A History of Research and a Review of Recent Developments

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Response, safety and evolution
calculation of strains and shear forces in the component a multi-degree of
freedom analysis needed to be used.
The single degree of freedom representation of structures was used in most
of the design handbooks for hardened military structures produced by agencies
in the USA in the period between 1960 and 1980. It was also used in the
assessment of the effects of accidental explosions on civilian structures. The
elasto-plastic responses of beams and plates in flexure are represented in this
way in the publications of the US Army and Air Force given in references
[9.6], [9.7] and [9.8], and in a related publication by Baker et al. [9.9]. Readers
wishing to read more of the background should consult a paper by Baker and
Spivey [9.10], who take the example of collapse stages in a simple beam with
clamped ends under dynamic flexural loading. Elastic linear response during
the early stages of loading changes to a second phase linear response after the
formation of plastic hinges at the end of the beam. This phase continues until
the formation of a central plastic hinge results in collapse. This ‘trilinear’
response and its equivalent ‘bilinear’ response is shown in Figure 9.1, taken
from reference [9.10]. The equivalent spring constant, K e, is R m/(y el) e, and the
equivalent system reaches its ultimate resistance at the equivalent limit of
elasticity (y el) e. A similar representation is also made for structures such as
rectangular plates with clamped edges under flexural loading, which have a
four stage or ‘quadrilinear’ response. The transformation factors, formulae
for maximum resistance and spring constants for various stages of deformation,
have been established for rectangular plates using a variety of boundary
Figure 9.1 The equivalent bilinear response for a clamped beam (from Baker and
Spivey, ref. 9.10).