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Design of Stiffeners Under Loads

As per AISC guidelines, the combined effect of forces from moment and shear should be considered while designing fasteners or welds for end connections of girders, beams, and trusses. When flanges or moment connection plates for end connections of beams and girders are welded to the flange of an I- or H-shape column, a pair of column-web stiffeners having a combined cross-sectional area Ast not less than that calculated from the following equations must be provided whenever the calculated value of Astis positive:

Ast = Pbf-Fyctwc(tb+5K) / Fyst
where
Fyc = Column yield stress, ksi (MPa)
Fyst= Stiffener yield stress, ksi (MPa)
K= Distance, in (mm), between outer face of column flange and web toe of its fillet, if column is rolled shape, or equivalent distance if column is welded shape
Pbf= Computed force, kip (kN), delivered by flange of moment-connection plate multi plied by 5/3 , when computed force is due to live and dead load only, or by 4/3, when computed force is due to live and dead load in conjunction with wind or earthquake forces
twc= Thickness of column web, in (mm)
tb= Thickness of flange or moment-connection plate delivering concentrated force, in (mm)

Notwithstanding the preceding requirements, a stiffener or a pair of stiffeners must be provided opposite the beam compression flange when the column-web depth clear of fillets dcis greater than

dc=( 4100t3wc(Fyc)½)/Pbf

and a pair of stiffeners should be provided opposite the tension flange when the thickness of the column flange tf is less than

tf=0.4(Pbf)½/Fyc

Stiffeners required by the preceding equations should comply with the following additional criteria:

1. The width of each stiffener plus half the thickness of the column web should not be less than one-third the width of the flange or moment-connection plate delivering the concentrated force.

2. The thickness of stiffeners should not be less than tb/2.

3. The weld-joining stiffeners to the column web must be sized to carry the force in the stiffener caused by unbalanced moments on opposite sides of the column.

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WEBS UNDER CONCENTRATED LOADS

Criteria for Buildings

The AISC specification for ASD for buildings places a limit on compressive stress in webs to prevent local web yielding. For a rolled beam, bearing stiffeners are required at a concentrated load if the stress f a , ksi (MPa), at the toe of the web fillet exceeds Fa= 0.66Fyw , where Fyw is the minimum specified yield stress of the web steel, ksi (MPa). In the calculation of the stressed area, the load may be assumed distributed over the distance.

For a concentrated load applied at a distance larger than the depth of the beam from the end of the beam:

Fa=R/fw(N+5K)
where
R=concentrated load of reaction, kip (kN)
tw = web thickness, in (mm)
N=length of bearing, in (mm), (for end reaction, not less than k)
K=distance, in (mm), from outer face of flange to web toe of fillet

For a concentrated load applied close to the beam end:
fa=R/tw(N+2.5 k)

To prevent web crippling, the AISC specification requires that bearing stiffeners be provided on webs where concentrated loads occur when the compressive force exceeds R, kip (kN), computed from the following:

For a concentrated load applied at a distance from the beam end of at least d/2, where d is the depth of beam:

R=67.5t2w[1+3(N/d)(tw/tf)1.5](Fywtf/tw)½

where
t f=flange thickness, in (mm)

For a concentrated load applied closer than d/2 from the beam end:

R=34r2w[1+3(N/d)(tw/tf)1.5](Fywtf/tw)½

If stiffeners are provided and extend at least one-half of the web, R need not be computed.

Another consideration is prevention of sidesway web buckling. The AISC specification requires bearing stiffeners when the compressive force from a concentrated load exceeds limits that depend on the relative slenderness of web and flange rwf and whether or not the loaded flange is restrained against rotation:

rwf=(dc/tw )/(l/bf)

where
l=largest unbraced length, in (mm), along either top or bottom flange at point of application of load
bf=flange width, in (mm)
dc= web depth clear of fillets = d – 2k

Stiffeners are required if the concentrated load exceeds R, kip (kN), computed from

R=6800t3w/h(1+0.4r3wf)
where
h=clear distance, in (mm), between flanges, and rwf
is less than 2.3 when the loaded flange is restrained against rotation. If the loaded flange is not restrained and rwf is less than 1.7,

R=0.4r3wf(6800t3w/h)
R need not be computed for larger values of rwf

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Combined Axial Compression Or Tension And Bending

The AISC specification for allowable stress design for buildings includes three interaction formulas for combined axial compression and bending.

When the ratio of computed axial stress to allowable axial stress f /F a exceeds 0.15, both of the following equations must be satisfied:

( f a / F a ) + ( C m x f b x ) / (1– f a /F e x ) F b x + C m y f b y / (1 – f a / F e y ) F b y ? 1

f a / 0.60F y + f b x /F b x + f b y / F b y ? 1

when f a /F a ? 0.15, the following equation may be used instead of the preceding two:

f a / F a + f b x / F b x + f b y / F b y ? 1

In the preceding equations, subscripts x and y indicate the axis of bending about which the stress occurs, and

In the preceding equations, subscripts x and y indicate the axis of bending about which the stress occurs, and

F a = axial stress that would be permitted if axial force alone existed, ksi (MPa)

F b = compressive bending stress that would be permitted if bending moment alone existed, ksi (MPa)

F e = 149,000 / ( Kl b / r b ) 2 , ksi (MPa); as for F a , F b , and 0.6F y , F e may be increased one-third for wind and seismic loads

Lb = actual unbraced length in plane of bending, in (mm)

r b = radius of gyration about bending axis, in (mm)

K = effective-length factor in plane of bending

f a = computed axial stress, ksi (MPa)

f b = computed compressive bending stress at point under consideration, ksi (MPa)

C m = adjustment coefficient

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Deflections of Bents and Shear Walls

Horizontal deflections in the planes of bents and shear walls can be computed on the assumption that they act as cantilevers. Deflections of braced bents can be calculated
by the dummy-unit-load method or a matrix method. Deflections of rigid frames can be computed by adding the drifts of the stories, as determined by moment distribution
or a matrix method.

Figure showing Building frame resists lateral forces with (a) wind bents or (g) shear walls or a combination of the two. Bents may be braced in any of several ways, including (b) X bracing, (c) K bracing, (d) inverted V bracing, (e) knee bracing, and (f) rigid connections.

For a shear wall (Fig) the deflection in its plane induced by a load in its plane is the sum of the flexural deflection as a cantilever and the deflection due to shear. Thus, for a wall with solid rectangular cross section, the deflection at the top due to uniform load is

where
w = uniform lateral load
H = height of the wall
E = modulus of elasticity of the wall material
t = wall thickness
L = length of wall

For a shear wall with a concentrated load P at the top, the deflection at the top is

Units used in these equations are those commonly applied in United States Customary System (USCS) and the System International (SI) measurements, that is, kip (kN), lb /in 2 (MPa), ft (m), and in (mm).

Where shear walls contain openings, such as those for doors, corridors, or windows, computations for deflection and rigidity are more complicated. Approximate methods, however, may be used.

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Load Distribution To Bents And Shear Walls

Provision should be made for all structures to transmit lateral loads, such as those from wind, earthquakes, and traction and braking of vehicles, to foundations and their supports
that have high resistance to displacement. For this purpose, various types of bracing may be used, including struts, tension ties, diaphragms, trusses, and shear walls.

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