Design of Chevron Gusset Plates
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1 017 SEAOC CONENTION PROCEEDINGS Desin of Chevron Gusset Plates Rafael Sali, Director of Seismic Desin Walter P Moore San Francisco, California Leih Arber, Senior Enineer American Institute of Steel Construction Chicao, Illinois Abstract The Chevron Effect is a term used to describe local beam forces in the usset reion of a chevron (also termed inverted- ) braced frame. These local forces are typically missed by beam analysis methods that nelect connection dimensions. Recent publications have shown how to correctly analyze for these forces (Fortney & Thornton, AISC Enineerin Journal, ol. 5, 015). This study adds desin solutions for addressin hih shears in the connection reion, includin reinforcement, proportionin, and innovative detailin. Introduction Chevron (also termed inverted-) braced frames are commonly used in steel structures. In these frames two braces connect to the beam midpoint. Typically the braces are ow the beam, formin an inverted, althouh they may be above, formin a, or both above and ow, formin a two-story X with the beam at the center. Fiure 1 shows these confiurations. Fi.. Typical chevron usset desin. Recent work by Fortney and Thornton (Fortney and Thornton, 015) hihlihts the importance of careful connection analysis in order to determine the local stresses induced by the usset connection in a chevron braced frame. In particular, Fortney and Thornton derive expression for the local moments and shears that result from distribution of brace forces over the usset-plate lenth. These forces can result in the need to supplement the beam web with a doubler plate. An example of such a condition is shown in the AISC Seismic Desin Manual (AISC, 01). This study applies the same concepts investiated by Fortney and Thornton, but with the aim of providin enineers with desin equations to enable the selection of beams that do not require reinforcement. Fi. 1. Chevron braced frame confiurations. These frames are typically desined usin centerline models, and the beam forces and brace forces are in equilibrium at the center connection. In typical desin, a substantial usset plate is provided at the center, and force transfer between braces and beams is accomplished over the lenth of the usset plate. Fiure shows such a usset plate. Consistent with ductile desin of braced frames, it is assumed that braces apply loads to the beams and do not provide support. These forces are typically equal to the capacity of the braces in the desin of ductile systems, but the desin equations derived here are equally applicable to chevron frames desined for wind or other cases that do not involve capacity desin. 1
2 017 SEAOC CONENTION PROCEEDINGS Symbols, Nomenclature, and Conventions This study employs the followin symbols and terms: H Horizontal component of brace forces (braces ow). L Gusset lenth. L beam Beam lenth. M Moment at beam-to-usset-interface due to brace forces (braces ow). N Concentrated force at beam flane. P 1 Left-hand (lower) brace. Tension is positive. P Riht-hand (lower) brace. Compression is positive. P 3 Left-hand (upper) brace. Compression is positive. P 4 Riht-hand (upper) brace. Tension is positive. R u Required strenth. b Maximum beam shear (within connection reion) due to brace forces. ba Beam shear (outside connection reion) due to brace forces. ertical component of brace forces (braces ow). ef Effective beam shear strenth. n Nominal beam shear strenth. z Beam shear from moment transfer (for concentratedstress approach). a Lenth of beam from support to usset ede (equal to half the difference between the beam lenth and the usset lenth). d Beam depth. d Gusset depth. e b Eccentricity from beam flane to beam centerline, equal to half the beam depth. e z Lenth of moment arm (for concentrated-stress approach). k Distance from outer face of flane to web toe of fillet. t Gusset thickness. t w Beam web thickness. x Distance from usset ede toward beam midpoint. z Lenth of concentrated stress reion at ends of usset (for concentrated-stress approach). φ Resistance factor θ Brace anle from horizontal. τ Horizontal shear stress. Beam Forces Fi 3. Chevron usset eometry. For clarity, brace forces are separated into vertical and horizontal components. Assumin two braces ow with forces P 1 and P, the horizontal component is: ( ) H P1 + P cos θ The vertical component is: P1 P sin θ ( ) These forces on the beam-to-usset interface are statically determined. In addition to these vertical and horizontal forces, there is a moment (required for static equilibrium): M H e b Fiure 4 shows free-body diarams of the usset plate. Fiure 3 shows dimensions noted on beam and usset-plate diarams. Fi. 4. Free body diaram of usset plate Uniform stress approach Typically, the stresses at the beam-to-usset interface are assumed to be distributed uniformly usin the full lenth for the vertical and horizontal forces and a plastic-sectionmodulus approach for the moment (Fortney and Thornton,
