Handbook of Civil Engineering Calculations

(c) Displocement diogrom

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 6. 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. 326, let rh = length of horizontal projection of OP; rv = length of vertical projec-
   tion of OP; A^ = horizontal displacement of P; Ay = 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 Ah and Av, considering only absolute values
   Since 6 is infinitesimally small, set PP" = rd; A^ = PP" sin a = rd sin a; \ = PP" cos a =
   r9 cos a. But r sin a = rv and r cos a = rh; therefore, AA = rv 6 and A^ = rhd.

(Q) Virtual displacement (b) Projected lengths

of point P of OP