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6. Analysis of Portal Frames
The frames are analysed for permanent and imposed load acting vertically, and wind causing horizontal load on the walls and uplift load on the sloping roof. The
moments as a result of wind load are generally in the opposite direction to those caused by the dead and imposed load. Permanent and imposed load is frequently
the critical load case for low-rise structures in South Africa and the rest of the Southern African region. The analysis is carried out using an elastic computer method
for the two loading cases with load factors of 1.2 and 1.6 for dead plus imposed load, and load factors of 0.9 and 1.3 for dead minus wind load, assuming fully rigid joints at the eaves and apex connections. Fully rigid joints were assumed for the eaves and apex joints, based on the moment-rotation curves of these connections (Dundu and Kemp, 2006). These curves did not show significant flexibility in the connections. A comparison of the bending moment diagrams in Figures 7 for the 12 m-frame shows the combination of permanent and imposed to be more critical than that of permanent and wind load. The comparison is possible because lateral restraint is provided to the top and bottom flange at each purlin for the two load cases.
7. Portal Frame Member Design
The rafter and column members are subjected to a bending moment in addition to a compressive load, and therefore behave as a beam-column. Thus the separate
cases of bending and compression has to be satisfied as well as the combined action of axial compression and flexure. The ultimate limit states require that local
buckling be prevented, a cross-sectional strength check, a shear and bending interaction check and a bearing check be carried out. The frames are designed for permanent and imposed load acting vertically, as this is the critical loading case. 7.1. Local buckling The design of the members (buckling resistance) is based on modified section properties, which include only the effective width of compression elements, to control local buckling (Tables 2 and 3). The effective widths of
different plate elements are specified in SANS 10162-2 (2005) and depend on the ratio of the actual flat width of the element to its thickness, the edge support conditions
of the plate element and the actual stress existing in the element. The effective area is calculated for the case of axial compression whilst the effective moment of inertia and section modulus are calculated for the case of
bending. These properties are determined for two cases:
• When the calculated stress (f) in the elements of the channel is equal to the yield strength (fy) and
• When the calculated stress (f) in the elements of the channel is equal to the compressive limit stress (fc) for bending or (fa) for axial compression.
Table 2 shows that there is little or no difference between the moment of inertia and section modulus calculated for the two cases of f=fy and f=fc. This is because these parameters are influenced by the calculated stresses (fy and fc), which are either equal or vary by a small margin. However, there is significant variation of
the calculated stresses and areas in Table 3. Axial compression is therefore more influential in reducing section properties of the channels than bending.
Excluding the material properties of the channels, the major parameters that influence the magnitude of the buckling resistances are the unbraced length of the
member and the moment gradient factor (Cb). In all the three frames investigated, the rafter has a longer lateral lyunsupported length (L) than the column. The rafter and column unbraced length have the same effective length factor (K) and the moment-gradient factor (Cb) of the rafter is larger than the one for the column. An effective length factor of 0.85 for minor axis buckling (assuming a partially restrained member at each end) and moment gradient factors of 1.67 (rafter) and 1.22 (column) are used in all lateral-torsional buckling moment calculations (see Fig. 8). The moment gradient factor for the rafter accounts for the distribution of the bending moment from the point of contraflexure (zero moment) in the rafter to the moment at purlin 1, whilst the one for the column accounts for the distribution of bending moment from the rail position (half the height of the column) to purlin 2.