BRACED FRAMES
5.22 BEAM DESIGN
–12Pu ⫽tF Ay g⫽ tF twy ⫽(0.9)(36)(0.1875)w⫽ 6.075w w⫽ 106 ⫽ 8.73 in. say 9-in.-wide plate
2(6.075)
Pu ⫽2tF UAu g⫽ 2(0.75)(58)(0.75)(0.1875)(9.0)⫽110 kips⬎ 106 kips The plates have sufficient strength.
The two––163 ⫻ 9 plates will be welded, one to each side of the column web.
To obtain the required plate length, the weld needs to be designed. By choos-ing a –18-in. weld size, we will determine the required weld length L and associated plate length. The design strength of the weld across the throat of an -in. fillet weld in shear per 1-in. length is given by–18
–1
0.707twFw ⫽0.707t(w)(0.6FEXX) ⫽0.707( )(0.75)(0.68 ⫻ 70)
⫽2.7838 k / in.
where t⫽ weld size. Therefore the required weld length with four lengths to match two splice plates is
L⫽ 106 ⫽ 9.52 in. say 10 in.
4⫻ 2.7838
The induced moment, magnified by 150%, will be counteracted by a -in., 8-–38 in.-long partial-penetration weld (the width of the upper column) with a flex-ural strength over 10 times larger than applied.
It was felt necessary to provide a weld size of–38 in. larger than mandated by strength considerations. The reader is reminded that the column flange thickness of the lower column is 0.9 in. and that specifying a too-small weld might result in a brittle, unsafe weld on account of the cooling-off effect of the mass of the column flange. Table J2.3, Part 6 of the Specification, gives a minimum weld size of––165 in. for a thickness of –1 in. Figure 5.14 shows–34 details of the column splice.
5.22 BEAM DESIGN
Design of Second-Floor Beam
Following the provisions of CHAP. 22, DIV. IV, 9.4.a.3: ‘‘A beam intersected by V braces shall be capable of supporting all tributary dead and live loads assuming the bracing is not present.’’ The beam will be designed as a simply
168 SEISMIC STEEL DESIGN: BRACED FRAMES
Figure 5.14 Column splice.
supported beam with L⫽ 19.0 ft to resist said load. Furthermore and follow-ing CHAP. 22, DIV. IV, 9.4.a.4, we will provide fully effective lateral support to the beam at half of its span, that is, the point of intersection with the V braces. With careful design selection,
Lb⫽ 9.5 ft⬍ Lp⫽ 10.7 ft
the full plastic moment of resistance Mp can be expected from the beam.
Factored gravity loads acting on the beam are
Dead load: 1.2D⫽ 1.2(0.8)⫽ 0.96 kip-ft Live load: 1.6L⫽ 1.6(0.7)⫽ 1.12 kip-ft 2.08 kip-ft
However, per CHAP. 22, DIV. IV, 9.4.a.1: ‘‘The design strength of the brace members shall be at least 1.5 times the required strength using Load
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5.22 BEAM DESIGN 169
binations 3-5 and 3-6.’’ We will multiply both moment and axial force in the brace member by 1.5. The maximum simply supported bending moment is
2 2
wL 2.08⫻ 19.0
Mu⫽ 1.5冉 冊 冉8 ⫽ 1.5 8 冊⫽ 141 kip-ft
The maximum associated axial force acting at the centerline of the beam is given as
Px⫽ 15 kips From load case [3], computer analysis Pux⫽ 1.5(15)⫽22 kips
The beam will be designed for combined bending and axial force about the major axis with
KL⫽ 9.5 ft
and for pure axial load about the potential out-of-plane buckling about its minor axis. Going back to the LRFD equations (H1-1a) and (H1-1b) and the LRFD moment magnification equation (C1-5), the properties of the W10 ⫻ 54 are
Ag⫽ 15.8 in.2 rx⫽ 4.37 in. ry⫽2.56 in.
KLx 9.5⫻ 12 KL
⫽ ⫽ 26 ⫽ 44.5
rx 4.37 ry
with
Pu 22
⫽ ⫽0.05 ⬍0.2
cPn 465 Use the LRFD equation (H1-1b).
