Classification by static stability criteria

An initial step that is important and significantly useful for establishing the correct equilibrium equations for the entire structure or any part of the structure (resulting from the sectioning) is to sketch the free body diagram (FBD). The free body diagram simply means the diagram showing the configuration of the structure or part of the structure under consideration and all forces and moments acting on it. If the supports are involved, they must be removed and replaced by corresponding support reactions, likewise, if the part of the structure resulting from the sectioning is considered, all the internal forces appearing along the cut must be included in the FBD. Figure 1.34(b) shows the FBD of the entire structure shown in Figure 1.34(a) and Figure 1.34(c) shows the FBD of two parts of the same structure resulting from the sectioning at a point B.

In particular, the fixed support at A and the roller support at C are removed and then replaced by the support reactions {RAX, RAY, RAM, RCY}. For the FBD shown in Figure 1.34(c), the internal forces {FB, VB, MB} are included at the point B of both the FBDs.

1.8 Classification of Structures

Idealized structures can be categorized into various classes depending primarily on criteria used for classification; for instance, they can be categorized based on their geometry into one-dimensional, two-dimensional, and three-dimensional structures or they can be categorized based on the dominant behavior of constituting members into truss, beam, arch, and frame structures, etc. In this section, we present the classification of structures based upon the following three well-known criteria: static stability, static indeterminacy, and kinematical indeterminacy. Knowledge of the structural type is useful and helpful in the selection of appropriate structural analysis techniques.

1.8.1 Classification by static stability criteria

Static stability refers to the ability of the structure to maintain its function (no collapse occurs at the entire structure and at any of its parts) while resisting external actions. Using this criteria, idealized structures can be divided into several classes as follows.

1.8.1.1 Statically stable structures

A statically stable structure is a structure that can resist any actions (or applied loads) without loss of stability. Loss of stability means the mechanism or the rigid body displacement (rigid translation

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and rigid rotation) develops on the entire structure or any of its parts. To maintain static stability, the structure must be properly constrained by a sufficient number of supports to prevent all possible rigid body displacements. In addition, members constituting the structure must be arranged properly to prevent the development of mechanics within any part of the structure or, in the other word, to provide sufficient internal constraints. All “desirable” idealized structures considered in the static structural analysis must fall into this category. Examples of statically stable structures are shown in Figures 1.3, 1.5 and 1.14-1.16.

Figure 1.34: (a) A plane frame subjected to external loads, (b) FBD of the entire structure, and (c) FBD of two parts of the structure resulting from sectioning at B.

1.8.1.2 Statically unstable structures

A statically unstable structure is a structure that the mechanism or the rigid body displacement develops on the entire structure or any of its parts when subjected to applied loads. Loss of stability in this type of structures may be due to i) an insufficient number of supports as shown in Figure 1.35(a), ii) inappropriate directions of constraints as shown in Figure 1.35(b), iii) inappropriate

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arrangement of member as shown in Figure 1.35(c), and iv) too many internal releases such as hinges as shown in Figure 1.35(d). This class of structures can be divided into three sub-classes based on how the rigid body displacement develops.

1.8.1.2.1 Externally, statically unstable structures

An externally, statically unstable structure is a statically unstable structure that the mechanism or the rigid body displacement develops only on the entire structure when subjected to applied loads.

Loss of stability of this type structure is due to an insufficient number of supports provided or an insufficient number of constraint directions. Examples of externally, statically unstable structures are shown in Figure 1.35(a) and 1.35(b).

1.8.1.2.2 Internally, statically unstable structures

An internally, statically unstable structure is a statically unstable structure that the mechanism or the rigid body displacement develops only on a certain part of the structure when subjected to applied loads. Loss of stability of this type of structure is due to inappropriate arrangement of member and too many internal releases. Examples of internally, statically unstable structures are shown in Figure 1.35(c) and 1.35(d).

Figure 1.35: Schematics of statically unstable structures 1.8.1.2.3 Mixed, statically unstable structures

A mixed, statically unstable structure is a statically unstable structure that the mechanism or the rigid body displacement can develop on both the entire structure and any part of the structure when subjected to applied loads. Examples of mixed, statically unstable structures are shown in Figure 1.36.

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Figure 1.36: Schematics of mixed, statically unstable structures

Figure 1.37 clearly demonstrates the classification of idealized structures based upon the static stability criteria.

Figure 1.37: Diagram indicating classification of structures by static stability criteria

Development of rigid body displacement?

Statically stable structures Statically unstable

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Rigid body displacement of entire structure?

Mixed statically unstable structures

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Internally statically unstable structures

Externally statically unstable structures Idealized structures

Rigid body displacement of part of structure?

Copyright © 2011 J. Rungamornrat