3 017 SEAOC CONENTION PROCEEDINGS 015). Followin this approach, the shear within the connection reion is described by the followin equation: ( 1 ) sin θ ( 1 ) sin θ 4( 1 + )( cos θ) P P P P P P eb ( x) x + x L L L for 0 x The maximum shear in the connection reion occurs at the beam midpoint and is equal to: M b L ( P1 + P )( cos θ) d L This shear is not equal to the vertical component of either of the brace forces (P 1sinθ or P sinθ); it may be reater or smaller than those values, dependin on the eometry of the connection. The difference between the two is the shear carried by the usset,, presented later. Note that this beam shear, b, is due only to the horizontal components. The unbalanced vertical component does cause shear in the beam, but this shear becomes zero at the beam midpoint. Fiure 5 shows a shear diaram for brace-induced shears in a typical pin-end beam. Fixed-end beams may have a sway-induced shear at midspan. Also, in certain loadin conditions ravity loadin may cause a non-zero shear at the midpoint. L M φ n ( P1 + P )( cos θ) φ n d The loner the usset plate, the reater the portion of shear that remains in the usset and the less that is transferred to the beam. In this sense, the usset plate can be used as external shear reinforcement for the beam, althouh the deree of reinforcement is limited by the connection eometry. Note that selection of a shallower beam reduces the required usset lenth. For beams with small moments due to vertical unbalanced forces it is often more economical to select a shallow beam rather than a deeper beam that would either have to be reinforced for shear or be heavier to preclude the need for reinforcement. The shear outside the connection reion is due to the unbalanced vertical components of the brace forces. The moment M due to the brace horizontal components produces no shear outside of the connection reion. ba 1 Beam moments are described by the followin equation: ( P1 P ) + cos θ x M ( x) eb x + L L ( P1 P ) sin θ ( P1 P ) sin θ ( x + a) x L A simplified equation can be used to provide a liberal estimate of the maximum brace-induced moment: Lbeam M Mb Fi. 5. Brace-induced shears in pin-end beam (uniform stress approach). The beam shear is the result of both the eccentricity (the beam depth) and the usset lenth. These can be adjusted (within reason) to provide a beam that does not require web strenthenin. Followin this approach, the minimum usset lenth is: This equation combines two maxima that do not occur in the same location, and nelects an offsettin term. The second component of this moment (which is a local effect of the connection eometry) is typically small, but may be the overnin moment in cases with no unbalanced vertical force from the braces. Gravity moments are typically at a maximum at the beam midspan and should be combined with these. 3
4 017 SEAOC CONENTION PROCEEDINGS Concentrated stress approach As noted above, the beam forces are a result of the assumed stress distribution on the usset-to-beam interface. The beam shear may be reduced by increasin the moment arm over which the moment is divided. Instead of increasin the usset lenth, however, the concentrated stress approach maximizes the moment arm within a iven usset lenth. In this approach, the moment (due to the braces horizontal components) is assumed to be transferred at the ends of the usset over lenths z. The unbalanced force (due to the braces vertical components) is transferred in the remainin center portion of the usset, between the end reions. Fiure shows such a stress distribution. Fi. 6. Stress distribution for concentrated stress approach. Fiure 7 shows a shear diaram correspondin to this stress distribution. The shear from the moment transfer is thus: M z e z Note that in this case the maximum shear does not occur at the beam midpoint. It is a combination of the shear due to the unbalanced force and the shear due to delivery of the moment. The maximum shear is iven by the followin equation: + 1 b z This shear may be set less than or equal to the desin shear strenth of the beam in order to preclude the need for shear reinforcement. For a iven usset lenth the maximum moment transfer can be achieved by the hihest concentration of stress at the ends. At a maximum, stiffeners at the usset edes and within the throat of the beam may be used to create a moment arm equal to the usset lenth L. Short of that, the concentrated stress may be limited by the web tensile strenth or the usset strenth. (Typically it is the former.) Assumin the usset lenth is optimized, the concentrated stress will be maximized such that the full beam shear strenth is utilized. Considerin that some of the beam shear strenth is utilized in resistin the unbalanced force, the remainin beam shear strenth that can be utilized for the moment transfer is: φn 1 Considerin these limits the minimum lenth over which this force can be transferred by the usset is: z φf y t The minimum lenth over which this force can be transferred by the beam in web local yieldin (AISC Specification Section J10.) is: z 5k φf t y w For simplicity, the latter term may be nelected. Hereafter it is assumed that the beam web is the limitin factor. This method sets the beam required shear strenth equal to its desin strenth: Fi. 7. Brace-induced shears in pin-end beam with nonuniform stress distribution at connection. The moment arm e z is: e L z z + 1 φ b ef The correspondin moment arm is: M ez n ef 4
