Civil Engineering Reference
In-Depth Information
3
7
3
7
Y
6
10
4
8
6
2
2
4
1
5
9
X
(b)
(a)
Figure 3.1
(a) Two-dimensional truss composed of ten elements. (b) Truss joint connecting five
elements.
2.
The physical characteristics (in this case, the stiffness matrix and element
force) of each element must be transformed, mathematically, to the global
coordinate system to represent the structural properties in the global system
in a consistent mathematical frame of reference.
3.
The individual element parameters of concern (for the bar element, axial
stress) are determined after solution of the problem in the global coordinate
system by transformation of results back to the element reference frame
(postprocessing).
Why are we basing the formulation on displacements? Generally, a design
engineer is more interested in the stress to which each truss member is subjected,
to compare the stress value to a known material property, such as the yield
strength of the material. Comparison of computed stress values to material prop-
erties may lead to changes in material or geometric properties of individual ele-
ments (in the case of the bar element, the cross-sectional area). The answer to the
question lies in the nature of physical problems. It is much easier to predict the
loading (forces and moments) to which a structure is subjected than the deflec-
tions of such a structure. If the external loads are specified, the relations between
loads and displacements are formulated in terms of the stiffness matrix and we
solve for displacements. Back-substitution of displacements into individual ele-
ment equations then gives us the strains and stresses in each element as desired.
This is the
stiffness
method and is used exclusively in this text. In the alternate
procedure, known as the
flexibility
method [1], displacements are taken as the
known quantities and the problem is formulated such that the forces (more gen-
erally, the stress components) are the unknown variables. Similar discussion ap-
plies to nonstructural problems. In a heat transfer situation, the engineer is most
often interested in the rate of heat flow into, or out of, a particular device. While
temperature is certainly of concern, temperature is not the primary variable of
interest. Nevertheless, heat transfer problems are generally formulated such that
temperature is the primary dependent variable and heat flow is a secondary,
computed variable in analogy with strain and stress in structural problems.
Returning to consideration of Figure 3.1b, where multiple elements are con-
nected at a global node, the geometry of the connection determines the relations
