CELE Structural Theory & Analysis — Indeterminate Structures: Displacement MethodsSummary

The number of unknown reactions or internal forces minus the number of independent equilibrium equations. For a planar structure: DI = (R + C) − 3n, where R = number of reactions, C = internal releases (hinges, etc.), and n = number of rigid bodies. DI also equals the number of redundants needed to fully solve the structure. A structure with DI = 0 is determinate (solved by equilibrium); DI > 0 is indeterminate; DI < 0 is unstable.

The determinate structure obtained by removing (releasing) all redundants. Removing a redundant means replacing it with a cut, roller, hinge, or simply deleting the force. The primary structure must be stable—you cannot remove a support that would make the structure unstable. The primary structure is used to compute deflections under the applied loads and under unit values of each redundant.

The deflection at a point due to a unit load (or unit moment) applied at that same point, or the cross-deflection due to a unit load at another point. δᵢⱼ is the deflection at location i due to a unit force at location j. For a beam element, δ = ∫(MM̄)/(EI) dx, where M is the moment diagram from the applied loading and M̄ is the moment diagram from the unit load. Flexibility coefficients are always positive; sign conventions for compatibility are handled explicitly.

The fundamental principle that deflections in the actual (indeterminate) structure must physically match the support conditions. If a support is fixed or simply supported, the actual deflection there is zero (or a known settlement Δ). The equation is: δ₀ + R·δ₁₁ = Δ, where δ₀ is the deflection at the released point caused by applied loads alone, R is the redundant reaction, δ₁₁ is the flexibility coefficient, and Δ is the support displacement (usually 0 for no settlement). For multiple redundants, this becomes a system of simultaneous linear equations.

For a continuous beam, relates the bending moments at three consecutive supports A, B, C over spans of lengths L₁ (span AB) and L₂ (span BC): M_A·L₁ + 2M_B·(L₁ + L₂) + M_C·L₂ = −6(A₁·x̄₁/L₁ + A₂·x̄₂/L₂). The right-hand side depends on the load on each span: A is the area of the simple-beam moment diagram, and x̄ is the distance from the support to the centroid of that area. For a UDL w: the load term per span = wL³/4. For a central point load P: the load term = 3PL²/8. Simply supported ends have M = 0; fixed ends have M ≠ 0 (unknown).

A redundant reaction is an extra support force (e.g., the prop of a propped cantilever, or an interior support reaction of a continuous beam). An internal redundant is a force or moment introduced by a cut or hinge (e.g., the shear force or moment at an interior cut). Both are treated identically in the force method: release them to get the primary structure, compute flexibility coefficients, and apply compatibility. The choice of which redundants to select is arbitrary but should lead to a simple primary structure.

A cantilever fixed at one end and supported by a roller (or reaction) at the free end. This is a DI = 1 structure. For a UDL w over span L: prop reaction R_prop = 3wL/8, fixed-end moment M_fix = wL²/8. These formulas are derived by setting δ₀ = wL⁴/(8EI) (free-end deflection of cantilever) equal to R·δ₁₁ = R·L³/(3EI) (deflection per unit redundant). The key insight: the prop carries only 3/8 of the total load; 5/8 is carried by the fixed support, creating a hogging moment.

When DI > 1, you release all redundants and get a system of compatibility equations: δ₀⁽ⁱ⁾ + Σⱼ(Rⱼ·δᵢⱼ) = Δᵢ for each released point i. In matrix form: {δ₀} + [δ]·{R} = {Δ}, which solves as {R} = [δ]⁻¹·({Δ} − {δ₀}). The flexibility matrix [δ] is symmetric (δᵢⱼ = δⱼᵢ by reciprocal theorem). Each diagonal element δᵢᵢ is always positive; off-diagonal elements δᵢⱼ can be positive or negative depending on geometry and load directions.

If a support moves by a known amount Δ (e.g., 10 mm downward settlement at an interior support of a continuous beam), the compatibility equation becomes δ₀ + R·δ₁₁ = Δ (instead of 0 for no movement). The right-hand side is now non-zero. A settlement creates additional moments and reactions in the structure, even with no applied loads. This is particularly important for long bridges and structures on compressible soils (common in Philippine construction on river deltas and soft clay).

While not the main topic of this chapter, moment distribution (or Hardy Cross method) is an alternative to force methods for continuous beams and frames. It directly distributes moments at joints rather than solving compatibility equations. Both methods are used in practice; moment distribution is often faster for hand calculations on small structures, while force methods (with matrix form) are more suitable for computer implementation and large structures.

