Structural Analysis Fundamentals / Statically indeterminate structures
Portal frames, braced beams and two-pinned arches
Three indeterminate structures that each need one idea you already have.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- An independent structural reviewer must check remaining derivations, diagrams, sign conventions and the interactive models across their full input ranges.
- Check full stiffness matrices, member release conventions, arch geometry/stiffness weighting and second-order applicability before design use.
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 notesWhy this matters
Portal frames carry most single-storey industrial buildings, braced beams turn up wherever a beam is propped by something that gives a little, and two-pinned arches are what you build when a crown hinge would be a nuisance. All three are indeterminate, and all three fall to methods already covered — slope-deflection, compatibility and least work respectively.
By the end of this lesson you should be able to
- Analyse a symmetric portal frame by slope-deflection
- Explain what sway is and when it can be ruled out
- Solve a braced beam where the brace is elastic
- Find the thrust in a two-pinned arch and compare it with the three-pinned case
What you should already know
- Slope-deflection and fixed-end moments (this module)
- The force method and compatibility (this module)
- Three-pinned arches (Module 6)
- Strain energy (Module 15)
A portal frame with fixed bases is statically indeterminate to the third degree, which sounds forbidding. Slope-deflection reduces it to a manageable number of unknown displacements — its kinematic indeterminacy, which is 3: the two joint rotations and the sway.
Sway is the horizontal movement of the top of the frame relative to its base. It enters the slope-deflection equation through ψ, the relative transverse displacement divided by the member length. Two situations rule it out:
- The frame and its loading are both symmetric, so there is no reason for it to lean either way.
- The frame is braced against horizontal movement by something else.
When sway can be ruled out, the analysis collapses. For a symmetric portal under a symmetric load, symmetry also makes the two joint rotations equal and opposite, and three unknowns become one.
Worked example
Symmetric portal frame under a UDL
Given
- Symmetric portal: beam span L = 8.00 m, columns height h = 4.00 m
- Both column bases fully fixed; uniform EI throughout
- UDL of 20.0 kN/m over the whole beam
Find
The moments at the top and base of the columns, and at mid-span of the beam.
Assumptions
- Symmetric structure and symmetric loading, so there is no sway
- Axial deformation of the members is neglected
A braced beam is a beam propped by a member that is itself elastic — a strut, a tie, a hanger. It is the propped cantilever with one change: the prop gives a little.
Compatibility now says that the deflection of the beam at the brace equals the shortening or stretching of the brace, rather than zero:
δbeam under load − δbeam under R = R × Lbrace/(Abrace Ebrace)
Rearranged, the brace force is the free deflection divided by the sum of the two flexibilities:
R = δfree / (fbeam + fbrace)
The two limits are worth checking. A rigid brace has zero flexibility, so R = δfree/fbeam — the fully propped tip reaction, 3wL/8 for a UDL. An infinitely flexible brace has infinite flexibility, so R → 0 and the beam is unpropped. Everything real sits in between, and the brace never quite achieves what a rigid prop would.
Predict first
A beam is propped at mid-span by a slender strut. Compared with a rigid prop, what force does the strut carry?
A two-pinned arch has an indeterminate horizontal thrust. Neglecting axial and shear deformation, compatibility gives
H = ∫[M₀y/(EI)] ds ÷ ∫[y²/(EI)] ds
The EI factors cancel only when EI is constant. For a shallow parabolic arch with constant EI and ds approximated by dx, a central point load gives H ≈ 25WL/(128h). The approximation should be checked for a relatively deep arch.
Under a full UDL per unit horizontal span, the bending-only funicular result is H = wL²/(8h). Axial shortening, support movement and other deformations can change the thrust.
Practice
A symmetric portal frame has a beam span of 8.00 m, columns 4.00 m high, fixed bases and uniform EI, carrying a UDL of 20.0 kN/m on the beam. What is the hogging moment at the top of each column, in kNm?
Practice
For the same frame, what is the sagging moment at mid-span of the beam, in kNm?
Practice
A two-pinned parabolic arch spans 30.0 m with a rise of 6.00 m and carries a single central point load of 100 kN. Using the shallow-arch, bending-only approximation H ≈ 25WL/128h, what is the horizontal thrust, in kN?
Summary
- A fixed-base portal is statically indeterminate to degree 3 but kinematically only to degree 3 as well — two rotations and a sway
- Symmetry of both structure and loading rules out sway and halves the remaining work
- For a symmetric portal under a UDL: θB(4/h + 2/L) = wL²/12EI
- A braced beam is a propped beam whose prop deforms: R = δfree/(fbeam + fbrace)
- A rigid brace recovers 3wL/8; an infinitely flexible one carries nothing
- Two-pinned arch, bending only: H = ∫[M₀y/(EI)] ds/∫[y²/(EI)] ds
- Under a full UDL a parabolic arch gives H = wL²/8h whether two- or three-pinned; under a point load they differ