, the CMDB system is able to force in-plane buckling (δx/δy b 1.0) for designs with γc b 1.0 by specifying a sufficiently high value of γe.

/ h = 0.5%

8 c

6 4

c

= 1.00 c

2 0 0.75

= 0.50

1

= 0.75

1.25

1.5

Balanced 2

1.75

2.25

e

Fig. 9. Effect of γc (Δ/h = 0.5%).

2.5

2.75

3

G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

4.3. Sensitivity to imperfection

4.0

The contours in Fig. 9 are from analyses of initially straight elements. The sensitivity of CMDB buckling control due to mill, fabrication and erection tolerances is now examined. The initial imperfection is introduced into the model by applying an initial displacement field with a maximum amplitude δo, in either orthogonal direction as shown in Fig. 10. The imported displacement field is the first (primary) buckling mode shape, determined through an eigenvalue analysis [11] of the FE model in its elastic state, and scaled to the allowable tolerance of L/480 as specified in AISC [16], equal to 1.6 cm for the evaluation frame. Consider first a CMDB design with γc = γm = 1.0 1.0 and γe varied from 0.25 to 3.0. The response is shown in Fig. 11 as displacement ratios (δx/δy), at a frame drift Δ/h of 0.5%. Results are for the maximum allowable in-plane (δoy = L/480) and out-of-plane (δox = L/480) imperfection. The initially perfect case is shown for reference. As seen, the range of possible initial imperfections shifts the δx/δy curves significantly. In this case, where a balanced condition is realized for an initially straight member for γe ≈ 0.95, an increase in γe to nearly 1.75 is required to preserve in plane buckling response for δox = L/480. The influence of γc and γm is now included in order to determine the limiting values for these parameters. Fig. 12 shows contours of (γc, γe) pairs required to produce the balanced condition. In Fig. 12 (a) the contours of balanced response are shown for a square HSS (γm = 1.0). A range of acceptable (γe, γc) designs, assuming out-ofplane imperfection at the tolerance limit, are seen to exist from (2.0, 1.0) to (4.0, 0.66). In Fig. 12(b), the balanced contour for maximum allowable out-ofplane imperfection (δ0x) is shown for different rectangular HSS section oriented strong-axis out-of-plane (γm > 1.0). As seen in Fig. 12 (b), the balanced condition is shifted to lower values of γe with increased γm. Thus an extended range for in-plane response is produced through the use of a rectangular HSS section. The possibility of CMDB designs with low γ EC and CC components is advantageous in terms of system post-buckling strength and in-plane energy dissipation [9]. Thus, the shaded area represents more desirable designs. Valid (γe, γc, γm) design sets for in-plane buckling in this region include: (1.0, 1.0, 3.0), (1.25, 0.75, 3.0), and (1.5, 0.75, 2.25).

3.0

79

max 0 x "perfect" max 0 y

2.0

/ h = 0.5% Balanced

1.0 0.0 0.25

1

1.75

2.5

e

Fig. 11. Imperfection sensitivity: γc = γm = 1.0, Δ/h = 0.5%.

the length of the ductile region, Lsd. In Fig. 13, this ratio is plotted versus local slenderness of the pertinent cruciform cross-sectional element. The local slenderness is the cast component element widthto-thickness ratio (maximum of d/2t and b/2w). Different ductile region lengths were as indicated by Λ. The local slenderness ffi qffiffifficonsidered, limit (λps ¼ 0:3 FEy ) for seismically compact sections as specified by the AISC Seismic Provisions [13] is referenced in the figure. The measurement is taken at a drift Δ/h = 0.75%. An increased ductile region length lowers the plastic curvature [9], and hence the propensity for local buckling for a given slenderness ratio (see Fig. 13). Increased local buckling is observed for higher local slenderness, as expected. An allowable local slenderness can be selected based on the amplification of Equivalent Plastic Strain (EPS) [11], measured as the ratio of EPS demand at the cruciform tip in an analysis free to local buckle to one where the local buckling is inhibited. Amplified EPS is plotted versus δxtip/Lsd, in Fig. 14(a), and local slenderness in Fig. 14(b). Using an amplified EPS demand limit of 20%, a local slenderness limit slightly stricter than the AISC Seismic provision [13] is proposed: qffiffiffiffi d/2t, b/2w = 7.5, or λ ¼ 0:28 FEY (see Fig. 14b). Inelastic torsional modes did not control any of the analyses in the study. Such modes were only observed for cases where the EC was given a polar moment of inertia less than 1.5% of the HSS member and the cast component axial strength less than 30% of the HSS member, values that are not used for typical CMDB designs [12]. 5. CMDB design buckling strength

