### Structural Elements in Series: Assessment of the Effect of Stiffness Ratios by Kotur Raghavan

**Structural Elements
in Series, Influence of Stiffness Ratios**

In a recent article mechanics of
structural elements which are either in series or in parallel was discussed ( https://www.fembestpractices.com/2020/10/load-transfer-in-series-and-parallel.html
). Components which are in series transfer the same load to one another. On the
other hand, components placed in parallel share loads coming from a common
contiguous component. In the current article, numerical assessment of the
influence of stiffness ratios of structural elements in series is carried out
with the help of a simple model.

The case study model consists of
a uniformly loaded beam supported at the ends on linear springs. The spring stiffness is a parameter in this
and is non-dimensionalized with respect to the stiffness of uniformly loaded
simply supported beam. The effects of non-dimensional stiffness, referred to as
SR (stiffness ratio) on static deflections, stresses and natural frequencies
are discussed.

**Static Deflections:**

Deflection patterns for two
representative stiffness ratios are shown in the figure below.

Fig. 2

Elastic deformation is seen to be
more pronounced for the higher value of stiffness ratio as is to be expected.
Maximum beam deflections, non-dimensionalized with respect to the deflection of
simply supported beam, are presented in the table below.

Table 1

When the stiffness ratio is 1000,
the spring influences the beam stiffness to the extent of only 0.8 per cent.
Thus, as an engineering approximation, series components having stiffness ratio
of 1000 or more can be deemed to be ‘very stiff’ or rigid.

**Stresses:**

The beam and the springs are in
series and hence the entire applied load is passing through the beam. Therefore
stresses in the beam are not expected to be different for different values of
stiffness ratios.

Fig. 3

Bending stresses for all the
cases were computed to be in the range of -750 to 750 MPa. In fact the relative
deflection of the centre of the beam with respect to the ends was uniformly
6.28 mm in all the cases.

Natural Frequencies

Comparative evaluations were
carried out for the first three natural frequencies. Representative mode shapes
are shown in the figure below.

Fig. 4

Computed values of natural
frequencies are presented in the table below. Here again the frequencies are
non-dimensionalized with respect to those of a simply supported beam.

Table 2

The results are identical to
those presented for static deflections. Stiffness ratios of 1000 or more are of
same significance.

**Key Takeaways**

In any complex assembly components which have relative stiffness of 1000 or more can replaced by displacement constraint. If it happens to be in the middle of the load path it can be replaced by ‘rigid region’ simulation using constraint equations or MPCs.

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