Engineering Rock Mechanics

Foundation instability 301

would be one of many upper bound solutions to the actual collapse load.
The geometry of the assumed plastic wedges would then require variation
in an attempt to produce increasingly lower values of collapse load, with
the result that each one would be closer to the actual plastic collapse load.
An alternative approach which is more concise and less prone to error
is to apply the concept of virtual work, allowing the equilibrium to be
established by considering a small amount of work done by the forces
involved. For example, in Fig. 17.15, we show three forces acting at a point.
Considering the imposition of an imaginary displacement of magnitude u
in the direction shown in Fig. 17.15, then

work done by force = (force magnitude) x (component of displacement in

direction of force)

and

virtual work = Z (work done by all forces).

The magnitude of the virtual work will be zero if the system is in
equilibrium-because the work done by the resultant force (which is zero
for a system in equilibrium) must be zero. For the forces shown in Fig. 17.15,
the inset table gives the calculation of the virtual work.
The application of the concept of virtual work to a more complex
foundation problem is illustrated in Fig. 17.16. Although this is intended
to represent a system of discrete blocks formed by discontinuties, it may
also be regarded as a refinement to the upper bound plastic problem shown
in Fig. 17.14. In this case the angle of friction is non-zero.
As a first stage in the analysis, the directions of the virtual displacements
associated with the forces arising from the strength of the discontinuities
are drawn on the diagram. These directions, shown by the vectors vl, vz,
v,, v12 and ~23 in Fig. 17.16(a), are drawn inclined at an angle 4, the angle
of friction, to the discontinuity. This results in each virtual displacement
being orthogonal to the resultant force on each discontinuity. To evaluate
the compatibility relations between the various virtual displacements, the
polygon of displacements shown in Fig. 17.16(b) is constructed. This is
initiated by assuming a unit magnitude for the virtual displacement VI, and

Force Angle with line Component of u in Work done

of virtual direction of force

Unit virtual displacement, u

displacement, u

I

/ / FI = 20.0 69.5" cos 69.5 = 0.3502 7.00

Fl = 20

F2 = 20.0 20.5" cos 20.5 = 0.9367 18.73

F3 = 36.4 F3= 36.4 - 135" COS -135 = -0.7071 -25.73

24p4450

24.5"

F2= 20

Sum of virtual 0.00

work components

Figure 17.15 The principle of virtual work applied to the analysis of equilibrium.