Transcript Eng. 6002 Ship Structures 1 Hull Girder Response Analysis Lecture 9: Review of Indeterminate Beams.

Eng. 6002 Ship Structures 1
Hull Girder Response Analysis
Lecture 9: Review of
Indeterminate Beams
Overview
The internal forces in indeterminate
structures cannot be obtained by statics
alone.
 This is most easily understood by
considering a similar statically determinate
structure and then adding extra supports
 This way also suggests a general
technique for analyzing elastic statically
indeterminate structures
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Statically Indeterminate Beams
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A uniformly loaded beam
is shown with three
simple supports.
If there had been only
two supports the beam
would have been
statically determinate
So we imagine the same
beam with one of the
supports removed and
replaced by some
unknown force X
Statically Indeterminate Beams
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If the center support were removed the beam
would sag as illustrated
The sag at the centre is counteracted by the
reaction force X1, providing an upward
displacement
Note: the subscript 0 is used to denote
displacements generated by the original external
load on the statically determinate structure
Statically Indeterminate Beams
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In the original statically
indeterminate structure
there is no vertical
displacement of the
centre due to the support
Thus, the force X1 must
have a magnitude that
exactly counteracts the
sag of the beam without
the centre support
Statically Indeterminate Beams
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There are two approaches for solving
indeterminate systems. Both approaches use
the principle of superposition, by dividing the
problem into two simpler problems,
soling the simpler problems and adding the two
solutions.
The first method is called the Force Method (also
called the Flexibility Method).
The idea for the force method is;
Statically Indeterminate Beams
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The idea for the force method is:
Step 1. Reduce the structure to a statically determinate
structure. This step allows the structure to displace where
it was formerly fixed.
Step 2. solve each determinate system, to find all
reactions and deflections. Note all incompatible deflections
Step 3. re-solve the determinate structures with only a set
of self-balancing internal unit forces at removed reactions.
This solves the system for the internal or external forces
removed in 1.
Step 4. scale the unit forces to cause the opposite of the
incompatible deflections
Step 5. Add solutions (everything: loads, reactions,
deflections…) from 2 and 4.