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Mn = Af ff (d − kd/3) (2)
where 2 ( )2 f f f f f f k = ρ n + ρ n − ρ n ; ρf is given by Af /bd;
and nf is the modular ratio Ef /Ec. The nominal moment found
using Eq. (2) will be slightly smaller than that from Eq. (1).
As a point of interest, recommendations provided in
Reference 1 result in even smaller moments than found using
Eq. (2). For either calculation, the capacity is extremely
sensitive to the position of the grid. If the cover is 1 in. (25 mm)
rather than 0.75 in., for example, the nominal values would be
lowered by 8%. Measurements made after the failures showed
that the product was not held in position within the flange, as
the cover and thus the effective depth varied widely (Fig. 5).
The moment capacities from three different assumptions
about the strand capacity are given in Table 1. Values are
computed using Eq. (1), with d = 2.72 in. and d = 2.50 in.
These are stated in the usual terms for slabs, in kip·in./ft.
The Load Side of the Equation
The design loads for parking structures are given in the
International Building Code (IBC)8 and ASCE 7-10.9 For
structures designed for vehicles carrying no more than nine
passengers, the design loads are either: ••Dead load plus 40 lb/ft2 (1.9 kN/m2) live load plus snow
load; or ••Dead load plus a single concentrated load of 3000 lb (13 kN).
The load combination producing the larger forces governs,
and the resultant forces are to be computed using the
appropriate load factors.
For Structure A, the 3.5 in. flange dead load is 43.75 lb/ft2
(2.1 kN/m2), and the specified snow load is 20 lb/ft2 (1 kN/
m2), leading to wu = 1.2(43.75) + 1.6(40) + 0.5(20) = 126.5 lb/
ft2 (6.1 kN/m2). For a cantilever length of 37-3/8 in., mu = 7.36
kip·in./ft (2.73 kN·m/m).
References 8 and 9 instruct the designer that the 3000 lb
concentrated load represents the force from a jack, and it is to
be applied to an area that is 4.5 in. (114 mm) square. The
moment M is simply force P times cantilever length l. We
would reasonably assume that a load applied at a joint
between flanges will spread out at a 45-degree angle, leading to
the moment being resisted by a width of 2l. Thus, the moment
becomes m = Pl/2l = P/2, where this represents a moment per
unit width, such as kip·ft/ft or kip·in./in. As noted in Reference 10,
this agrees quite well with the maximum local moment
obtained from an elastic plate theory solution of m = 0.509P.
At the time Structures A, B, and C were designed, the sixth
edition of the Prestressed Concrete Institute (PCI) Design
Handbook11 was widely used in the industry, and it provides
an example illustrating the calculation of the moment due to
the concentrated load in flanges of prestressed DT members.
The example indicates that the moment is to be resisted by a
width of 2l + 8 in., where the 8 in. (203 mm) is the assumed
width of a tire. Using the PCI Handbook recommendations,
the tributary width for the DTs in Structure A is 6.896 ft (2.1 m).
The 3000 lb concentrated load is assumed to be shared evenly
between two adjacent flanges, and the factored moment from
this component of the load can be calculated as
mu = 1.5 kip (1.6 × 37.375 in./6.896 ft) = 13.01 kip·in./ft
Fig. 5: Effective depth d measured on DTs installed on Level 4 of
Structure A. Note that more than half of the measurements indicate
that d is at or below 2.5 in. (63 mm)
Summary of moment capacities with different strand force assumptions based on a 3.5 in. thick flange and
effective depths of 2.72 and 2.50 in. Moment values for the latter effective depth are provided in parentheses ( )
Case Strand force, lb
Value listed in Reference 2
830 = Avg. − 2σ
Based on data published by the grid producer
683 = Avg. − 3σ
Based on ACI 440.1R1 recommendations
Note: 1 lbf = 4.4 N; 1 kip/in.2 = 0.007 kN/mm2; 1 kip·in./ft = 0.37 kN·m/m