Engineers classify structures as statically determinate or statically indeterminate to decide what analysis tools are needed. A statically determinate structure can be solved using only the equations of static equilibrium. A statically indeterminate structure has more unknown reactions or internal forces than equilibrium equations can determine.
This distinction matters because it affects safety checks, material efficiency, deflection prediction, and how a structure responds when supports settle or members change temperature.
For a plane structure, equilibrium gives three independent equations: sum Fx = 0, sum Fy = 0, and sum M = 0. If the number of unknown reactions and internal force unknowns matches the number of useful equilibrium equations, the structure is determinate. If there are extra unknowns, engineers must also use compatibility conditions that describe how the structure deforms.
Indeterminate structures are often stronger and stiffer, but their analysis requires material properties, geometry, and deformation relationships such as stress strain laws and beam deflection formulas.
Key Facts
- For a 2D rigid body in static equilibrium: sum Fx = 0, sum Fy = 0, and sum M = 0.
- A plane simply supported beam with one pin and one roller has 3 reaction unknowns and is usually statically determinate.
- A fixed ended beam has 6 reaction unknowns in 2D and is statically indeterminate to degree 3.
- Degree of external indeterminacy for a plane structure can be estimated by DI = r - 3 for one rigid body, where r is the number of external reaction components.
- For plane trusses, a common determinacy check is m + r = 2j, where m is members, r is reactions, and j is joints.
- Indeterminate analysis requires both equilibrium and compatibility, such as deformation consistency at supports and joints.
Vocabulary
- Statically determinate
- A structure is statically determinate when all support reactions and internal forces can be found from equilibrium equations alone.
- Statically indeterminate
- A structure is statically indeterminate when there are more unknown forces or reactions than can be solved using equilibrium equations alone.
- Support reaction
- A support reaction is a force or moment supplied by a support to prevent a structure from moving in a constrained direction.
- Compatibility
- Compatibility is the requirement that structural deformations fit the support and connection constraints without gaps or impossible motion.
- Degree of indeterminacy
- Degree of indeterminacy is the number of extra unknown force quantities beyond those that can be found by equilibrium alone.
Common Mistakes to Avoid
- Counting supports instead of reaction components is wrong because a pin gives two reaction components, a roller gives one, and a fixed support gives three in 2D.
- Assuming every beam with three reactions is determinate is wrong because internal hinges, multiple spans, and connection details can change the number of useful equilibrium equations.
- Ignoring compatibility in indeterminate structures is wrong because equilibrium alone cannot determine how redundant reactions split the load.
- Treating indeterminate structures as always safer is wrong because extra restraint can create large forces from settlement, temperature change, or fabrication errors.
Practice Questions
- 1 A 2D beam has a pin support at A and a roller support at B. It carries a 12 kN downward point load at midspan. How many reaction components are unknown, and is the beam statically determinate?
- 2 A fixed ended beam in 2D has a fixed support at each end. Count the reaction components and calculate its external degree of indeterminacy using DI = r - 3.
- 3 A continuous beam passes over three supports instead of two. Explain why equilibrium alone is not enough to determine all support reactions, and state what additional idea is needed.