Example Problem 10.2

Given: A floor system similar to that shown in BGSCM Figure 10.1.1. The slab is 4" thick and is made from concrete with f'c = 3,000 psi. The floor beams span 50 ft and are spaced 5 ft apart. Assume that the beams are shored at third points during construction. The beams only have lateral support at shoring and the ends during the initial construction. The imposed floor dead load is 25 psf , the construction live load is 20 psf, and the occupancy floor live load is 50 psf. Consider studs of 1/2", 5/8", 3/4", and 7/8" diameter.

Wanted: Select the lightest I-shaped section for the application considering both ASD and LRFD. Also determine the number and spacing of shear studs needed for full composite action.

Solution:

The problem statement implies that a complete solution is required. This means that we need to check all applicable limit states. For each potential member, the following limits are considered:

Flexural capacity of just the steel beam during construction (before the concrete sets).
Flexural capacity of the composite section under occupancy loads.
Shear capacity under occupancy loads.
Deflection during construction.
Live Load only deflection during occupancy.

The general approach is to select a steel section and see if it works. If not, or if it provides too much excess capacity, another section is chosen. The solution provided here works off a list sorted by size. The process begins with the lightest section then works up to the first section that satisfies all the limit state criteria. This search process is easily programmed with VBA code in Excel.

Basic Section Information

The first section after the results summary includes the computation of the effective flange width for flexural calculations, b_E, and a listing of the relevant section properties to be used in later calculations.

Load Computations

The given loads are "imposed" loads. In other words, these are loads that are in addition to the self weight of the beam and slab. Consequently, it is necessary to re-compute the loads with every change in steel section and slab thickness. This is a common exercise in beam design. The distributed loads are computed as the beam weight plus the pressure loads times the beam's tributary width. The tributary width for load calculations is the beam spacing, not the effective width for flexural calculations, b_E.

Since we are considering both the construction and occupancy conditions, we need to compute the loads at those stages.

Steel Beam Capacity During Construction

The major concern during construction is the flexural capacity of the section without composite action. The loads supported include the beam weight, slab weight, and the construction live loads. In this case, the beam is shored at third points and acts as a continuous beam over three spans, substantially reducing moment and deflection. The beam is also considered to have lateral support at the ends and at the shoring supports. This results in a long laterally unbraced length that must be considered when computing the LTB limit state.

Composite Beam Capacity During Occupancy

After the concrete sets, the member becomes composite and the shoring can be removed. The moment capacity calculation is based on the location of the plastic neutral axis, PNA. As there are three possible conditions for setting up the equilibrium equations for finding the PNA, we solve for all three then select the one that is applicable. This is easy to do in a spreadsheet. Not so easy when doing a hand solution.

Once the PNA is located, then the internal forces and resulting moment capacity are computed. The final result is compared against the demand. The demand at this point includes the beam and slab weights plus the imposed occupancy dead and live loads.

Shear Capacity During Occupancy

As the shear capacity is not affected by the composite action, it need only be considered under the worst case loading, which is during occupancy with the span is longer and the loads larger. It is also interesting to note that there is significant reserved capacity for shear, as is normally the case for longer span beams.

Shear Connectors

The problem statement asks for four different sizes of studs to be considered. The shear stud capacity is primarily a function of stud diameter and concrete strength. The computation in the spreadsheet shows results. Spacings are provided for both single and double rows of studs. The final design is to use one of the computed solutions.

Deflection Calculations

Two deflections are considered in this problem. Total load deflection during construction and live load only deflection during occupancy. The total load deflection must be kept to a minimum during construction or there may be a ponding effect that occurs during the concrete pour that results in a slab that is thicker than designed. During occupancy, the primary concern is the differential deflection caused by transient loads.

As we see, deflections are not an issue for the selected beams.

Summary

Of the various limit state, the controlling limit state is composite flexural strength during occupancy. This is not always the case, but appears to be so in this one.

Also, note that the best solutions are different between ASD and LRFD, though both members are essentially the same weight. The best choice for both is the W16x26 because you get some extra depth (increases strength and stiffness) without any extra weight over the W14x26 that is adequate for LRFD.

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