## P

The design procedure follows the criteria of the AISC LRFD Specification [3]. The applicable resistance factor is given for the case of gross cross-section yielding. This is because the preliminary design is primarily needed to find the bottom chord size and this component is primarily loaded in tension [4-6,11]. The load factors are given by the ASCE 7 load standard [12] for the combination of dead plus live load.

The load computations follow the choice of the layout of the floor framing plan, whereby girder and floor beam spans are determined. This gives the tributary areas that are needed to calculate the dead and live loads. The load intensities are governed by local building code requirements or by the American Society of Civil Engineers (ASCE) recommendations, in the absence of a local code.

Reduced live loads should be used wherever possible. This is especially advantageous for stub girder floor systems, since spans and tributary areas tend to be large. ASCE 7 [12] makes use of a live load reduction factor, RF, which is significantly simpler to use and also less conservative than that of earlier codes. The standard places some restrictions on the value of RF, to the effect that the reduced live load cannot be less than 50% of the nominal value for structural members that support only one floor. Similarly, it cannot be less than 40% of the nominal live load if two or more floors are involved.

Proceeding with the preliminary design, the stub girder and its floor beam locations determine the magnitudes of the concentrated loads that are to be applied at each of the latter locations. The following illustrative example demonstrates the steps of the solution.

Figure 33.4 shows the layout of the stub girder for which the preliminary sizes are needed. Other computations have already given the sizes of the floor beam, the slab, and the steel deck. The span of the girder is 40 ft, the distance between adjacent girders is 30 ft, and the floor beams are located at the quarter points. The steel grade remains to be chosen; the concrete is lightweight, with wc = 120 pcf and a compressive strength of fc = 4000 psi.

Loads:

Estimated dead load: 74 psf Nominal live load: 50 psf

Live load reduction factor:

Reduced live load:

Load factors (for D + L combination):

For dead load: 1.2 For live load: 1.6

Factored distributed loads:

Dead load, DL = 74 x 1.2 = 88.8 psf Live load, LL = 30 x 1.6 = 48.0 psf Total = 136.8 psf

Concentrated factored load at each floor beam location: Due to the locations of the floor beams and the spacing of the stub girders, the magnitude of each load, P, is

Maximum factored midspan moment: The girder is symmetric about midspan and the maximum moment therefore occurs at this location

Estimated interior moment arm for full stub-girder cross-section at midspan (refer to Figure 33.2 for typical details): The interior moment arm, that is, the distance between the compressive stress resultant in the concrete slab and the tensile stress resultant in the bottom chord is set equal to the distance between the slab centroid and the bottom chord (wide flange shape) centroid. This is simplified and conservative. In the example, the distance is estimated as

This is based on having a 14-series W-shape for the bottom chord, W16 floor beams and stubs, a 3 in. high steel deck, and 34 in. of lightweight concrete over the top of the steel deck ribs (this allows the deck to be used without having sprayed-on fire protective material to the underside). These are common sizes for the stub-girder system.

In general, the interior moment arm varies between 24.5 and 29.5 in., depending on the heights of the bottom chord, floor beams/stubs, steel deck, and concrete slab.

Slab and bottom chord axial forces, F (these are the compressive and tensile stress resultants) F _ Mmax/d = (820 x 12)/27.5 = 357.9 kip

Required cross-sectional area of bottom chord, As: The required cross-sectional area of the bottom chord can now be found. Since the chord is loaded in tension, the j-value is 0.9.

It is also important to note that in the Vierendeel analysis that is commonly used in the final evaluation of the stub girder, the member forces will be somewhat larger than those determined through the simplified preliminary procedure. It is therefore recommended that an allowance of some magnitude be given for the Vierendeel action. This is done most easily by increasing the area As by a certain percentage. Based on experience [6,11], an increase of one third is suitable, and such has been done in the computations that follow.

On the basis of the data that have been developed, the required area of the bottom chord is

Mmax = 1.5 x P x 20 - P x 10 = 820 kft which gives As-values for 36 and 50 ksi steel of

Conclusions: If 36 ksi steel is chosen for the bottom chord of the stub girder, the wide-flange shapes W12 x 50 and W14 x 53 will be suitable. If 50 ksi steel is the choice, the sections may be W12 x 40 or W14 x 38.

The final decision is made by the structural engineer. However, since W12-series shapes will save approximately 2 in. in net floor system height, per story of the building, this would mean significant savings if the overall structure is 10 to 15 stories or more. The differences in stub girder strength and stiffness are not likely to play a role [7,12].

