Page 148 - Handbook of Civil Engineering Calculations, Second Edition

STRUCTURAL STEEL DESIGN 1.131

FIGURE 32

THEOREM OF VIRTUAL DISPLACEMENTS

In Fig. 32a, point P is displaced along a virtual (infinitesimally small) circular arc PP
centered at O and having a central angle . Derive expressions for the horizontal and ver-
tical displacement of P in terms of the given data. (These expressions are applied later in
analyzing a gable frame by the mechanism method.)

Calculation Procedure:
1. Construct the displacement diagram
In Fig. 32b, let r h length of horizontal projection of OP; r v length of vertical projec-
tion of OP; h horizontal displacement of P; v vertical displacement of P.
In Fig. 32c, construct the displacement diagram. Since PP
is infinitesimally small, re-
place this circular arc with the straight line PP that is tangent to the arc at P and therefore
normal to radius OP.
2. Evaluate h and v , considering only absolute values
Since is infinitesimally small, set PP r ; h PP sin r sin ; v PP cos
r cos . But r sin r v and r cos r h ; therefore, h r v and v r h .