Civil Engineering Reference
In-Depth Information
2.9 Summary
1. Forces and stresses in any body of material must be in equilibrium: this means that
the polygon of forces acting on the body, or on an element inside the body closes.
2.
Strains and displacements in any distorting body must be compatible: this means
that the material does not tear or overlap and the displacement diagram closes.
3.
States of stress or strain at a point can be analysed using the Mohr circle con-
struction so that the stresses or strains on any plane can be calculated from the
geometry of the circle.
4.
If slip surfaces develop, their directions correspond to the directions of zero exten-
sion lines and the relative movement across a slip surface is at an angle
ψ
to its
direction.
Worked examples
Example 2.1: Equilibrium of forces using a force polygon
Figure 2.11(a) shows forces
acting on a rigid triangular block of soil with a slip surface; two of the forces are
known to be
W
60 kN. Figure 2.11(b) shows the corresponding
polygon of forces. Scaling from the diagram, or by calculation,
P
=
160 kN and
T
=
=
75 kN.
Figure 2.11
Example 2.2: Compatible displacements using a displacement diagram
Figure 2.12(a)
shows two rigid blocks separated by slip surfaces where all the angles are 45
◦
or
90
◦
; the left-hand block moves with a vertical component of displacement 1 mm as
Figure 2.12
