CELE Structural Theory & Analysis — Indeterminate Structures: Force MethodsRevision Notes

Formulas

Exam Tips

• On board exams, always state DI before starting any solution — it signals to the examiner you understand the problem type.
• For quick DI checks: propped cantilever = 1, two-span continuous beam (simple ends) = 1, fixed-fixed beam = 3, fixed-propped beam = 2.
• If a problem gives EI, it likely requires the flexibility method — this is your cue to apply force methods.

Key Points

• The Degree of Indeterminacy (DI) equals the number of unknown reactions/internal forces minus the number of independent equilibrium equations available.
• For a beam: DI = total reactions − 3 (for 2D structures with no internal hinges). Add 1 for each internal hinge.
• A redundant is any reaction or internal force that can be removed WITHOUT making the structure geometrically unstable — it is the 'extra' unknown.
• Removing all redundants from the original structure produces the PRIMARY (released) structure, which must be statically determinate and stable.
• The number of compatibility (deformation) equations required equals the degree of indeterminacy.
• Common redundants: prop reaction of a propped cantilever, interior support reaction of a continuous beam, one fixed-end moment of a fixed-fixed beam.
• Choice of redundant is not unique — different choices lead to the same final answer but different solution complexity. Choose for easiest deflection calculations.

Definitions

Section Title

1. Degree of Static Indeterminacy and Redundants

Common Mistakes

• Counting reactions incorrectly: a pin gives 2 reactions, a roller gives 1, a fixed support gives 3.
• Releasing a reaction that makes the structure unstable (e.g., releasing the only horizontal support when horizontal loads are present).
• Confusing degree of indeterminacy with degree of kinematic indeterminacy (which is used in the stiffness/displacement method).
• Forgetting that an internal hinge reduces the DI by 1 (each hinge adds one equation: moment = 0 at hinge).

Formulas

Exam Tips

• Memorize the five most common beam deflection formulas: cantilever (UDL and point load at free end), simply supported beam (midspan point load and UDL midspan deflection).
• R_prop = 3wL/8 for propped cantilever under UDL is a direct board-exam answer — recognize the problem type instantly.
• When EI is not given numerically, it cancels in the compatibility equation — no need to substitute actual EI values.
• For settlement problems: δ₀ is downward (positive), δ₁₁ is upward per unit R (positive), Δ is the settlement value — write carefully.
• Draw the primary structure and label all forces clearly before calculating any deflections.

Key Points

• Core principle: the deflection at the point of the redundant in the original structure equals a known value (usually zero for a rigid support, or Δ for a settled support).
• Step 1 — Release: Remove the redundant to obtain the primary structure. Calculate deflection δ₀ at the released point due to APPLIED LOADS ONLY.
• Step 2 — Unit load: Apply unit redundant (R = 1) to the primary structure. Calculate deflection δ₁₁ at the released point.
• Step 3 — Compatibility: Set up the equation δ₀ + R·δ₁₁ = Δ (where Δ = 0 for no settlement, or Δ = support settlement value).
• Step 4 — Solve for R, then use equilibrium to find all remaining reactions and draw shear/moment diagrams.
• For n redundants: write n compatibility equations simultaneously — δᵢ₀ + Σⱼ Rⱼ δᵢⱼ = Δᵢ for each released coordinate i.
• All deflection calculations use the PRIMARY (released) determinate structure — apply standard beam deflection formulas or virtual work.
• Sign convention: define positive direction consistently. Typically, downward deflection is positive. If loads deflect the point downward and redundant pushes it upward, the compatibility equation correctly yields a positive R.

Definitions

Section Title

2. Method of Consistent Deformation

Common Mistakes

• Using the wrong deflection formula — always verify whether the load is on the primary structure (cantilever, simply supported, etc.).
• Sign error: if δ₀ and δ₁₁ act in the same direction, the compatibility equation may give a negative R — this is valid, it means the redundant acts opposite to the assumed direction.
• Forgetting the compatibility equation changes to δ₀ + R·δ₁₁ = Δ (not zero) when support settlement is given.
• Using the original indeterminate structure's deflection formulas instead of the primary structure's formulas.
• Not verifying the answer by checking equilibrium after finding R.

