Structural Determinacy and Stability of Beams, Trusses and Frames

NCEES FE Civil spec: Structural Engineering, D. Structural determinacy and stability analysis of beams, trusses, and frames. FE Reference Handbook: Civil Engineering chapter, Structural Analysis section (Stability, Determinacy, and Classification of Structures).

What the FE Civil Spec Covers

Section D asks you to classify a beam, truss or frame as unstable, stable and statically determinate, or stable and statically indeterminate, and to give the degree of indeterminacy. These are quick counting questions, but they reward care: you must count reactions, members, joints and internal hinges correctly and then check the arrangement for hidden instability.

Key Concepts

Reactions: roller or link = 1, pin = 2, fixed support = 3.
Condition equations c: an internal hinge where n members meet releases moment and adds n − 1 equations. A hinge joining two segments adds 1.
External vs internal: a structure can have extra reactions (externally indeterminate), extra members (internally indeterminate), or both. The total degree is what the FE usually asks for.
Geometric instability: parallel reactions (no horizontal restraint), concurrent reactions (all pass through one point, so no moment resistance), or a panel that can deform without stretching any member.

Core Formulas

Let m = members, j = joints, r = reaction components, c = condition equations.

Plane truss:

m + r < 2 j : unstable; m + r = 2 j : determinate; m + r > 2 j : indeterminate

Plane frame (also works for beams, treating segments between joints as members):

3 m + r < 3 j + c : unstable; 3 m + r = 3 j + c : determinate; 3 m + r > 3 j + c : indeterminate

Degree of indeterminacy:

n_{t r u ss} = (m + r) - 2 j, n_{f r am e} = (3 m + r) - (3 j + c)

Shortcut for beams:

n_{b e am} = r - 3 - c

Spotting Instability Quickly

Before counting, look at the supports. A beam on rollers only has no way to resist horizontal force, so it is unstable no matter how many rollers it has. A body supported by three links whose lines of action meet at a single point can rotate about that point. For trusses, check that every panel is triangulated: a square panel with no diagonal is a four-bar mechanism even if the overall count looks fine, because a member is missing in one place and an extra member sits somewhere else.

A good habit is to build the truss in your head from a base triangle, adding two members and one joint at a time. A truss assembled this way (a simple truss) is internally stable and determinate, and any extra member makes it internally indeterminate. Then check the supports separately: three reactions that are neither parallel nor concurrent give external determinacy.

Why Indeterminacy Matters

Indeterminate structures need compatibility (deflection) conditions in addition to equilibrium, and they are sensitive to support settlement and temperature change, which a determinate structure simply accommodates without internal force. They also offer redundancy: losing one member or support does not necessarily cause collapse.

Where It Is in the FE Reference Handbook

The Civil Engineering chapter, Structural Analysis section, has a short table titled Stability, Determinacy, and Classification of Structures with the plane truss and plane frame criteria. Remember its note that stability also requires a proper arrangement of members and reactions.

Worked Example: Classify Four Structures

1. Truss with 13 members and 8 joints on a pin and a roller

m + r = 13 + 3 = 16 = 2 (8) \Rightarrow determinate (if properly arranged)

2. Two-span continuous beam on a pin and two rollers

n = r - 3 - c = 4 - 3 - 0 = 1 \Rightarrow indeterminate to the first degree

3. Portal frame with fixed column bases

Two columns and one beam: m = 3, j = 4 (two bases, two top corners), r = 3 + 3 = 6, c = 0.

3 m + r = 9 + 6 = 15, 3 j + c = 12 \Rightarrow n = 3

4. Three-hinged portal frame

Pinned bases (r = 4) and an internal hinge at the top of one column joining column and beam (c = 1):

3 m + r = 9 + 4 = 13, 3 j + c = 12 + 1 = 13 \Rightarrow determinate

Without the hinge, the pinned-base portal frame would be indeterminate to the first degree (13 vs 12). Adding one internal release removes exactly one degree of indeterminacy.

Common Mistakes

Counting a fixed support as 2 reactions instead of 3.
Forgetting to count the support points as joints in the frame equation.
Adding only 1 condition for a hinge where three or more members meet (it should be n − 1).
Calling a beam on three rollers determinate. It has no horizontal restraint, so it is unstable.
Trusting the count alone. Always look for parallel or concurrent reactions and mechanisms.

Once you know a structure is indeterminate, see how to solve it in elementary statically indeterminate structures; determinate ones are covered in analysis of determinate structures.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A planar propped cantilever is fixed at one end and supported by a vertical roller at the other. What is its degree of static indeterminacy? See the worked solution or practice the Structural Basics deck
A continuous beam rests on a pin support and two roller supports, forming two spans. Neglecting axial force, the degree of static indeterminacy is: See the worked solution or practice the Structural Basics deck
A planar truss has 12 members and 7 joints. It is supported by a pin at one joint and a roller at another. Neglecting the arrangement of members, the truss is classified as: See the worked solution or practice the Trusses deck
An ideal triangular truss has joints A (0, 0), B (4, 0) and C (2, 3), in meters, with members AB, BC and CA. A has a pin support. At B a roller bears against a vertical wall, so it supplies only a horizontal reaction. Using small-displacement first-order analysis, is the support arrangement externally stable, given that m + r = 3 + 3 = 6 = 2j? See the worked solution or practice the Trusses deck

Related Calculators and Examples

Frequently Asked Questions

How do I find the degree of indeterminacy of a beam?

Subtract the number of equilibrium equations from the number of unknown reactions. A planar beam has 3 equations, plus 1 condition for each internal hinge joining two segments. A propped cantilever has 4 reactions and 3 equations, so it is indeterminate to the first degree.

Can a structure satisfy the counting equation and still be unstable?

Yes. The counting rules are necessary but not sufficient. If all reactions are parallel, all reaction lines pass through one point, or part of a truss forms a mechanism, the structure is unstable even when the count says determinate.

Does the determinacy check depend on the loads?

No. Determinacy and stability depend only on the geometry, members, supports and internal releases. The loads do not change the classification.

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