22 2. LET’S GET STARTED
Example 2.1: Simple Truss Analysis (continued)
are u D 6:00 10
4
in and v D 6:74 10
3
in. e deformed shape can be illustrated by
post-processing the finite element results, as shown in Fig. 2.10.
Figure 2.10: e structure deforms and point P displaces as the load is applied. e finite ele-
ment result matches the exact result for nodal displacements and bar forces.
is example is adapted from Papadopoulos et al. [2013] with permission.
2.4 WHAT DIMENSION ARE YOU IN?
e distribution of stress, strain, and displacement in an elastic body subject to prescribed forces
requires consideration of a number of fundamental conditions relating material constitutive laws,
material properties, geometry, and surface forces.
1. e equations of equilibrium must be satisfied throughout the body.
2. A constitutive law relating stress and strain must apply to the material, e. g., linear elastic
Hooke’s law.
2.4. WHAT DIMENSION ARE YOU IN? 23
3. Compatibility must hold, i. e., the components of strain must be compatible with one an-
other or the strain must be consistent with the preservation of body continuity. is is a
critical matter for FEA that is not always discussed in mechanics of materials.
4. e stress, strain, and deformation must be such as to conform to the conditions of loading
imposed at the boundaries.
Realistically, all problems are three-dimensional, but satisfying all the conditions outlined
above can quickly become intractable. Indeed, closed form solutions to three-dimensional bound-
ary value problems in linear elasticity can be very involved or even impossible. When possible,
it is wise to take advantage of simplifications in which the displacement , stress, or strain fields
take on a one- or two-dimensional nature. ese opportunities afford themselves when a lower
dimensional model captures enough of the essential behavior.
For instance, in Example 2.1, we tacitly recommended that the 3-bar structure be modeled
with beam or bar elements. is was natural enough, but to elaborate, we assumed that behaviors
such as lateral contraction of the bars via the Poisson effect, bending, or other stresses not directed
along the axes of the bars were negligible. us, a model that resembles the behavior of a simple
axial bar, and its correspondingly simple behavior as described in Section 2.3.1, is sufficient. It is
unnecessary to develop a ‘true’ three-dimensional model that is more complicated.
In general, when modeling, the metaphor to not ‘throw the baby out with the bathwater’
is apt. e ‘baby’ is that which is essential, i. e., the dominant mechanics that we choose to keep
in the model, such as the dominant axial behavior of the bars in Example 2.1. e ‘bathwater’
is all of the other mechanics that we choose to neglect, such as the lateral effects in the bars of
Example 2.1.
ere are several other important situations in which it is appropriate to simplify the dimen-
sionality of a problem. is is evidenced when we realize that simple beam deflection solutions
resolve only the deformed shape of the neutral axis of the beam cross section. Indeed, in the
simplest beam bending theory that was reviewed in Section 2.3.2, referred to as Euler-Bernoulli
theory, the formulae for axial bending stress and maximum deflection are sufficient in the limit as
the beam length dominates over the remaining two cross-sectional dimensions. In other words,
Euler-Bernoulli beam theory holds only in the limit as the beam becomes “long and slender.” e
simplest bending relations become progressively more insufficient as the cross-sectional dimen-
sions grow and are no longer small compared with the beam’s length. In this limit, one can argue
that the beam becomes ‘hopelessly three-dimensional.’
Other opportunities afford themselves when two dimensions, say in a plane, are either com-
monly large or small compared with an out-of-plane dimension. In this limit, we have been taught
two-dimensional planar solutions for plane stress, plane strain, and axisymmetric conditions. We
explore these situations in the following sections.
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