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

Papadopoulos.

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 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

3

=3EI ).

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.

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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