From Table 8 of the Specification with KL / rx⫽ 26, P Ae1 g⫽ 423.4 kips
Cm 1.0
B1⫽ ⫽ ⫽1.055
1⫺ P /Pu e1 1⫺ 22 / 423.4
Mux⫽ 1.055⫻ 141⫽ 149 kip-ft From LRFD equation (C1-5) By applying the LRFD interaction equation (H1-1b) with
170 SEISMIC STEEL DESIGN: BRACED FRAMES
Lb ⫽9.5 ft ⬍Lb ⫽10.7 full plasticity is achieved and
bMp ⫽180 kip-ft From LRFD selection table 4-19, Part 4, Specification
cPn ⫽465 kips
Remember thatcPncomes from the column design table 3-27 of the Spec-ification but with a KLxmodified by the rx/ryfactor for the major axis.
The beam is safe in carrying the factored gravity loads without support from the braces.
Magnitude of Postbuckling Effect
Figures 5.9 and 5.10 illustrate the design-detailing requirements for the out-of-plane buckling of the brace.
The next issue is the design of beams for inverted-V or chevron bracing.
In this structural system the braces intersect at the midspan of the floor beam.
There are two important aspects for consideration:
1. The 1997 UBC clearly reiterates C707.4.1 of the SEAOC ‘‘Recom-mended Lateral Force Requirements and Commentary’’ that the beam should be capable of supporting both dead and live loads without the help of the brace system ‘‘in the event of a loss in brace capacity.’’ The 1997 UBC provisions for safeguarding such an event state: ‘‘The beam intersected by braces shall be continuous between columns’’ (CHAP.
22, DIV. IV, 9.4.a.2), ‘‘A beam intersected by V braces shall be capable of supporting all tributary dead and live loads assuming the bracing is not present’’ (CHAP. 22, DIV. IV, 9.4.a.3), and ‘‘The top and bottom flanges of the beam at the point of intersection of V braces shall be designed to support a lateral force equal to 1.5 percent of the nominal beam flange strength (Fybctƒ)’’ (CHAP. 22, DIV. IV, 9.4.a.4).
2. At the postbuckling stage, when the internal compressive force in the compression brace element will be smaller than the tensile force in the stiffer tensile brace, the unbalanced vertical component will impact on the horizontal connecting beam, which could cause a plastic hinge and large vertical beam deformation at the intersection.
Unlike the 1994 UBC, the 1997 edition is silent about the magnitude of such unbalanced force and beam stiffness / strength requirement to counter such unbalanced force. In reviewing research data the issue appears complex.
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5.22 BEAM DESIGN 171
Among important parameters impacting on one another are the driving function of the earthquake (frequency, magnitude, relationship between p and s waves), the relative stiffness of brace and beam elements, and the slender-ness ratio of the braces.
A rough and somewhat simplified picture of the mechanism is as follows:
A marked deterioration of compression brace member stiffness develops after a number of cycles. The stiffness / resistance of the compression brace can drop as low as 30% of the initial stiffness of the brace. A great number of researchers and the SEAOC Recommendations agree on this value (C708.4).5 There is a consensus among researchers and design engineers that the tensile brace retains, during the postbuckling phase, a stiffness and resistance greater than the compression brace. However, there seems to be less agree-ment over quantitative determination of the exact value of the tensile force in the tensile brace that is coupled with a reduced compression in the corre-sponding compressive brace.6–9
Some options for the analysis are as follows:
(a) Energy methods.
(b) Compatibility analysis based on inelastic deformations.
(c) Strength evaluation taking into account the maximum moment of re-sistance Mp of the beam at the brace intersection. What makes this option viable and relatively easy to apply in an actual design is that researchers and SEAOC agree on a 30% compression brace resistance as the lower bound.
We will apply method (c) to the floor beams of our project.
Assume the story drift is still small in relation to the dimensions of the structure. Then the unknown force in the diagonal tension brace and its ver-tical component can be calculated from the relationship
1 1
– –
Mp ⫽ 4(T sin␣ ⫺ 0.3P sinn ␣)L⫽ 4[sin ␣(T ⫺0.3P )L]n
where ␣ is the angle measured from the horizontal to the centerline of the inclined brace, T is the axial force developed in the tension brace, and Pn is the nominal strength of the compression brace before postbuckling.
