## Info

Figure 3.47. Typical bracing-to-column connection, SCBF.

Notes: 1. Yield line intersects free edge of gusset plate and not beam flange.

2. Whitmore effective width Ww = tube width + 2lw (tan 30).

3. Welding of beam flanges is for resisting drag forces.

(Courtesy of Louis Choi, S.E., John A. Martin & Associates Structural Engineers, Los Angeles, CA.)

Figure 3.48. X-brace-to-beam connection. (Courtesy of Louis Choi, S.E., John A. Martin & Assoc. Structural Engineers, Los Angeles, CA.)

Imaginary yield line (Typ)

Slotted tube (Typ)

Figure 3.48. X-brace-to-beam connection. (Courtesy of Louis Choi, S.E., John A. Martin & Assoc. Structural Engineers, Los Angeles, CA.)

3.11.2. Eccentric Braced Frame (EBF)

The design principles of an eccentric braced frame are perhaps best understood by considering the tensile strength of the chain shown in Fig. 3.51a. Using the well-known adage that the strength of a chain is controlled by the strength of its weakest link, the ductility of the entire chain may be controlled by the ductility of a single link, L (Fig. 3.51a). The nominal, or ideal, tensile strength of this ductile link is T1, while the other links, presumed to be brittle, are designed to have a strength in excess of the maximum feasible strength of the weak link. Observe that if the other links were designed to have the same nominal strength as the ductile link, the randomness of strength variation between all links, including the ductile link, would imply a high probability that failure would occur in a brittle link and the chain would not have the intended ductility.

In an eccentric braced system, the segment e of the beam is our ductile link. The segment outside of e, the brace, and the columns are the other links presumed to be brittle and designed to have a strength in excess of the strength of the weak link to account for the normal uncertainties of material strength and strain-hardening effects at high strains.

TS brace

Figure 3.49. Column and brace connection at foundation. (Courtesy of Louis Choi, S.E., John A. Martin & Assoc. Structural Engineers, Los Angeles, CA.)

Foundation mat, spread footing or pile cap

Imaginary yield line

Anchor bolts designed to resist tension. (Bearing plates not shown for clarity)

### TS brace

Figure 3.49. Column and brace connection at foundation. (Courtesy of Louis Choi, S.E., John A. Martin & Assoc. Structural Engineers, Los Angeles, CA.)

Figure 3.50. Chevron brace-to-beam connection.

Notes: 1. Chevron braces designed to SCBF requirements are not subject to the load amplification factor of 1.5 imposed on chevron braces in OCBF systems.

2. Beam depth shown is not to scale. (Courtesy of Louis Choi, S.E., John A. Martin & Assoc. Structural Engineers, Los Angeles, CA.)

### Figure 3.50. Chevron brace-to-beam connection.

Notes: 1. Chevron braces designed to SCBF requirements are not subject to the load amplification factor of 1.5 imposed on chevron braces in OCBF systems.

2. Beam depth shown is not to scale. (Courtesy of Louis Choi, S.E., John A. Martin & Assoc. Structural Engineers, Los Angeles, CA.)

The link e is to be designed to carry an earthquake-induced force Pv = PE. Hence, the ideal strength of the link Pi needs to be greater than Pi = PE. Having chosen an appropriate segment of the beam as the eccentric link e, its overstrength, which becomes the design force, can be readily established and hence, the required strength for the strong and presumed brittle braces and columns. This illustrates the important relationship between the ductility potential of the entire bracing system and the corresponding ductility demand of a single ductile link.

Essential design features of the EBF are as follows:

• An eccentrically braced frame dissipates energy by controlled yielding of its link. It is a framing system in which the axial forces induced in the braces are transferred either to a column or another brace through shear and bending in a small segment of the bem called the link. The links in EBFs act as

Brittle links (typ) pOuctiie link L pwith strength »T,

Brittle links (typ) pOuctiie link L pwith strength »T,

Tensile strength of ductile link = T,

Figure 3.51a. Chain with ductile and brittle links.

Tensile strength of ductile link = T,

Figure 3.51a. Chain with ductile and brittle links.

structural fuses to dissipate earthquake-induced energy in a stable manner. To do so, a link needs to be properly detailed such that it has adequate strength and stable energy dissipation characteristics. All other structural components such as beam segments outside of the link, braces, columns, and connections are proportioned following capacity design provisions to remain essentially elastic during the design earthquake. They are designed for the forces generated by the actual (or expected) capacity of the links rather than the code-specified design seismic forces. The capacity design concept thus requires that the computation of the link strength be based not only on the expected yield strength of the steel, but also on considerations of strain-hardening and overstrength due to composite action of the slab.

