II II II II II I - Waste Isolation Pilot Plant - U.S. Department of Energy

The second integral in the preceding equation is used to define the element internal force vector fiI as
3uiI ‘iI = ~Ae ‘ij aui,j ‘A “
(3.1 .9)
The first and third integrals define the external force vector, and the fourth integral defines the inertial response.
We perform one-point integration by neglecting the nonlinear portion of the element displacement field, thereby
considering a state of uniform strain and stress. The preceding expression is approximated by
(3.1.10)
—
where we have eliminated the arbitrary virtual displacements, and Tij represents the assumed uniform stress tensor.
By neglecting the nonlinear displacements, we have assumed that the mean stresses depend only on the mean strains.
Mean kinematic quantities are defined by integrating over the element as follows:
(3.1.11)
We now define the discrete gradient operator as
‘iI = ~A$I,i ‘A “
(3.1.12)
The mean velocity gradient, applying Equation (3. 1.5), is given by
(3.1.13)
Combining Equations (3. 1.10) and (3. 1.12), we may express the nodal forces by
fil = ~ij Bjl . (3.1.14)
Computing nodal forces with this integration scheme requires evaluation of the gradient operator and the
element area. These two tasks are linked since
‘M
= Sij (3.1.15)
where ~ij is the Kroneker delta. Equations (3.1.1), (3.1.12), and (3.1. 15) yield
XiI BjI = ~V(XiI $I),j dA =A~ij . (3.1.16)
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