4.4. Local and torsional buckling The use of a cruciform shape for the cast components requires the elimination of local and torsional buckling modes. Inelastic local buckling is measured here using the ratio of the out-of-plane displacement of the cruciform element tip δxtip, see inset Fig. 10(b), to

Design expressions are developed for the CMDB. It is noted that the EC, CC and the HSS member can possess different axial yield

a 4

b

/ h = 0.5 % "perfect"

max

ox

2 1 max

0 0.50

oy

0.75

1.00

c

b 4

m

=2.25 m

3 e

a

e

3

/ h = 0.5 % = 1.0

2 1 0 0.50

m =3.0

0.75

1.00

c

Fig. 10. Initial imperfection: (a) in-plane, δ0y; (b) out-of-plane, δ0x.

Fig. 12. Contours of balanced response: (a) γm = 1.0; (b) γm > 1.0.

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0.3

a

ps

= 2% = 3%

0.2

= 5%

"

0.1 0 5

6

7

8

9

10

11

12

b

d/2t or b/2w Fig. 13. Inelastic local buckling: Λ varying.

strengths. The relative axial strength of these elements has an effect on the CMDB post-buckled plastic mechanism [9]. The minimum value of the three is the controlling axial strength, termed as the CMDB yield strength, PY, and is used to normalize the results to follow. Fig. 15. Expression calibration: (a) effective length; (b) equivalent moment of inertia.

5.1. Effective slenderness and equivalent moment of inertia The influence of the EC and CC components on the compression response of the CMDB elements are summarized as follows: (1) the EC component has a direct influence on the CMDB element end rotational restraint; (2) the CC component has a non-negligible influence on the effective flexural rigidity of the CMDB element, since it is located at midspan where maximum secondary bending moment occurs. Accordingly the EC properties are incorporated in an effective length factor, k; while the CC properties are incorporated in an equivalent moment of inertia for the CMDB element, Ieq. Both effects are calibrated from the elastic response obtained in Section 3. The effective length factor, k, is then expressed using a curve fit of Section 3 results as (see Fig. 15(a)):     Λ k ¼ β 0:53 þ 0:045 þ e ðκ e −1Þ 3

a)

ð4Þ

The coefficient β represents the flexibility due to the bolted interface, estimated as 1.07 based on nonlinear FE pushover analyses of the CMDB system including the bolted interface [17]. The equivalent moment of inertia, Ieq is expressed using a curve fit of the Section 3 results as (see Fig. 15(b)): 0:7 Ieq 1:25Λc ¼ 2−κ c Im

Note that once in-plane buckling response is enforced through the γe–γc–γm values, the in-plane section properties need only be considered. As such, γm will not influence κ and Ieq, and κ = γ for γ > 1.0, κ = 1 for γ b 1.0, provided Case 1 is used (as will occur in typical CMDB design). 5.2. Inelastic buckling strength Most CMDB designs will be controlled by inelastic buckling strength. For inelastic buckling, the influence of residual stresses must be considered. The residual stress patterns introduced in the model as initial stress states are shown in Fig. 16 for the (a) HSS and (b) for the cast component, based on [18,19]. Fig. 17 shows the critical load for analyses of CMDB designs with γm = 3.0, and γe varied between 1.0 and 2.0 and γc = 1.0 (γc ≠ 1.0 is not shown for clarity but produces similar response). The brace element length is varied to change CMDB slenderness within typical brace ranges [10]. The slenderness parameter, kL/r, is calculated based on the section properties of the controlling section, where the minimum yield strength PY occurs. In addition to the residual stress

b)

Fig. 14. Amplification on EPS versus: (a) δxtip/Lsd; (b) local slenderness.

ð5Þ

Fig. 16. Residual stresses patterns: (a) HSS; (b) cruciform.

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81

6. Conclusions The buckling behavior of a new bracing concept for seismic-resistant steel frames has been presented. The system, which is intended to provide ductile response in an earthquake, requires control of the buckling direction to produce the desired mechanism, and an estimate of its buckling strength for design. The following conclusions are made:

Fig. 17. Compressive strength for varying CMDB slenderness: γc = 1.

pattern in Fig. 16, an initial in-plane imperfection, δ0y = L/1000 [2] is introduced. The results in Fig. 17 are normalized by the CMDB yield strength PY. The bounding elastic/plastic cases and the code nominal compressive strength curve [16] are shown for reference. As seen the CMDB inelastic strength is similar to but not exactly fit by the code expressions [16] and at low slenderness is governed by the controlling axial strength. The curves on different γe are seen to be close, indicating that the effect of γe is reasonably captured by k. 5.3. Expression to predict CMDB critical load An expression to predict the CMDB element critical buckling load is proposed based on the results shown in Fig. 17. This expression is a modified version of the code compressive strength curve [16]. The nominal compressive strength, PN-CMDB is: 0