If we ignore the effects of axial force and the interaction between moment and shear in the link, flexural hinges always form at two ends of the link when both MA and MB reach the plastic moment, Mp. A shear hinge is said to form when the shear reaches the plastic shear capacity, Vp (see Fig. 3.51). The presence of an axial force in a link reduces not only the flexural and shear capacities, but also its inelastic deformation capacity. When the axial force Pn exceeds 15% of the yield force, Py=AgFy, the P-M interaction formula for plastic design can be used to compute the reduced plastic moment. Mpa =

Composite action due to the presence of slabs can significantly increase the link shear capacity during the first few cycles of large inelastic deformations. However, composite action deteriorates rapidly in subsequent cycles due to local concrete floor damage at both ends of the link. For design purposes, it is conservative to ignore the contribution of composite action for calculating the link shear strength. But the overstrength produced by the composite slab effect must be considered when estimating the maximum forces imposed by the link on other structural components.

When detailing a link, full-depth web stiffeners must be placed symmetrically on both sides of the link web at the diagonal brace ends of the link. These end stiffeners are required to have a combined width not less than (bf - 2tw)

and a thickness not less than 0.75tw or 3/8 in., whichever is larger. The link section must satisfy the same compactness requirement as the beam section for special moment frames. Further, the link must be stiffened in order to delay the onset of web buckling and to prevent flange local buckling. The stiffening requirement is dependent on the length of link. Intermediate link web stiffeners must be full depth. Whereas two-sided stiff-eners are required at the end of the link where the diagonal brace intersects the link, intermediate stiffeners placed on one side of the link web are sufficient for links less than 25 in. in depth. Fillet welds connecting a link stiffener to the link web are to have a design strength to resist a force of Ast Fy, where Ast is the stiffener area. The design strength of fillet welds fastening the stiffener to the flanges shall be adequate to resist a force of AstFy /4. To ensure stable hysteresis, a link must be laterally braced at each end to avoid out-of-plane twisting. Lateral bracing also stabilizes the eccentric bracing and the beam segment outside the link. The concrete slab alone cannot be relied upon to provide lateral bracing. Therefore, both top and bottom flanges of the link beam must be braced. The bracing should be designed for 2% of the expected link flange strength. See Figs. 3.52 and 3.53 for a graphic representation of stiffener requirements.

The required axial and flexural strength of the diagonal brace shall be those generated by the expected shear strength of the link increased by 125% to account for strain-hardening. Although braces are not expected to experience buckling, the AISC-Siesmic provisions take a conservative approach by requiring that a compact section be used for the brace. At the connection between the diagonal brace and the beam, the intersection of the brace and beam centerlines shall be at the end of the link or within the length of the link. If the intersection point lies outside of the link length, the eccentricity together with the brace axial force produces additional moments in the beam and brace, which should be accounted for in the design. The diagonal brace-to-beam connection at the link end of the brace is also to be designed to develop the expected strength of the brace. No part of this

Figure 3.53. An EBF with W-shape bracing; stiffener requirements. Notes: 1. See Fig. 3.52 for items not noted.

2. Refer to AISC-Seismic, Sec. 15.3, and Sec. 3.11.2.1 of this text for required spacing of stiffeners.

Figure 3.53. An EBF with W-shape bracing; stiffener requirements. Notes: 1. See Fig. 3.52 for items not noted.

2. Refer to AISC-Seismic, Sec. 15.3, and Sec. 3.11.2.1 of this text for required spacing of stiffeners.

connection shall extend over the link length to reduce the link length, e. If the connection is designed as a pin, the gusset plate must be properly stiffened at the free edge to avoid local buckling.

It is highly desirable to use a split V-braced EBF to avoid a moment connection between the link and column. Test results have shown that a fully-restrained welded connection between a column and a link, particularly if the link is relatively long, is vulnerable to brittle fracture similar to those found in the beam-to-column moment connections after the Northridge earthquake. Therefore, AISC-Seismic provisions require that the deformation capacity of the link-to-column connections be verified by qualifying cyclic tests to demonstrate that the link inelastic rotation capacity is at least 20% greater than the calculated values.

When cover plates are used to reinforce a link-to-column connection, the link over the reinforced length must be designed such that no yielding takes place in this region. In this context, the link is defined as the segment between the end of the reinforcement and the brace connection. Cyclic testing is not needed when: 1) the shortened link length does not exceed e0 = 2Mp/Vp; and 2) the design strength of the reinforced connection is equal to or greater than the force produced by a shear force of 1.25 RyVn in the link. For the preferred EBF configuration, where the link is not adjacent to a column, a simple connection between the beam and column is considered adequate if it provides some restraint against torsion in the beam. Provisions of AISC-Seismic stipulate that the magnitude of this torsion be calculated by considering perpendicular forces equal to 2% of the beam flange nominal strength, Fybftf, applied in opposite directions on each flange. Although the link end moment is distributed between the brace and the beam outside of the link according to their relative stiffness, in preliminary design, it is conservative to assume that all the link end moment is resisted by the beam. Because a single member is generally used for both the link and the beam outside the link, it is too conservative to use the expected yield strength, RyFy, for estimating the force demand produced by the link while the beam strength is based on the nominal yield strength, Fy. Therefore, AISC-Seismic provisions allow designers to increase the design strength of the beam by a factor Ry.

• The horizontal component of the brace produces a significant axial force in the beam, particularly if the angle between the diagonal brace and the beam is small. Therefore, the beam outside the link must be designed as a beam-column. When lateral bracing is used to increase the capacity of the beam-column, this bracing must be designed to resist 2% of the beam flange nominal strength, Fybftf.

