Module 16 · Lesson 16.1
Indeterminacy, compatibility and the force method
Where the missing equations come from when statics runs out.
Why this matters
Almost every real structure is statically indeterminate. Continuous beams, portal frames, fixed-ended members, most bridge decks — none of them can be solved by equilibrium alone. The extra equations have to come from somewhere else, and where they come from is the shape the structure is forced to take.
By the end of this lesson you should be able to
- Count the degree of static indeterminacy of a beam or frame
- State a compatibility condition in words before writing any algebra
- Solve a propped cantilever by releasing the redundant
- Explain why indeterminate structures respond to settlement and temperature
What you should already know
- Equilibrium and support reactions (Module 2)
- Determinacy and stability (Module 4)
- Standard beam deflection formulae (Module 13)
A planar structure gives you three equations of equilibrium: ΣFx = 0, ΣFy = 0 and ΣM = 0. If the unknown reactions and internal forces number three, statics settles the matter. If there are more than three, it cannot.
The degree of static indeterminacy is simply the excess:
degree = (number of unknowns) − (number of independent equilibrium equations)
A simply supported beam has three reaction components and three equations: degree 0, determinate. Add a prop at mid-span and you have four unknowns and still three equations: degree 1. Build both ends in and you have six unknowns: degree 3.
The extra unknowns are called redundants — an unfortunate name, since they are anything but useless. What they are redundant for is equilibrium: the structure would stand up without them. What they provide is stiffness, alternative load paths and robustness.
Predict first
One support of a beam settles by 10 mm. In which case does this produce bending moments in the beam?
So where do the missing equations come from? From compatibility — statements about how the structure must fit together, which are true regardless of the forces.
The force method (also called the flexibility or consistent-deformation method) turns that into a procedure:
- 1.Choose a redundant and release it, producing a determinate structure you can solve.
- 2.Analyse that released structure under the real loads. Where the redundant used to be, there is now a violation of compatibility — a gap, an overlap, a rotation that should not be there.
- 3.Analyse the released structure under the redundant alone, as an unknown.
- 4.Write the compatibility condition: the two effects must cancel. Solve for the redundant.
- 5.Superpose to get everything else.
Step 4 is where the extra equation appears. It is not an equilibrium statement — it is a geometric one.
From first principles
The propped cantilever, from compatibility
We want to show: that the prop of a propped cantilever under a full UDL carries 3wL/8.
Imagine the prop is not there yet. The cantilever sags, and its free end drops by some amount — call it δ₁. Now slide the prop into place. It is a rigid support, so it will not let the beam sit that low; it has to push the end back up until the deflection is exactly zero. So the prop force is whatever force it takes to push the tip of a cantilever back up by δ₁. That single sentence is the entire analysis. Everything after this is arithmetic on two standard deflection formulae you already know. The compatibility condition — 'the deflection at the prop is zero' — is the extra equation that equilibrium could not supply.
Worked example
A propped cantilever with numbers
Given
- Propped cantilever, span L = 6.00 m
- Built in at the left-hand end, propped on a roller at the right
- Uniformly distributed load w = 24.0 kN/m over the whole span
- Uniform EI
Find
The prop reaction, the wall reaction and the fixing moment.
Practice
A beam is built in at one end and rests on a roller at the other. Counting reaction components against the equations of equilibrium, what is its degree of static indeterminacy?
Practice
A propped cantilever spans 6.00 m and carries a uniformly distributed load of 24.0 kN/m over its whole length. What is the prop reaction, in kN?
Practice
For the same propped cantilever, what is the fixing moment at the built-in end, in kNm?
Summary
- Degree of indeterminacy = unknowns − equilibrium equations
- The missing equations are compatibility statements about geometry
- Force method: release, find the incompatibility, restore it with the redundant
- Propped cantilever with a UDL: R = 3wL/8, Mwall = wL²/8
- Indeterminate structures develop forces from settlement, temperature and lack of fit
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint