Superposition for Indeterminate Beams

Addition of fraction and likewise subtraction Requireth that first they all have like bases Which by reduction is brought to perfection And being once done as ought in like cases, Then add or subtract their tops and no more Subscribing the base made common before.  -- Thomas Hylles, The Art of Vulgar Arithmetic, 1600

 

solution technique

1. determine the degree of indeterminacy
2. select the redundant reactions
3. write equilibrium equations
   • these will relate the other unknown reactions to the redundant reactions and loads
4. release the structure
5. write force-displacement equations
   • these are the beam deflection equations
6. write compatibility equations
   • these will require that the deflections in the released beam match those in the original beam
7. substitute the force-displacement equations into the compatibility equations to determine the redundant reactions
8. all other reactions can now be solved using the equilibrium equations

propped cantilever

replace roller with force and set deflection equal to zero there

or

replace wall with pin and moment and set slope equal to zero there

 

beam on three supports

replace a pin or roller with force and set deflection equal to zero there

 

elastic supports

replace elastic support with force and set deflection equal to δ = NL/EA (watch signs)