Rock Mechanics.pdf - Mining and Blasting

Figure 6.7 Development of a finite
element model of a continuum problem,
and specification of element geometry
and loading for a constant
strain, triangular finite element.
METHODS OF STRESS ANALYSIS
Thus, since strain components are defined uniquely in terms of various derivatives
of the displacements, the imposed displacement variation defines the state of strain
throughout an element. These induced strains and the elastic properties of the medium
together determine the induced stresses in an element. Superposition of the initial and
the induced stresses yields total stresses in the element.
The assumption in the finite element method is that transmission of internal forces
between the edges of adjacent elements can be represented by interactions at the
nodes of the elements. It is therefore necessary to establish expressions for nodal
forces which are statically equivalent to the forces acting between elements along
the respective edges. Thus the procedure seeks to analyse the continuum problem
(Figure 6.7a) in terms of sets of nodal forces and displacements for the discretised
domain (Figure 6.7b). The solution procedure described here, for purposes of illustration,
considers triangular element geometry, linear variation of displacement
with respect to element intrinsic co-ordinates, and resultant constant stress within an
element.
6.6.1 Displacement variation
In Figure 6.7c, induced nodal displacements are uxi, u yi, etc., and displacements
[u] at any point within the element are to be obtained by suitable interpolation from
the nodal values. Introducing a matrix of interpolation functions, [N], a suitable
interpolation formula is
ux
[u] = = ⎡ ⎤
ui
[Ni][ui] = [Ni, N j, Nk] ⎣ u j ⎦ (6.36)
where
184
u y
[ui] =
= [N][u e ]
uxi
u yi
Ni 0
[Ni] = Ni =
0 Ni
uk