1. Fundamental Concepts

1.5: Small Rigid Body Displacements

It is frequently important to visualize a structure as it undergoes small rigid body displacements. We are using the term ‘rigid body’ in a slightly unusual sense here that really refers to articulated bodies. Individual parts of a structure remain rigid, but relative displacements may be allowed between the parts.

Rigid Body Displacements

Rigid body simply means that we assume members are infinitely stiff – they neither bend, nor shear, nor change in length. On the other hand, compatible deformations are allowed at special conditions: relative rotations are allowed at internal articulations such as hinges, and supports are allowed to move in any direction in which they are not constrained. For example, a beam end connected to a pin support is allowed to rotate, and one connected to a roller support is allowed to rotate and translate perpendicularly to the direction of constraint.

Small Rotations

Fig. 5-1: Small Rotation

The characteristics of small rotations are illustrated in Fig. 5-1. Let straight beam element a-b rotate about point a a small amount $d\theta$. Point b travels along the straight line (not an arc) that is perpendicular to the radius from a to b (in its original location). Point b ends up at point b’ along that line. The rotated member is now the line a-b’ and its length is still r, unchanged from its original length.

In the limit, as $d\theta \rightarrow 0$, this is valid and without error.

Fig. 5-2: Small Rotation + Translation

When a line member undergoes small rigid body translations and a rotation, such as that shown in Fig. 5-2, the length of the projection of the line on an axis parallel to the original line does not change.

Fig. 5-3: Resulting Displacements

Fig. 5-3 shows the displacements that result when an inclined line undergoes a small rotation about one end – point a.

Point b moves along the perpendicular to b’. Its horizontal movement is equal to its vertical distance from a times the angle change and its vertical movement is equal to the horizontal distance from a times the angle change.

Systems of Small Displacements

Finally, Fig. 5-4 shows all of these concepts together on one structure.

Fig. 5-4: Small Rigid Structural Displacements

Articulated frame a-b-c-d has an internal hinge at b. The section b-c-d is rigid; let it rotate clockwise a small amount $\theta$ about the pin at point d - this is our starting point.

• The rigid joint c translates horizontally (perpendicular to c-d) a distance $4\theta$, and rotates clockwise an amount $\theta$.

• The portion b-c-d rigidly retains its shape. The right angle at c is maintained.

• Point b then translates horizontally a distance $4\theta$ (the same as c) and vertically a distance $3\theta$ (its horizontal distance from the centre of rotation is 3).

• Point a translates horizontally a distance of $4\theta$, as does every point on the straight line a-b-c.

• Line a-b must rotate ${3\over 2}\theta = 1.5\theta$ in order to have the correct vertical displacement of $3\theta$ at b.

• The projection of a-b-c on the horizontal does not change in length during the displacement.