# ANALYTICAL MODELS

An analytical model is a simplified representation, or an ideal, of a real structure for the purpose of analysis. The objective of the model is to simplify the analysis of a complicated structure. The analytical model represents, as accurately as practically possible, the behavioral characteristics of the structure of interest to the analyst, while discarding much of the detail about the members, connections, and so on, that is expected to have little e¤ect on the desired characteristics. Establishment of the analytical model is one of the most important steps of the analysis process; it requires experience and knowledge of design practices in addition to a thorough understanding of the behavior of structures. Remember that the structural response predicted from the analysis of the model is valid only to the extent that the model represents the actual structure.
Development of the analytical model generally involves consideration of the following factors.

# Line Diagram

The analytical model of the two- or three-dimensional body selected for analysis is represented by a line diagram. On this diagram, each member of the structure is represented by a line coinciding with its centroidal axis. The dimensions of the members and the size of the connections are not shown on the diagram. The line diagrams of the bridge truss of Fig. 1.13(a), and the rigid frame of Fig. 1.14(a) are shown in Figs. 1.13(b) and 1.14(b), respectively. Note that two lines ( * * ) are sometimes used in this text to represent members on the line diagrams. This is done, when necessary, for clarity of presentation; in such cases, the distance between the lines does not represent the member depth.

# Connection

Two types of connections are commonly used to join members of structures: (1) rigid connections and (2) flexible, or hinged, connections. (A third type of connection, termed a semirigid connection, although recognized by structural steel design codes, is not commonly used in practice and, therefore, is not considered in this text.)

A rigid connection or joint prevents relative translations and rotations of the member ends connected to it; that is, all member ends connected to a rigid joint have the same translation and rotation. In other words, the original angles between the members intersecting at a rigid joint are maintained after the structure has deformed under the action of loads. Such joints are, therefore, capable of transmitting forces as well as moments between the connected members. Rigid joints are usually represented by points at the intersections of members on the line diagram of the structure, as shown in Fig. 1.14(b). A hinged connection or joint prevents only relative translations of member ends connected to it; that is, all member ends connected to a hinged joint have the same translation but may have di¤erent rotations. Such joints are thus capable of transmitting forces but not moments between the connected members. Hinged joints are usually depicted by small circles at the intersections of members on the line diagram of the structure, as shown in Fig. 1.13(b).

The perfectly rigid connections and the perfectly flexible frictionless hinges used in the analysis are merely idealizations of the actual connections, which are seldom perfectly rigid or perfectly flexible (see Fig. 1.13(c)). However, actual bolted or welded connections are purposely designed to behave like the idealized cases. For example, the connections of trusses are designed with the centroidal axes of the members concurrent at a point, as shown in Fig. 1.13(c), to avoid eccentricities that may cause bending of members. For such cases, the analysis based on the idealized connections and supports (described in the following paragraph) generally yields satisfactory results.

# Supports

Supports for plane structures are commonly idealized as either fixed supports, which do not allow any movement; hinged supports, which can prevent translation but permit rotation; or roller, or link, supports, which can prevent translation in only one direction. A more detailed description of the characteristics of these supports is presented in Chapter 3. The symbols commonly used to represent roller and hinged supports on line diagrams are shown in Fig. 1.13(b), and the symbol for fixed supports is depicted in Fig. 1.14(b).

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