Page 184 - Buried Pipe Design

158 Chapter Three

modulus of elasticity increases by a factor of 2 or 3. This would indi-
cate that a soil will have very little lateral deformation with changing
vertical stress if the soil has seen a stress condition greater than the
existing stresses.
The behavior of the soil parameters on primary loading and unload-
ing was investigated using triaxial soil tests at Utah State University.
The results of this testing program indicate that the bulk modulus
behavior is very unpredictable on loading and reloading. It is difficult
to make any definite observations on the behavior of the bulk modu-
lus. However, the elastic modulus exponent, in some cases, is depen-
dent on stress history. Consequently, the soil model has been modified
to use both the unloading elastic modulus constant and the unloading
elastic modulus exponent.

Magnitude of unloading modulus constant. As mentioned, Duncan et al. 6
recommend that the unloading modulus constant be approximately
1.2 times higher than the primary loading constant for stiff soils and
3.0 times higher for soft soils. These approximate factors appear to
work relatively well, in view of the results of the triaxial testing pro-
gram. In fact, the modulus constant has been as much as 4 times higher
on unloading than on primary loading. This leads to the phenomena of
small or even negative values of Poisson’s ratio.

Construction of the stiffness matrix. The stiffness matrix is composed of
several parts. In the isoparametric soil elements that are used, the
stiffness matrix is recomputed at every iteration. One component is a
constitutive matrix relating stress to strain through the elasticity para-
meters. Another component relates element strains to nodal displace-
ments through the strain-displacement matrix. This matrix is computed
based on element types, shape functions, and nodal coordinates. It is not
within the scope of this book to derive the above-mentioned relation-
ships. The intent is merely to describe how the global stiffness matrix
is computed during the analysis.
Beam, rod, and soil elements have their own particular stiffness matri-
ces. A beam element can transmit axial and transverse forces and a bend-
ing moment, and a rod element can only transmit axial forces. Both beam
and rod elements are called one-dimensional elements. For these ele-
ments, the strain-displacement matrix is derived based on the appropri-
ate shape functions and their cross-sectional area, length, and angle of
inclination of the element. A soil element is a two-dimensional element. It
does not transmit moments at the nodes. The strain-displacement matrix
is derived using the x and y coordinates of each node that comprise the
element and the shape functions that are used to describe the deforma-
tion characteristics of the soil elements.