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since the stub is welded to the bottom chord, a portion of its flexural stiffness should be added to that of the moment of inertia of the wide-flange shape [4,5,7]. This approach is identical to treating the bottom chord W-shape as if it has a cover plate on its top flange. The area of this ''cover plate'' is the same as the area of the bottom flange of the stub. This should be done only in the areas where the stubs are placed. In the regions of the interior and exterior stubs it is therefore realistic to increase the moment of inertia of the bottom chord by the parallel-axis value of Af x d2, where Af designates the area of the bottom flange of the stub and df is the distance between the centroids of the flange plate and the W-shape.

The bending stiffness of the top Vierendeel chord equals that of the effective width portion of the slab. This should include the contributions of the steel deck as well as the reinforcing steel bars that are located within this width. In particular, the influence of the deck is important. The effective width is determined from the criteria in the AISC LRFD Specification [3]. These were originally developed on the basis of analyses and tests of prismatic composite beams. The approach has been found to give conservative results [4,5], but should continue to be used until more accurate criteria are available.

In the computations for the slab, the cross-section is conveniently subdivided into simple geometric shapes. The individual areas and moments of inertia are determined on the basis of the usual transformation from concrete to steel, using the modular ratio n _ E/Ec, where E is the modulus of elasticity of the steel and Ec is that of concrete. The latter must reflect the density of the concrete that is used, and can be computed from [13]

The shear connectors used for the stub are required to develop 100% interaction, since the design is based on the computed shear forces, rather than the axial capacity of the steel beam or the concrete slab, as is used for prismatic beams in the AISC Specification [3]. However, it is neither common nor proper to add the moment of inertia contribution of the top flange of the stub to that of the slab, contrary to what is done for the bottom chord. The reason for this is that dissimilar materials are joined and some local concrete cracking and/or crushing can be expected to take place around the shear connectors.

The discretization of the stubs into vertical Vierendeel girder components is relatively straightforward. Considering the web of the stub and any stiffeners, if applicable (for exterior stubs, most commonly, since interior stubs usually can be left unstiffened), the moment of inertia about an axis that is perpendicular to the plane of the web is calculated. As an example, Figure 33.6 shows the stub and stiffener configuration for a typical case. The stub is a 5 ft long W16 x 26 with 5j x 2 in. end-plate stiffeners. The computations give

Moment of inertia about the z-z axis:

Izz _ [0.25 x(60)3]/12 + 2 x 5.5 x 0.5 x(30)2 _ 9450 in.4

Depending on the number of Vierendeel truss members that will represent the stub in the model, the bending stiffness of each is taken as a fraction of the value of Iz. For the girder shown in Figure 33.5, where the stub is discretized as three vertical members, the magnitude of Ivert is found as

Moment of inertia of vertical member:

Ivert _ Izz/(no. of verticals) _ 9450/3 _ 3150 in.4

The cross-sectional area of the stub, including the stiffeners, is similarly divided between the verticals

Area of vertical member:

_ [0.25 x(60 - 2 x 0.5)+ 2 x 5.5 x 0.5]/3 6.75 in.2

Stiffener

Web of W16 x 26 stub

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