Rock Mechanics.pdf - Mining and Blasting

Figure 6.8 A simple finite element
structure to illustrate the relation between
nodal connectivity and construction
of the global stiffness matrix.
METHODS OF STRESS ANALYSIS
Introducing equation 6.41,
[q e
] = [B] T [D][B][u e
]dV +
Ve
Ve
[B] T [ 0
]dV −
Ve
[N] T [b]dV (6.42)
Examining each component of the RHS of this expression, for a triangular element
the term
Ve [B]T [D][B] dV yields a 6 × 6 matrix of functions which must be integrated
over the volume of the element, the term
Ve [B]T [0 ]dV a6× 1 matrix, and
Ve [N]T [b]dV a6× 1 matrix. In general, the integrations may be carried out using
standard quadrature theory. For a constant strain triangular element, of volume Ve,
the elements of [B] and [N] are constant over the element, and equation 6.42 becomes
[q e ] = Ve[B] T [D][B][u e ] + Ve[B] T [ 0 ] − Ve[N] T [b]
In all cases, equation 6.42 may be written
[q e ] = [K e ][u e ] + [f e ] (6.43)
In this equation, equivalent internal nodal forces [q e ] are related to nodal displacements
[u e ] through the element stiffness matrix [K e ] and an initial internal load vector
[f e ]. The elements of [K e ] and [f e ] can be calculated directly from the element geometry,
the initial state of stress and the body forces.
6.6.4 Solution for nodal displacements
The computational implementation of the finite element method involves a set of
routines which generate the stiffness matrix [K e ] and initial load vector [f e ] for all
elements. These data, and applied external loads and boundary conditions, provide
sufficient information to determine the nodal displacements for the complete element
assembly. The procedure is illustrated, for simplicity, by reference to the two-element
assembly shown in Figure 6.8.
Suppose the applied external forces at the nodes are defined by
188
[r] T = [rx1 ry1 rx2 ry2 rx3 ry3 rx4 ry4]