Structural Analysis Fundamentals / module 16
Statically indeterminate structures
When equilibrium alone is not enough: compatibility, the force method, slope-deflection, moment distribution and the matrix stiffness method.
What you will understand
- Recognise indeterminacy and count its degree
- Use compatibility to supply the equations equilibrium cannot
- Analyse a propped cantilever by the force (flexibility) method
- Use fixed-end moments and the slope-deflection equations
- Distribute moments at a joint using stiffness and carry-over
- Explain how the matrix stiffness method assembles and solves a structure
Lessons
Indeterminacy, compatibility and the force method
Where the missing equations come from when statics runs out.
Fixed-ended beams and fixed-end moments
The building blocks every hand method for frames is assembled from.
The slope-deflection method
Write every end moment in terms of the joint rotations, then solve for them.
Moment distribution and the matrix stiffness method
Balancing joints by hand, and the matrix method that replaced it.
Kinematic indeterminacy and indeterminate trusses
Counting the other kind of unknown, and using the force method on a truss.
Portal frames, braced beams and two-pinned arches
Three indeterminate structures that each need one idea you already have.
Module summary
- A structure is indeterminate when it has more unknowns than equilibrium equations
- The extra equations come from compatibility — statements about how the structure must fit together
- Force method: release redundants, calculate the incompatibility, restore it with the redundant forces
- Propped cantilever with a UDL: prop reaction 3wL/8, fixed moment wL²/8
- Fixed-ended beam with a UDL: end moments wL²/12, mid-span wL²/24
- Slope-deflection expresses end moments in terms of rotations and sway
- Moment distribution balances joints iteratively using distribution factors and carry-over
- Stiffness matrices for every element are assembled into one global matrix and solved — the basis of every analysis program
- Indeterminate structures have alternative load paths and are sensitive to support movement and temperature
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Practice
A propped cantilever of span 8.00 m carries a UDL of 15.0 kN/m. What is the prop reaction, in kN?
Practice
A fixed-ended beam of span 5.00 m carries a UDL of 30.0 kN/m. What is the magnitude of the end moment, in kNm?
Check yourself
In the force method, what kind of equation supplies the information that equilibrium cannot?
Practice
Three members meet at a rigid joint with stiffnesses 2.0, 3.0 and 5.0 (same units). What is the distribution factor for the stiffest member?
Check yourself
Why does a support settlement produce bending moments in an indeterminate beam but not in a determinate one?