CHAPTER 8 RESISTANCE TO PROGRESSIVE
8.6 EXAMPLE SUMMARY
Based on the example presented in this chapter, the follow- ing conclusions can be drawn:
• The steel framing acting alone as tension members in catenary action can provide a solution but may not be reasonable due to the large connection and restraint forces required.
• Steel framing action in tension and bending, in concert with the concrete slab acting in compression, provides a reasonable solution. The steel headed stud anchors tie the two materials together.
• The concrete/metal deck slab, properly reinforced, can be designed as a membrane to span across the failing bay on its own. This method is simple and efficient.
• If a more sophisticated analysis is employed that incorporates the contribution of all the structural ele- ments present—beams, studs, deck, concrete and reinforcement—the engineer may obtain a more eco- nomical design. The engineer does not need to resort to a full nonlinear dynamic analysis, but can utilize the nonlinear static pushover method coupled with the energy balance approach to compute the pushover and capacity curves. Additionally, the engineer should try to use the smallest steel beams possible thus creating more demand on the concrete and causing the stresses in the concrete to be significant. Progressive collapse is avoided if the initial collapse is prevented.
The examples discussed in this chapter are only applicable to the floors that are not directly affected by the blast that
Fig. 8-41. Axial force in girder connection.
Fig. 8-43. Steel tension strain in composite slab: steel layers start yielding at 44% of the load.
breaches the column. The floors that have not suffered dam- age will support themselves after the removal of the column preventing the failure of all the slabs that are supported by the column removed. The blast that breaches the column will produce uplift, damage the slab above the column, and limit
the capability of this slab to subsequently develop membrane action. Corner and perimeter columns, due to their geometry, cannot develop the membrane action that an interior column develops; hence, other engineering approaches are necessary to account for the partial loss of membrane action.
SYMBOLS
(Note: ms = milliseconds)A Surface area exposed to the pressure wave, in.2
Ag Gross area of the element, in.2
Ak Load or load effect resulting from an extraordinary
event A, kips
Aw Area of the web, in.2
B Blast load, kips
Cr Ratio of reflected pressure to free-field pressure
D Dead load, kips
DCR Demand-to-capacity ratios
DIF Dynamic increase factor
DLF Dynamic load factor
Ec Modulus of elasticity of concrete, ksi
Fcr Critical stress, ksi
Fmax Maximum resisting force the structure would experi-
ence if it were capable of remaining elastic, kips
Fpeak Peak blast load, kips
Fu Specified minimum tensile strength, ksi
Fy Specified minimum yield stress, ksi
Fyield Force that would cause the structure to yield, kips
I Impulse, psi-ms
Ir Reflected impulse, psi-ms
Iso Side-on impulse, psi-ms
K Structure stiffness, kip/in.
KL Load factor
KLM Load mass factor
KM Mass factor
L Span, in.
L Live load, kips
Mp Plastic moment, kip-in.
P Pressure, psi
Po Atmospheric pressure; peak pressure, psi
Pr Reflected pressure, psi
Pso Free-field pressure, psi; side-on peak pressure, psi
QUD Acting force (demand) determined in component or
connection/joint (moment, axial force, shear, and possible combined forces)
QCE Expected ultimate, unfactored capacity of the com-
ponent and/or connection/joint (moment, axial force, shear and possible combined forces)
R Stand-off distance, ft
Rgpu Shear rupture capacity of the gusset plate underneath
the weld, kip/in.
Rgpy Shear yield capacity of the gusset plate underneath
the weld, kip/in.
Rm Maximum resistance, kips
Rtu Tube rupture capacity under conventional shear, kip/
in.
Rtvu Shear rupture capacity of the tube underneath the
weld, kip/in.
Rtvy Shear yield capacity of the tube underneath the weld,
kip/in.
Rty Tube yield capacity under conventional shear,
kip/in.
Rw Shear capacity of the weld material, kip/in.
S Elastic section modulus, in.3
S Snow load, kips
SIF Strength increase factor
SR Strength ratio
T Natural period of structure, s
Tx Force in the reinforcement in the x-direction corre-
sponding to the strain, εx, kips
Ty Force in the reinforcement in the y-direction corre-
sponding to the strain, εy, kips
U Shock front velocity, ft/ms
V Velocity of the system, ft/ms
W TNT equivalent charge weight, lb
Wk Kinetic energy, joule
WP Energy produced by the load pulse, joule
WS Strain energy absorbed by the system, joule
Z Plastic section modulus, in.3
Z Scaled distance, ft/lb1/3
c Viscous damping
cc Critical damping
dbg Depth of the bolt group, in.
f Cyclic frequency, cycles per second
f ′c Minimum compressive strength of concrete, ksi
fconc Compressive strength of the concrete, ksi
fi Force per floor used to obtain the displacement per
floor, kips
f ′dc Dynamic strength of concrete, ksi
fds Dynamic design stress, ksi
fdv Dynamic design stress for shear, ksi
g Acceleration due to gravity, 386 in./s2
gk Specified dead load for the floor or roof, kips
kDRF Dynamic reduction factor for impulsive loads where
T > td
m Mass of the structure, lb
me Mass of the system, lb
ps Density of air behind shock front, lb/ft3
qk Specified imposed load (live load) for the floor or
roof, kips
qo Peak dynamic pressure, psi
st Mean transverse spacing between ties, in.
t Thickness of the slab, in.
t2 Time to peak pressure, ms
ta Time of arrival, ms
td Load duration, ms
te Load duration, ms
tr Rise time to peak pressure, ms
us Particle velocity, ft/ms
vi Initial velocity, ft/ms
w Distributed load, kips
wi Weight per floor, kips
xmax Peak displacement, in.
xo Arbitrary displacement, in.
Δ Displacement, in.
ΔPL Permanent deformation of the system, in.
ΔT Axial deformation at expected yielding load, in.
Δc Axial deformation at expected critical stress, in.
Δel Elastic displacement, in.
Δi Displacement per floor, in.
Δm Maximum displacement, in.
Δmax Peak displacement, in.
Δyield Yield displacement, in.
εx Strain in the x-direction
εy Strain in the y-direction assumed to be equal to
εx
(
Lx2 Ly2)
μ Ductility
ϕ Resistance factor
ω Undamped natural frequency, rad per unit of time ωd Damped natural frequency for the structure, rad per
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