PNCMDB ¼ F cr Ag

ð6Þ

where the area, Ag′, is modified to account for the axial strength reduction in the castings and set equal to: 0

Ag ¼

Ag ≤Ag Ωa

ð7Þ

where Ωa [9] is equal to: Ωa ¼

PYHSS PYcast

ð8Þ

The radius of gyration, r, is calculated as: r¼

sffiffiffiffiffiffi Ieq

ð9Þ

0

Ag

The flexural critical stress is determined using code procedures for compression members. In Fig. 18, the ratio between the predicted strength, PN-CMDB, and the strength from FE results, PN, is plotted with respect to the CMDB element slenderness, kL/r, showing good agreement and slightly conservative behavior for most cases.

1. The effect of design parameters, γe, γc and γm on the control of buckling direction has been shown, including the effects of initial imperfection. 2. γe, γc, γm design sets producing a controlled in-plane mechanism with assumed maximum out of plane imperfection are given in Fig. 12(b). 3. A local slenderness (d/2t, b/2w) of 7.5 is recommended to mitigate the effect of inelastic local buckling. 4. A design expression for the CMDB buckling load is developed using the current code compression member strength procedures [16] in conjunction with a CMDB effective length factor (Eq. (4)), and equivalent moment of inertia (Eq. (5)). In combination with design recommendations for developing a ductile plastic mechanism [9], the results from this paper are being used to develop a physical CMDB prototype for full-scale experimental work. A family of modular bracing elements will be proposed that covers the range of strengths needed in design. Notation The following symbols are used in this paper E EPS FY I J k L

Elastic modulus; Equivalent Plastic Strain; yield stress; moment of inertia; polar moment of inertia; equivalent effective length factor; length of the brace between working point (member centerlines); Mp plastic moment; PY axial yield load; Pcr Euler critical load; PCMDB, PSCBF Elastic critical load of the CMDB element, equivalent SCBF; PN strength of the CMDB element; PN-CMDB predicted strength of the CMDB element; Z plastic modulus; β bolted interface coefficient for equivalent effective length factor; δ displacement of the brace midspan; δΟ initial out-of-straightness of the brace midspan; Δ frame drift; γ bending stiffness ratio; κ bending stiffness factor; λps local slenderness limit; Λ length ratio; ρcr Euler critical loads ratio; Ωb bending overstrength factor; Ωa axial overstrength factor;

Acknowledgements

Fig. 18. CMDB strength prediction verification: γc = 1.

This research was supported by NSF Award Grant No. CMS-0324664. Supplemental funds were provided by the American Institute of Steel Construction (AISC) and the Steel Founder's Society of America (SFSA). The writers are grateful for this support. Any opinions, findings, and

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conclusions or recommendations expressed in this material are those of the writers and do not necessarily reflect the views of the National Science Foundation. References [1] Uriz P, Mahin S. Seismic performance assessment of concentrically braced steel frames. Proceedings of the 13th world conference on earthquake engineering; 2004. [2] J. Yoo, (2006). “Analytical investigation on the seismic performance of special concentrically braced frames”, Ph.D. thesis, University of Washington. [3] Khatib I, Mahin S, Pister KS. Seismic behavior of concentrically braced steel frames. Report No. UCB/EERC-88-01. Berkeley, California, USA: Earthquake Engineering Research Center; 1988. [4] Uriz P, Mahin S. Toward earthquake-resistant design of concentrically braced steel-frame structures. Pacific Earthquake Engineering Research Center, Report No. PEER 2008/08; 2008. [5] M. Rezai, H. Prion, R. Tremblay, N. Boutatay, P. Timler, “Seismic performance of brace fuse elements for concentrically steel braced frames”. Behaviour of steel structures in seismic areas: proceedings of the third International Conference STESSA 2000, 21–24 August 2000, Montreal, Canada, Taylor & Francis, 2000. p. 39. [6] De Oliveira JC, Packer JA, Christopoulos C. Cast steel connectors for circular hollow section braces under inelastic loading. J Struct Eng 2008;134(3):374–83. [7] Gray MG, Christopoulos C, Packer JA. Cast steel yielding fuse for concentrically braced frames. Proceedings of the 9th U.S. National and 10th Canadian Conference on Earthquake Engineering. July 25–29 2010, Toronto, Ontario, Canada. Paper No. 595; 2010.

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