Applying the method for the second-floor beam of the SCBF prototype of our project,
sin␣ ⫽ 128 ⫽ 0.79 162.6
The nominal compression strength of the 7 ⫻ 7 ⫻ –12 structural tubing is derived as
172 SEISMIC STEEL DESIGN: BRACED FRAMES
The nominal plastic moment of resistance Mpof the W10⫻ 54 beam (LRFD selection table 4-19, Part 4, Specification) is
bMp 180
Mp ⫽ ⫽ ⫽200 kip-ft
b 0.9
L⫽18.94 ft (clear span between columns)
Therefore the maximum tensile force that can develop in the tensile chord simultaneous to postbuckling of the compression brace and full plastic yield-ing of the connectyield-ing beam is
4Mp 4(200)
T⫺132.5 ⫽ ⫽ ⫽53.5
(sin ␣)(L) (0.79)(18.94) T⫽53.5 ⫹132.5 ⫽186 kips
and the vertical unbalanced component impacting at the midspan of the beam is given as
Pu
sin␣冉T⫺ 0.3c冊⫽ 0.79(186⫺132.5) ⫽ 42.2 kips
The reaction by the relatively light dead and live loads on the still functioning braces was ignored.
It is worth noting that:
1. Full lateral support to beam flanges has been provided for the value of 0.015 Fybƒtƒat the midspan of the beam where the CG of braces inter-sects the CG of the beam.
2. By proper choice of the beam the unsupported length of the W12⫻54 beam Lb⫽ 9.5 ft ⬍ Lp⫽ 10.7 ft, full development of plastic moment Mp can be expected at the beam–brace intersection, facilitating forma-tion of a true plastic hinge.
The results of research work aimed at identifying weaknesses in the conventional CBF design and developing a much improved SCBF adapted to strong ground motion seismic areas have been briefly de-scribed earlier in this chapter in Section 5.10.
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5.22 BEAM DESIGN 173
Figure 5.15 Variations of concentric braced frame (CBF).
Associated with the research activities, studies have also been conducted about the magnitude and nature of internal forces induced in the family of the conventional inverted-V braced frame under static loading.10
Six configurations were studied involving different brace arrangements plus the addition of vertical struts at intersecting points of the brace. Among these are the arrangements termed STG and its ZIP variant, a ‘‘zipper’’ type of configuration. (See Figures 5.15a, b, and c.) The magnitude of the internal forces was smaller than those carried by its cousins, in particular the V-braced frame, named VREG (conventional V CBF).
Despite modest savings achieved using the STG or ZIP, the problem of energy dissipation remains at the anchor points of the structure where an elastic medium, the steel structure, meets a virtually infinitely stiff medium—
concrete footings or basement wall—demanding a high concentration of en-ergy dissipation. The sudden change in stiffness—slender steel members meeting large masses of unyielding concrete—will cause even a well-designed structure to undergo brittle fracture failure.
Such was the case for the Oviatt Library at the California State University in Northridge, where 4-in.-thick steel base plates of the concentrically braced frame structure shattered in brittle fracture, in addition to other visible dam-age: bending of 1 -in. anchor bolts and punching, shear-type failure cracks–34 around the perimeter of columns solidly welded to base plates. This was most likely due to lack of sufficient energy dissipation at the crucial anchor points in an otherwise conservatively designed structure.
This brings us to a new field of development: energy-dissipating mecha-nisms. The purpose of these is to improve energy dissipation of the structure.
Following the Loma Prieta and Northridge earthquakes it became evident that, contrary to general belief, steel structures possess relatively small inherent damping unless heavy—nonengineered architectural or engineered stiff ele-ments such as concrete or masonry walls—help in reducing the prolonged swaying and time-dependent story drifts that occur during an earthquake11:
During the Loma Prieta earthquake (M ⫽7.1), the East Wing of the 13-story Santa Clara Civic Center Office Building exhibited strong, prolonged building
174 SEISMIC STEEL DESIGN: BRACED FRAMES
response. Results from a study conducted after the earthquake showed a lack of inherent damping to be the primary cause for the building’s poor dynamic be-havior. . . . researchers determined the building had very low inherent damping (⬍1%) and that bracing alone would not solve the problem experienced during the Loma Prieta Earthquake. One unusual feature was the long duration of strong vibration (available records measured response in excess of 80 seconds with little sign of decay and a torsional beat measured at 100 seconds). [From ref.
12.]
Although the consensus among engineers is still 2–3% inherent critical damping, researchers involved with the evaluation of the dynamic response of structures during an earthquake believe that the actual inherent damping of steel structures having only light curtain walls built in the last decades is considerably lower and seldom exceeds %. The general thought about the–12 causes for such low actual damping values is the successful elimination of massive internal and external masonry and concrete architectural walls, stiff features to reduce the mass and impurity of structural response in modern earthquake engineering. It seems paradoxical that such commendable engi-neering effort leads to another issue: how to provide added damping to de-prived structures. In Chapter 13 of this book we will discuss some of the new trends in engineering and project design.13–15
5.23 COLUMN BASE-PLATE DESIGN