Superposition & Statically
Indeterminate Beams
Method of Superposition
Statically Indeterminate Beams
Method of Superposition
If a beam has several concentrated or distributed
loads on it, it is often easier to compute the slope
and deflection caused by each load separately.
The slope and deflection can then be determined
by applying the principle of superposition and
adding the values of the slope and deflection
corresponding to the various loads.
Method of Superposition
Assumptions
– material obeys Hooke's law
– deflections and slopes are small
– the presence of the deflections does not alter the
actions of the applied load
Statically Indeterminate Beams
Recall, Statically indeterminate beams are ones in which
the number of reactions exceeds the number of
independent equations of equilibrium
Most of the structures we encounter in everyday life,
automobile frames, buildings, aircraft, are statically
indeterminate.
4 unknowns, 3 equilibrium equations
Types of Indeterminate Beams
Usually identified by the beams support system
– Propped cantilever beam
– Fixed-end beam
– Continuous beam
The number of reactions in excess of the number of
equilibrium equations is called the Degree of Static
Indeterminacy
– A propped cantilever beam is statically indeterminate to the
first degree.
Types of Indeterminate Beams
Excess reactions are called static redundants and
must be selected in each particular case.
– In the case of a propped cantilever beam, the support at the
end may be selected as the redundant reaction
– This reaction is in excess of those needed to maintain
equilibrium, so it can be removed.
– Structure that remains when redundants are released is
called the released structure or the primary structure.
Types of Indeterminate Beams
The released structure must be stable and must be
statically determinate.
A special case: all loads action on the beam are vertical
– Horizontal reaction at A vanishes and three reactions remain
– Only two independent equations of equilibrium are available
– Beam is still statically indeterminate to the first degree.
Analysis by the deflection curve
Statically indeterminate beams may be analyzed
by solving any one of the equations of the
deflection curve
Procedure is essentially the same as for
statically determinate beams.
Illustrated by example
Method of Superposition
Fundamental in the analysis of statically indeterminate bars, trusses, beams, frames, and other structures.
First note the degree of static indeterminacy and selecting the redundant reactions
Having identified the redundants, write equations of equilibrium that relate the other unknown reactions to the redundant and the loads.
Method of Superposition
Next, assume both the original loads and the redundants act on the released structure.
– Find the deflections in the released structure by superposing the separate deflections due to the loads and redundants.
– The sum of these deflections must match the deflections in the original beam
– Since the deflections in the original beam at the restraints are 0 or a known value• We can write equations of compatibility (or equations of superposition)
Method of Superposition
The released structure is statically determinate
The relationships between loads and the
deflections of the released structure are called
Force-Displacement relations.
When these relations are substituted into the
equations of compatibility
– Unknowns are the redundants.
Method of Superposition Procedure
Study the boundary
conditions and sketch the
expected deflection curve.
Determine the degree of statical indeterminacy
Select and label redundant
forces and/or moments
Break problem into statically
determinate subproblems
– One for each load on the beam
and one for each of the
selected redundants.
Write compatibility equations– One for the deflection for each redundant force (or moment)
Write force-deflection equations
Substitute force-deflection equations into compatibility equations and solve for unknown redundants.
Write superposition equations for any additional quantities that are required by the problem statement
Complete solution (max deflection, etc.)