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308 Chapter 5
For an elastic body that is acted upon by external loads, the small-
displacement theory provides the Cauchy strain-displacement relationships
given in Eqs. (1.27) of Chapter 1. However, those equations are just keeping
the linear terms of the following more complete equations:
which are given, for instance, by Boresi, Schmidt and Sidebottom [3]. An
example will be analyzed next in order to better contrast the differences
between the small- and large-displacement theories.
Example 5.18
Consider a fixed-free bar of constant cross-section that is acted upon by
an axial force at its free end. Compare the maximum displacements
corresponding to small- and large-displacement theories. Given are the force
the cross-sectional area, the length of the bar,
and Young’s modulus, E = 160 GPa.
Solution:
The differential equation given by the small-displacement theory is:
which is a linear first-order differential equation whose solution gives the
well-known maximum displacement at the free end:
The first Eq. (5.124) can be expressed for large displacements by means
of the non-linear first-order differential equation:
