The Force Method

1. 1Find the degree of indeterminacy, D_s = r − 3 externally, or the truss equivalent.
2. 2Choose redundants and remove them, leaving a stable released structure that is determinate.
3. 3Compute Δ₀, the deflection at each released point under the real loads.
4. 4Compute δᵢⱼ, the deflection at i caused by a *unit* redundant at j, with the real loads removed.
5. 5Impose compatibility: Δ₀ + Σ δᵢⱼ·Rⱼ = 0, and solve for the redundants.
6. 6Superpose, and finish with ordinary statics.

Both deflection calculations are on a *determinate* structure, which is why the unit load method had to come first. Nothing in the method is new — it is the earlier topics arranged to supply the equation statics could not.

The choice is free provided the released structure remains stable — releasing the fixed support of a propped cantilever would leave a mechanism, which is not permitted.

Different valid choices give different arithmetic and the same final answer. A good choice makes the released structure something whose deflections you already know, such as a simple cantilever or a simply supported beam.

For one redundant: Δ₀ + R·δ₁₁ = 0, so R = −Δ₀/δ₁₁. For a propped cantilever under a uniform load this gives R = 3wL/8.

Note that EI cancels for a uniform member, so the redundant depends on geometry alone. It reappears the moment stiffness varies along the structure — which is why a haunched beam cannot be solved from a standard table.

With n redundants this becomes [δ]{R} + {Δ} = {0}. The flexibility matrix is symmetric by Maxwell's reciprocal theorem, which halves the coefficients you must compute and provides a check on the rest.

• Support settlement. Instead of setting the deflection to zero, set it to the known settlement. Everything else is unchanged.
• Temperature and lack of fit. These contribute to Δ₀ through the unit load method, exactly as before.
• Prestress. A deliberate lack of fit, introduced to produce a chosen force distribution.

This is where indeterminate structures earn their reputation: settlement and temperature induce real stresses with no load applied at all, because the structure has nowhere to move to.

The force method usually has fewer unknowns, which suited hand calculation. But it requires judgement in choosing a release, and a poor choice makes the arithmetic much worse — judgement a computer does not have. The stiffness method is entirely mechanical, so it scales to a hundred thousand unknowns, which is why every analysis program is built on it.