Formulae for Rigid Frames - Steel Structures.

General
The formulae given in this section are based on Professor Kleinlogel’s Rahmen-formeln  and Mehrstielige Rahmen.

The formulae are applicable to frames which are symmetrical about a central vertical axis, and in which each member has constant second moment of area.

Formulae are given for the following types of frame:

Frame I Hingeless rectangular portal frame.
Frame II Two-hinged rectangular portal frame.
Frame III Hingeless gable frame with vertical legs.
Frame IV Two-hinged gable frame with vertical legs.

The loadings are so arranged that dead, snow and wind loads may be reproduced on all the frames. For example, wind suction acting normal to the sloping rafters of a building may be divided into horizontal and vertical components, for which appropriate formulae are given, although all the signs must be reversed because the loadings shown in the tables act inwards, not outwards as in the case of suction.

It should be noted that, with few exceptions, the loads between node or panel points are uniformly distributed over the whole member. It is appreciated that it is normal practice to impose loads on frames through purlins, siderails or beams. By using the coefficients in Fig. 11.1, however, allowance can be made for many other symmetrically placed loads on the cross-beams of frames I and II shown, where the difference in effect is sufficient to warrant the corrections being made. The inde- terminate BMs in the whole frame are calculated as though the loads were uniformly distributed over the beam being considered, and then all  are adjusted by multiplying by the appropriate coefficient in Fig. 11.1. It may be of interest to state why these adjustments are made. In any statically indeterminate structure the inde- terminate moments vary directly with the value of the following quantity:


Where the loaded member is of constant cross section, EI may be ignored.

Conversion coefficients for symmetrical loads
Fig. 11.1 Conversion coefficients for symmetrical loads

Consider, as an example, the case of an encastré beam of constant cross section and of length L carrying a UDL of W. Then the area of the free BM diagram is


If, however, W is a central point load, the area of the free BM diagram is

The fixed end moments (FEM) due to the two types of loadings are WL/12 and WL/8 respectively, thus demonstrating that the indeterminate moments vary with the area of the free BM diagram and proving that the indeterminate moments are in the proportion of 1 : 1.5.

No rules can be laid down for the effect on the reactions of a change in the mode of application of the load, although sometimes they will vary with the indeterminate moments. Consider a simple rectangular portal with hinged feet. If a UDL placed over the whole of the beam is replaced by a central point load of the same magnitude, then the knee moments will increase by 50% with a corresponding increase in the horizontal thrusts H, while the vertical reactions V will remain the same.

Although the foregoing remarks relating to the indeterminate moments resulting from symmetrical loads apply to all rectangular portals, the rule applies for asym- metrical loads imposed upon the cross-beam of a rectangular portal frame with hinged feet. If a vertical UDL on the cross-beam is replaced by any vertical load of the same magnitude, then the indeterminate moments vary with the areas of the respective free BM diagram.

No doubt readers who use the tables frequently will learn short cuts, but it is not inappropriate to mention some. For example, if a UDL of W over the whole of a single-bay symmetrical frame is replaced by a UDL of the same magnitude of W over either the left-hand or right-hand half of the frame, the horizontal thrust at the feet is unaltered. If the frame has a pitched roof then the ridge moment will also be unaltered.

The charts in the Appendix have been prepared to assist in the design of rectangular frames or frames with a roof pitch of 1 in 5.

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