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 beam’s 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
Follow the directions at https://confluence.cornell.edu/display/
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.
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