Assumptions about Distribution
+m ( ∑ MB = 0 Ay (L) – Ph = 0 Ay
By using these approximate reactions. The approximate shean bending moment. And axial force diagrams for the frame can be constructed by considering the equilibrium of it members and joints. The bending moment diagrams for the members of the frame are shown in fig. 12.1 (c)
Assumptions about Distribution of Forces among Members and /or Reactions
Approximate analysis of indeterminate structures is sometimes performed by making assumptions about the distribution of forces among the members and/or reaction of the structures. The number of such assumption required for the analysis of a structure is equal to the degree of indeterminacy of the structure. With each assumption providing an independent equation relating the unknown member forces and/or reaction. The equations base on these assumtions are then solved simultaneously with the equilibrium equations of the structure to determine its approximate reactions and internal forces. For example the portal frame of fig. 12.1(a) can alternatively be analyzed by assuming that the horizontal reactions Ax And Bx are equal : that is , Ax = Bx. By solving this equation simultaneously with the three equilibrium equations of the frame. We obtain the same reactions as previously determined by assuming an inflection point at the midpoint of the girded CD of the frame.
The two types of assumptions described in this sections can either be used individually of they can be combined with each other and/or with other types of assumptions based on the engineering judgment of the structural response to develop methods for approximate analysis of various types of structures. In the rest of this chapter, we focus our attention on the approximate analysis of rectangular building frames.
12.2 ANALYSIS FOR VERTICAL OADS
Recall from section 5.5 that the degree of indeterminacy of a rectangular building frame with fixed supports is equal to three times the number of girders in the frame provided that the frame does not contain any internal hinges or rollers. This in an approximate analysis of such a rigid frame, the total number of assumptions required is equal to three times the number of girders frame.
A commonly used procedure for approximate analysis of rectangular building frames subjected to vertical (gravity) loads involves making three assumtions about the behavior of each girder of the frame. Consider a frame subjected to uniformly distributed loads w,as shown Fig. 12.2 (b). The free body diagram of a typical girder DE of the frame is shown in Fig. 12.2(b). From the deflected shape of the girder sketched in the figure, we observe that two inflection points exits near both ends of the girder. These inflection points develop because the columns and the adjecent girder connected to the ends of girder DE offer partial restraint or resistance againts rotation by exerting negative moments MDE dan MED at the girder ends D dan E, respectively. Although the exact location of the inflection points depends on the relative stiffnesses of the frame members and can be determited only from an exact analysis, we can establish the regions along the girder in which these points are located by examming the two extreme counditions of rotational restraint at the girder ends shown in fig. 12.2(c) and (d). If the girder ends were free to rotate , as in the case of a simply supported girded (Fig.12.2(c)), the zero bending moments and this the inflection points would occur at the ends. On the other extreme, if the girder ends were completely fixed againts rotation, we can show by the exact analysis presented in subsequent chapters that the inflection points would occur at a distance of 0.211L from each end of the girder, as ilustrated in fig. 12.2 (d). Therefore, when the girder ends are only partially restrained against rotation (Fig 12.2(b)), the inflection poit must occur somewhere within a distance of 0.211L from each end. For the purpose of approximate analysis, it is common practice to assume that the inflection points are located about halfway between the two extremes that is, at a distance of 0,1L from each end of the girder. Estimating the location of two inflection points in volves making two assumption about the behavior of the girder. The third assumption is based on the experience gained from the exact analyses of rectangular frames subjected to vertical loads only, which indicates that the axial forces in girders of such frames are usually very small. Thus, in an approximate analysis, it is reasonable to assume that the girder axial forces and zero.
To summarize the foregoing discussion, in the approximate analysis of a rectangular frame subjected to vertical loads the following assumptions are made for each girder of the frame :
1. The inflection points are located at one-tenth of the span from each endof the girder.
2. The girder axial force is zero.
The effect of these simplifying assumptions is that the middle eight-of the span (0,8L) of each girder can be considered to be simply supported on the two ends portions of the girder, each of which is of the length equal to one lenth of the girder span (1,1L), as shown in Fig. 12.2 (e). Note that the girders are now statically determinate and their end forces and moments can be determined from static, as shown in the figure, it should be realized that by making.
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