• Using a capacity design approach, columns in braced bays are designed to have sufficient strength to resist the gravity-load actions, moments, and axial forces generated by 1.1 times the expected nominal strength, RyVn, of the link.

Based on the results of limited tests, this design procedure may be appropriate for low-rise buildings and the upper stories of medium- and high-rise buildings, but may be too conservative in other instances. Therefore, an alternative design procedure is permitted by AlSC-Seismic provisions. The method consists of amplifying the design seismic axial forces and moments in columns by the overstrength factor, Q o = 2.0. The computed column forces need not exceed those computed by the first procedure. Therefore, the first design procedure will generally produce a more conservative design for columns.

### 3.11.2.1. Link Design

• To ensure stability of the link during inelastic deformations, compact sections shall be used, complying with the flange width-thickness ratios of bf /2tf = 52/( Fy )05

where bf = flange width tf = flange thickness

• Doubler plates on the web of the link are not allowed because they are ineffective during inelastic deformation.

• Holes are not allowed in the web of the link because they affect the inelastic deformation of the link web.

• To ensure ductile behavior, the specified minimum yield stress of steel used for links shall not exceed 50 ksi.

• The effect of axial force on the link design shear capacity need not be considered when

Pu = required axial strength

Ag = gross area of section

• If plastic hinges form at the ends of the link (see Fig. 3.51), a point of inflection occurs at the center of the link and the nominal required shear capacity is given by

Mp = nominal plastic flexural strength = ZFy

Z = plastic section modulus e = length of link Vp = nominal shear strength of link

db = depth of link tf = flange thickness tw = web thickness and fVp = design shear capacity of link f = resistance factor = 0.9

A balanced shear condition exists when flexural and shear hinges occur simultaneously for a link length of ey = 2Mp/Vp. For lengths less than ey, a shear mode predominates and for lengths greater than ey, a flexural mode predominates.

1. The nominal required shear capacity of the link is given by

pa where

Mpa = reduced nominal plastic flexural capacity

Vpa = reduced nominal shear capacity of link = Vp[1 - ( PU /Py )2]05

and fVna = reduced design shear capacity of link f = resistance factor = 0.9

2. From AlSC-Seismic, Eqs. (15.3) and (15.4), the length of the link is limited to the lesser of e = [1.15 - 0.5p'(Aw/A)]1.6Mp/ p'(//) > ^^^ or

where

Vu = required shear strength Ag = gross area of link

Figure 3.54. Link rotation angle gp. The link rotation angle gp is estimated by assuming the EBF bay rotates as a rigid body. By geometry, gp is related to plastic story drift angle 0p, which in turn is related to plastic story drift = Ap /h. Conservatively, Ap may be taken to equal design story drift. Refer to AISC 341-02, C15.2 for additional information.

Figure 3.54. Link rotation angle gp. The link rotation angle gp is estimated by assuming the EBF bay rotates as a rigid body. By geometry, gp is related to plastic story drift angle 0p, which in turn is related to plastic story drift = Ap /h. Conservatively, Ap may be taken to equal design story drift. Refer to AISC 341-02, C15.2 for additional information.

• For the maximum inelastic story drift, the elements of the frame may be considered rigid and the link rotation angle gp is derived as shown in Fig. 3.54 and is given by gp = LA / he = LQp / e where

L = beam length between column centers A = maximum inelastic story drift h = story height e = length of link Qp = story drift angle gp = link rotation angle

To limit the inelastic deformation of the frame, the link rotation angle is limited to the following values:

g < 0.080 radian... for short links of length e < 1.6 Mp /VP gp < 0.020 radian... for long links of length e > 2.6Mp/Vp

These limits are illustrated in Fig. 3.55 and linear interpolation may be used for intermediate link lengths. To ensure stable behavior of the link under cyclic loading, AlSC-Seismic specifies the following detailing requirements:

• To prevent web instability under cyclic loading, full-depth web stiffeners shall be provided on both sides of the link web at the brace end of the link. As shown in Fig. 3.56, the stiffeners shall have a combined width of

Figure 3.55. Permissible link rotation angle gp versus link length, e. The permissible link plastic rotation angle gp is the primary variable that describes the inelastic link deformation. It is strongly influenced by the length of the link, e, and its Mp/Vp ratio. When e < 2.6Mp/Vp, shear yielding dominates the inelastic response, and when e > 2.6Mp/Vp, flexural yielding governs the inelastic response.

Figure 3.55. Permissible link rotation angle gp versus link length, e. The permissible link plastic rotation angle gp is the primary variable that describes the inelastic link deformation. It is strongly influenced by the length of the link, e, and its Mp/Vp ratio. When e < 2.6Mp/Vp, shear yielding dominates the inelastic response, and when e > 2.6Mp/Vp, flexural yielding governs the inelastic response.

where bf = link flange width tw = link web thickness

The weld between the stiffener and the web is required to develop the full strength of the stiffener, as shown in Fig. 3.56. The weld must be adequate to resist the force as given by

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