If there are two redundant supports, then the beam is indeterminate to the second
degree. If there are three redundant supports, then it is a third degree indeterminate
beam (and so forth).
For example, the beam at the left could be considered a cantilever beam with
two redundant supports. This is an indeterminate beam to the second degree. To
solve the problem, release two redundant supports. It does not matter which supports
are released, but there are generally better ones than others. The key is to
have a basic beam that is listed in the appendix with beam equations.
The beam without the redundant supports can be solved for the deflection at
each support location. Usually, beam equation tables are used, similar to the
ones listed in the
Beam Equation appendix. But integration method could also be used.
Next, one at a time, place the two unknown reaction forces
on the cantilever beam. In this case, R_{B} and R_{C} are unknown
reactions. Each unknown reaction causes a deflection at both support locations.
The final step (and the key to solving indeterminate beams) is to apply the
compatibility equation(s). There must be one compatibility equation per redundant
support. In this case, there are two, giving
v_{B-load} = v_{B-B} + v_{C-B}
v_{C-load} = v_{B-C} + v_{C-C}
where v_{B-B} is the deflection at B due to the unknown reaction at
B and v_{C-B} is the deflection at B due to the unknown reaction
at C. Similarly, v_{C-C} is the deflection at C due to the unknown
reaction at C and v_{B-C} is the deflection at C due to the unknown
reaction at B. All these terms will have one of the two unknowns,
R_{B} and R_{C}, in there expressions.
The deflection due to all loads are v_{B-load} and v_{C-load}.
There will be two equations, and two unknowns. For each indeterminate degree,
there will be one equation and one unknown. |