5 017 SEAOC CONENTION PROCEEDINGS The correspondin minimum usset lenth is: L e + z z M + φ F t ef y w ( 1 )( cos ) d 1 P + P θ φn + φ φf t 1 n y w For usset lenths reater than this value, the lenth z is: L L M z 4 φf t For the case with stiffeners used to transfer the vertical force ef, the second term becomes zero. Shorter ussets may be used, but only if ef is increased (e., the beam is reinforced). Once aain, note that usin a shallower beam can be effective in reducin the required usset lenth. Local beam limit states such as web cripplin should be evaluated. The concentrated force to be considered is: Ru φn 1 or M Ru L z The bearin lenth may be taken equal to z. Beam Moment The approach to beam moment described earlier assumes that the vertical stresses are as shown in Fiure 6, and that the horizontal stresses are uniform and equal to: ( P1 + P ) cos θ τ Lt Fortney and Thornton (015) describe conditions in which the beam moment determined usin these assumptions (if not considered in desin) may necessitate reinforcement. In this study the authors propose an alternative approach to beam moment employin the lower-bound theorem to allow an alternative (non-uniform) shear stress distribution and thereby demonstrate the adequacy of the beam. In this approach is the horizontal shear stress at each point x alon the usset lenth is determined such that its effect on moment (due to eccentricity from the beam centerline) neates the incremental moment due to vertical shear: e τ ( x) dx ( x) dx b τ ( x) ( x) e b y w This horizontal stress distribution results in zero moment due to connection forces. This stress should be considered in usset analysis and in the weld sizin. For the uniform-stress approach (with non-uniform horizontal shear) the maximum shear stress is: ( P1 + P )( cosθ) d τ max Lt This maximum stress occurs at the beam midpoint. For the concentrated-stress approach the maximum horizontal shear stress (correspondin to the shear-stress distribution that results in zero moment) is: τ φn max dt This stress is quite hih and thus neatin the moment is not enerally a suitable approach if the concentrated-stress approach is used to limit beam shear. In such cases the horizontal stress may be assumed to be transferred over the lenth e z. The shear stress would thus be: ( P1 + P ) cosθ τ ezt In this case brace-induced moment in the beam would be: z Lbeam M b + 4 The first term is the connection-induced moment and is at a maximum at a point alon the usset a distance z from its ede. The beam should be evaluated considerin this moment in combination with the axial force. Note that the accumulation of axial force in the beam is a consequence of this horizontal stress and is thus at a maximum at the location of maximum moment. Weld sizin Under either the uniform-stress approach or the concentratedstress approach the weld adequacy should be evaluated usin AISC 360 methods, such as the instantaneous center of rotation, which represents both weld strenth and the limits on weld ductility (assumin the weld connects riid elements). Forces across the usset-to-beam interface are H,, and M. However, for desins employin the concentrated-stress approach stresses may redistribute alon the weld due to beam inelasticity. Conformance with the desin methods described above indicates adequacy of the system under those conditions. For the concentrated-stress approach the weld in the zones z should be evaluated for the vertical force ef. Welds in the center reion (L z), the vertical force is and the horizontal force is H. 5
6 017 SEAOC CONENTION PROCEEDINGS Combinations of forces The beam forces derived are for braces ow with opposite forces (one brace in tension and the other in compression). These forces may be combined with ravity-induced forces in the beam, and with shear due to flexural restraint for fixed-end beams. While the diarams show the left brace in tension and the riht brace in compression, forces correspondin to the opposite case are easily determined by usin neative values for the brace forces. For the two-story-x confiuration brace induced shears and moments will be additive for the typical case in which the story shears are in the same direction. Gusset plates may be of different lenths above and ow, but for simplicity they may be set to be equal. The modified equation for minimum ussetplate lenth to preclude the need for reinforcement is: L ( )( ) d P + P + P + P cos θ φ n + φ 1 φf t n y element analysis. His work will study the distribution of shear between the beam and the usset, the impact of usset yieldin on the connection reion, and the behavior of existin frames which were desined without consideration of the chevron effect. The research project will conclude in 018. References AISC (016). Specification for Structural Steel Buildins, ANSI/AISC , American Institute of Steel Construction, Chicao, IL, July 7. AISC (01), Seismic Desin Manual, nd ed., American Institute of Steel Construction, Chicao, IL. Fortney, Patrick J. and William A. Thornton. (015), The Chevron Effect Not an Isolated Problem. AISC Enineerin Journal, 015, Qtr. The vertical unbalance force includes the effects of all braces. Gusset forces Statics require that certain forces be transferred across the midpoint of the usset. These forces are: H M For the uniform-stress method: ( P1 P ) 1 cos θ 1 d H ( 1 ) 1 P + P sin θ For the concentrated-stress method: b Conclusions ( 1 ) 1 P + P sin θ ef This study provides desin equations that can be used in the selection of beams in chevron braced frames that will have sufficient shear strenth without the need for web reinforcement. The desin method allows enineers to use the usset plate as external reinforcement for the beam web. These equations can be used to assess the effects of beam depth and usset lenth on the beam shear demand in order to optimize beam selection. Dr. Paul Richards of Briham Youn University is currently investiatin the chevron effect throuh inelastic finite 6
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