1. Superposition & Statically Indeterminate Beams
Method of Superposition
Statically Indeterminate Beams

2. 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.

3. 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

7. 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

8. 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.

9. 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.
Could have selected the reactive moment at A as the redundant. Then when the moment restraint at A is removed, the released structure is a simple beam with a pin support at one end and a roller support at the other.

10. 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.

11. 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

16. 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.

17. 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)

18. 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.

19. 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.)