Deformations and Displacements

When a body is subjected to external loads, internal forces are developed in order to maintain equilibrium between the internal segments. These forces produce stresses which in turn produce strains that cause the body to change its shape and displace from its unloaded position.

Consider the member shown in Fig. 1.25. We apply an axial force which generates the axial stress, s, equal to

where A is the cross-sectional area. The resulting strain depends on E, the modulus of elasticity for the material.

Extensional strain is defined as the change in length

Then,

Strains are generally referred to as deformations since they relate to a change in shape. This example illustrates that displacements are a consequence of deformations which are due to forces. Note that deformations are dimensionless quantities whereas displacements have geometric units such as either length (translation) or angle (rotation). The coefficient of F in (1.19) has units of displacement/ force. We interpret this coefficient as a measure of the flexibility of the member. Here we are defining flexibility as displacement/unit force. The inverse of flexibility is called stiffness. Stiffness relates the force required to introduce a unit displacement. Inverting (1.19) leads to

It follows that the stiffness of an axial loaded member is equal to AE/ L .

Stiffness and flexibility are important concept in Structural Engineering. We use them to reason qualitatively about the change in behavior of a structure when we introduce modifications to the geometry and structural members. Obviously, to reduce displacements, one makes the structure stiffer. How this is achieved is one of the themes of this text.

Structural Behavior; Structural Analysis

When a structure is subjected to an external loading, it responds by developing internal forces which lead to internal stresses. The stresses generate strains, resulting in displacements from the initial unloaded position. Figure 1.26 illustrates the displacement process for a beam type member subjected to a transverse loading. This process is continued until the internal stresses reach a level at which the external loading is equilibrated by the internal forces. The final displacement profile corresponds to this equilibrium state.

Structural analysis is concerned with quantifying the response of structures subjected to external loading. The scope includes determining the magnitude of

the reactions, internal forces, and displacements. The analysis is generally carried out in the following order.

Study Forces

In the study of forces, we apply the equilibrium equations to various Free Body Diagrams. We work initially with the FBD for the structure treated as a single body, and determine the reactions. Once the reactions are known, we select various cutting planes and determine the corresponding internal forces. This phase involves some heuristic knowledge as to “the best” choice of cutting planes.

Study Displacements

Displacements are the geometric measures that define the position of the structure under the applied external loading. Displacements are a consequence of internal stresses and are usually expressed in terms of the internal forces. The form of the “Force-displacement” relations depends on the type of structural member, e.g., a truss member, a beam, etc.