Formulas

Exam Tips

• The two key RHS terms to memorize: UDL → wL³/4; central point load → 3PL²/8. These appear on almost every board exam problem involving the three-moment equation.
• For two-span equal beams with equal UDL and simple ends: M_B = −wL²/8. Write this down immediately when you see this configuration.
• Always draw the continuous beam, label spans L₁ and L₂, identify interior supports, and write the three-moment equation systematically.
• After solving for M_B, compute reactions span by span: treat each span as a simply supported beam with the computed end moments as additional applied couples.
• Always verify reactions: ΣR = total load applied. If the beam has symmetry, exploit it to reduce unknowns.

Key Points

• The three-moment equation is the most efficient method for continuous beams on multiple supports — it is the standard board-exam tool for multi-span beams.
• It relates bending moments at THREE consecutive supports (A, B, C) across spans L₁ (AB) and L₂ (BC).
• For each interior support, write one three-moment equation → n interior supports = n equations for n unknown interior moments.
• Boundary conditions: for simple (pin/roller) end supports, M = 0; for fixed end supports, M ≠ 0 (add a virtual span of zero length).
• The right-hand side (RHS) of the equation uses the 6Ax̄/L terms, computed from the SIMPLE-BEAM moment diagrams of each span loaded independently.
• For equal spans with UDL on all spans, the three-moment equation gives M_interior = −wL²/8 for a two-span beam (classical result).
• After finding all support moments, reactions are determined span-by-span using free body diagrams.
• The method can handle different span lengths, different loads per span, and support settlements.

Definitions

Section Title

3. Three-Moment Equation (Clapeyron's Theorem)

Common Mistakes

• Using 6Ax̄/L = wL³/4 but forgetting it represents the term for ONE span only — add both spans' terms on the RHS.
• Applying the three-moment equation at end supports instead of interior supports only.
• Setting M ≠ 0 at simply supported ends — simple supports have M = 0, always.
• For a fixed end, failing to add a fictitious zero-length span to correctly handle the boundary condition.
• Computing reactions directly from total load without accounting for the effect of support moments on each span.
• Not checking: the sum of all reactions must equal the total applied load on the entire beam.

Formulas

Exam Tips

• Create a quick reference card with all five standard deflection formulas — board exams often require direct substitution.
• The formula δ = Pa²(3L−a)/(6EI) for an interior load on a cantilever is frequently tested; derive it mentally from integration if you forget.
• When EI cancels in the compatibility equation, you do not need the actual EI value — this is intentional in most board exam problems.

Key Points

• In the force method, you MUST know deflection formulas for determinate beams — these are used for both δ₀ and δ₁₁.
• All formulas apply to the PRIMARY (released, determinate) structure, NOT the original indeterminate structure.
• The five most critical formulas: (1) cantilever free-end deflection under UDL, (2) cantilever free-end deflection under point load at free end, (3) cantilever free-end deflection under point load at interior point, (4) simply supported midspan deflection under UDL, (5) simply supported midspan deflection under central point load.
• EI typically cancels in the compatibility equation — keep it symbolic.
• For fixed-fixed or other complex primary structures, use superposition or virtual work (unit load method) to compute deflections.

Definitions

Section Title

4. Standard Beam Deflection Formulas for Force Methods

Common Mistakes

• Using midspan deflection formulas when the actual point of interest is not at midspan.
• Confusing δ₀ (due to applied loads) with δ₁₁ (due to unit redundant) — these must be computed separately.
• Applying cantilever formulas to the primary structure when the primary is actually a simply supported beam (wrong primary structure choice).

Formulas

Exam Tips

• For any propped cantilever exam problem: immediately write R_prop = 3wL/8 (UDL) or R_prop = 5P/16 (midspan point load) if you recognize the configuration.
• For any two-span equal beam with UDL: write M_B = −wL²/8 immediately, then compute reactions.
• Time management: these standard results should be recalled in under 30 seconds on the board exam.

Key Points

• Example 1: Propped cantilever, UDL — direct application of consistent deformation.
• Example 2: Two-span continuous beam, equal spans UDL — direct three-moment equation application.
• Example 3: Propped cantilever, interior point load — demonstrates non-standard δ₀ formula.
• Understanding the full solution process is critical — board exams may ask for reaction, moment, shear, or deflection at specific points.
• Always verify answers using equilibrium checks.

Section Title

5. Worked Board-Exam Problems (Fully Solved)

Common Mistakes

• In Example 2, forgetting that R_B gets contributions from BOTH spans — use R_B = 2wL − R_A − R_C.
• In Example 3, using δ₀ = wL⁴/(8EI) for a point load instead of the correct Pa²(3L−a)/(6EI) formula.
• After finding support moments, not redrawing free body diagrams span by span to correctly compute reactions.