### 33.3.3 Choice of Stub Girder Component Sizes

Some examples were given in the earlier sections for the choices of chord and floor beam sizes, deck height, and slab configuration. These were made primarily on the basis of acceptable geometries, deck size, and fire protection requirements, to mention some examples. However, construction economy is critical and the following guidelines will assist the user. The data that are given are based on actual construction projects.

Economical span lengths for the stub girder range from 30 to 50 ft, although the preferable spans are 35 to 45 ft. Fifty-feet span girders are erectable, but these are close to the limit where the dead load becomes excessive, which has the effect of making the slab govern the overall design. This is usually not an economical solution. Spans shorter than 30 ft are known to have been used successfully; however, this depends on the load level and the type of structure, to mention the key considerations.

Depending on the type and configuration of steel deck that has been selected, the floor beam spacing should generally be maintained between 8 and 12 ft, although larger values have been used. The decisive factor is the ability of the deck to span the distance between the floor beams.

The performance of the stub girder is not particularly sensitive to the stub lengths that are used, as long as these are kept within reasonable limits. In this context, it is important to observe that it is usually the exterior stub that controls the behavior of the stub girder. As a practical guideline, the exterior stubs are normally 5 to 7 ft long; the interior stubs are considerably shorter, normally around 3 ft, but components up to 5 ft long are known to have been used. When the stub lengths are chosen, it is necessary to bear in mind the actual purpose of the stubs and how they carry the loads on the stub girder. That is, the stubs are loaded primarily in shear, which explains why the interior stubs can be kept so much shorter than the exterior ones.

The shear connectors that are welded to the top flange of the stub, the stub web stiffeners, and the welds between the bottom flange of the stub and the top flange of the bottom chord are crucial to the function of the stub girder system. For example, the first application of stub girders utilized fitted stiffeners at the ends and sometimes at midlength of all of the stubs. Subsequent research demonstrated that the midlength stiffener did not perform any useful function and that only the exterior stubs needed stiffeners in order to provide the requisite web stability and shear capacity [5,6]. Regardless of the span of the girder, it was found that the interior stubs could be left unstiffened, even when they were made as short as 3 ft [6].

Similar savings were realized for the welds and the shear connectors. In particular, in lieu of all-around fillet welds for the connection between the stub and the bottom chord, the studies showed that a significantly smaller amount of welding was needed, and often only in the vicinity of the stub ends. However, specific weld details must be based on appropriate analyses of the stub, considering overturning, weld capacity at the tension end of the stub, and adequate ability to transfer shear from the slab to the bottom chord.

### 33.3.4 Modeling of the Stub Girder

The original work of Colaco [1,2] utilized a Vierendeel modeling scheme for the stub girder to arrive at a set of stress resultants, which in turn were used to size the various components. Elastic finite element analyses were performed for some of the girders that had been tested, mostly to examine local stress distributions and the correlation between test and theory. However, the finite element solution is not a practical design tool.

Other studies have examined approaches such as nonprismatic beam analysis and variations of the finite element method [5]. The nonprismatic beam solution is relatively simple to apply. On the other hand, it is not as accurate as the Vierendeel approach, since it tends to overlook some important local effects and overstates the service load deflections [4,5].

On the whole, therefore, the Vierendeel modeling of the stub girder has been found to give the most accurate and consistent results, and the correlation with test results is good [1,4,5,7]. Finally, it offers the best physical similarity with actual girders; many designers have found this to be an important advantage.

There are no ''simple'' methods of analysis that can be used to find the bending moments, shear forces, and axial forces in Vierendeel girders. Once the preliminary sizing has been accomplished, a computer solution is required for the girder. In general, all that is required for the Vierendeel evaluation is a two-dimensional plane frame program for elastic structural analysis. This gives moments, shears, and axial forces, as well as deflections, joint rotations, and other displacement characteristics. The stress resultants are used to size the girder and its elements and connections; the displacements reflect the serviceability of the stub girder.

Once the stress resultants are known, the detailed design of the stub girder can proceed. A final run-through of the girder model should then be done, using the components that were chosen, to ascertain that the performance and strength are sufficient in all respects. Under normal circumstances no alterations are necessary at this stage.

As an illustration of the Vierendeel modeling of a stub girder, the girder itself is shown in Figure 33.5a and the Vierendeel model in Figure 33.5b. The girder is the same as the one used for the preliminary design example. It has four stubs and is symmetrical about midspan; therefore, only one half is illustrated. The boundary conditions are shown in Figure 33.5b.

The bottom chord of the model is assigned a moment of inertia equal to the major axis I-value, Ix, of the wide-flange shape that was chosen in the preliminary design. However, some analysts believe that

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