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3.2. THE LINES IN THE SAND 37
dimensional simulations are required. Because multi-dimensional analysis is required to capture
stress concentrations, it is also required in numerical design considerations of how to alleviate
such stress risers. For instance, in the case of the stepped shaft, one might ask the question “Is
there any way to alleviate the stress concentration at the ﬁllet without changing the diameter on
either side or increasing the radius of the ﬁllet?” A three-dimensional analysis reveals that this
is actually possible by undercutting the larger diameter portion of the shaft in the vicinity of the
original step, as shown in Fig. 3.4.
Figure 3.4: It is possible to alleviate a stress riser without changing either diameter of a stepped shaft.
A multi-dimensional ﬁnite element analysis is required to capture these phenomena. is solution
is reproduced from [Papadopoulos et al., 2011] with permission, with particular credit due to Jim
3.2.2 A SHORT, STUBBY BEAM
Euler-Bernoulli beam theory, as introduced in strength of materials courses, accounts for trans-
verse deﬂection due to bending only. Bending deﬂections can be said to dominate the deforma-
tion response when the span-to-depth ratio of the beam exceeds, say, 15. For progressively shorter
beams, the assumption that shear deformation can be neglected when compared with the bend-
ing deformation is no longer warranted. In these limits, the shear deformation should be taken
into account. Timoshenko beam theory accounts for explicit contributions of deformation due to
38 3. WHERE WE BEGIN TO GO WRONG
shear. is is the theory one should apply when the span-to-depth ratio of the beam falls below
some prescribed limit.
SimCafe Tutorial 3: Stress and Deﬂection in a Timoshenko Beam
e purpose of this tutorial is to showcase where simple beam theory begins to break
down. In some commercial codes, simple one-dimensional cubic beam elements that capture
bending deﬂection do not capture shear deﬂection. Alternatively, Timoshenko beam theory
may be used by default in the element formulation (as with the BEAM188 element in ANSYS
v14). When shear deﬂection is accounted for in the one-dimensional element formulation,
results for the beams tip deﬂection will not agree with tip deﬂections predicted by simple
Euler-Bernoulli beam theory when the beam is relatively short. Again, attempts to capture
this eﬀect with h-convergence will ultimately fail when the necessary physics is not contained
in the element formulation. When it is and the results are compared to simpler theory, the
disagreement may be substantial. Once again, h-convergence captures no more of the solu-
tion than does a coarser discretization. is tutorial is meant to highlight when it is relatively
straightforward to apply three-dimensional FEA and resolve a solution correctly that belies
analytical treatment with simple formulae (such as bending tip deﬂection v D PL
3
=3EI ).
SIMULATION/Stubby+Beam to complete the tutorial.
Example 3.2: Large Depth-to-Span Ratio Beams
Consider a relatively short tip-loaded cantilevered I-beam, as shown in Fig. 3.5.
x
y
P
L
2c
Figure 3.5: A simple cantilever beam is loaded under transverse point tip load P .
e behavior of relatively short beams can be numerically approximated by either one-
dimensional beam elements that account for shear deﬂection or a fully three-dimensional
analysis. One should note, however, that while one-dimensional Timoshenko beam ele-
ments have interpolation functions for shear deformation, they do not capture the complete
three-dimensional state of stress within the beam. For instance, in short cantilever beams the
I
3.2. THE LINES IN THE SAND 39
Example 3.2: Large Depth-to-Span Ratio Beams (continued)
normal stress component at the clamped edge can no longer be predicted with the simple
bending formula in Chapter 2.
JO
 JO
JO
 JO
 JO
Figure 3.6: Cross section of a short I-beam and a corresponding three-dimensional solid model
that can be imported into many commercial ﬁnite element software packages.
e solid model is meshed for an I-beam whose span is 24 in. With a span-to-depth
ratio of only 3, the actual deformation and stress response will not be modeled well by Euler-
Bernoulli beam theory. ree-dimensional ﬁnite element simulations indicate that the shear
deﬂections are on the order of those from simple bending theory and the wall normal stresses
deviate substantially from those predicted by simple bending theory. Typical contours of
displacement and stress for the three-dimensional model are shown in Figs. 3.7 and 3.8 for
a tip load of 1000 lb.
I
40 3. WHERE WE BEGIN TO GO WRONG
Example 3.2: Large Depth-to-Span Ratio Beams (continued)
Figure 3.7: Both shear and bending contribute to the total transverse deformation of short
beams.
Figure 3.8: e axial stress at the ﬁxed wall deviates substantially from that predicted by one-
dimensional beam theory.

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