The degree of indeterminacy (DI) determined from reaction and connectivity analysis directly tells you how many forces/moments are redundant. Each redundant requires one compatibility equation. A two-span continuous beam has DI = 1 (one interior support is redundant); you write one three-moment equation. A fixed-fixed beam has DI = 3 (two fixed-end moments and one reaction are redundant); you write three compatibility equations. This relationship structures the entire solution.

Different primary structures yield different flexibility coefficient values, but the final redundant forces and moments are always the same—they must be, because the actual structure is unique. However, a primary structure that is simple (e.g., cantilever) gives simpler deflection integrals and fewer computational errors. This cascades: simple δ values → simpler compatibility equations → smaller chance of arithmetic mistakes → correct final answer.

The superposition principle is implicit in the force method. The deflection at any point in the actual structure is the sum of (1) the deflection from loads alone and (2) the deflection from redundants. This linear superposition is valid because we assume small deflections and linear elastic material (valid for most steel and concrete under service loads per NSCP 2015).

If Δ = 0 (no settlement), the compatibility equation is δ₀ + R·δ₁₁ = 0, which directly solves for R. If Δ ≠ 0 (known settlement), the equation becomes δ₀ + R·δ₁₁ = Δ, which still solves uniquely for R. The physical meaning: the actual deflection at the support is forced to equal the known displacement, creating additional (or reduced) redundant forces.

The three-moment equation is a shortcut specific to continuous beams. It arises from applying the force method with a specific choice of primary structure (usually, cutting the beam at interior supports and treating moments as unknowns). The equation encodes the compatibility conditions in a compact, moment-focused form. For beams, three-moment is faster; for frames, you return to the general force method.

The force method ensures compatibility (deflections match supports). Once you have the redundants, you apply equilibrium to find all other forces and reactions. Together, equilibrium and compatibility fully determine the structure's response, consistent with the principle of statically indeterminate analysis: you need both to find all unknowns.

The flexibility coefficient δᵢⱼ = ∫(MᵢMⱼ)/(EI) dx is computed by multiplying the moment diagrams from two load cases, dividing by EI, and integrating over the length. This comes from the relationship between bending moment and curvature (M = EI·d²y/dx²). The integral captures the cumulative effect of curvature on deflection—a key link between bending and deflection in the force method.

Highway and railway bridges in the Philippines commonly use continuous girders spanning multiple piers (e.g., 30 m + 35 m + 30 m). The three-moment equation allows rapid calculation of moments at piers and in each span under moving loads (simplified as UDL + point loads). These moments determine the prestressing pattern, reinforcement, and capacity checks per NSCP 2015. Engineers use the three-moment equation or computer analysis to optimize girder proportioning and material cost while ensuring adequate safety margins.

Multi-story building frames with rigid connections are indeterminate. The force method (often combined with moment distribution or computer analysis) finds moment and shear distributions in beams and columns under dead load, live load, and wind. These internal forces drive the design of beam-column connections and reinforcement, ensuring the structure meets NSCP 2015 flexural and shear strength requirements. For example, a typical 4-story frame has DI ≈ 6–12, requiring systematic method to avoid errors.

In low-lying areas of the Philippines (e.g., Metro Manila, where subsoil is soft clay), bridge piers and building foundations may settle differentially. Using the force method with non-zero settlement (Δ) in the compatibility equation, engineers calculate the additional moments and shears created by settlement. For example, a 5 mm settlement at one pier of a two-span bridge creates hogging moments at supports and affects the serviceability (crack width) and fatigue behavior. This analysis is crucial for long-term safety in settlement-prone regions.

During construction of cantilevered balconies, architectural overhangs, or temporary formwork support, a cantilever is often propped (supported at a temporary point). The propped cantilever formulas (R = 3wL/8, M = wL²/8) directly give the magnitude of the temporary prop reaction and the permanent fixed-end moment. This is essential for safe design of the prop (usually a temporary steel prop or adjustable shore) and the connections at the fixed end. The formula quickly shows the benefit of the prop: it reduces the fixed moment from wL²/2 (fully cantilever) to wL²/8.

Long pipelines crossing multiple valleys or with intermediate supports are indeterminate beams. Temperature changes, internal pressure, and soil settlement all create stresses. The force method with settlement terms models how soil movement (subsidence, heave, or liquefaction) alters pipe stresses. This is critical in seismic and geotechnical engineering in the Philippines, where soil liquefaction and ground displacement are common hazards.

A beam fixed at both ends (DI = 3) requires the force method or moment distribution to find end moments and reactions. Once you have these moments, the design of the fixed connections (welded end plates for steel, reinforced concrete embedments, or special fasteners) is straightforward. The method also reveals how end moments reduce the interior span moment, allowing lighter span sections compared to a simply supported span of the same length and load—this is an important design economy.

When a structure experiences periodic loading (machinery, traffic, earthquakes), its dynamic response depends on natural frequencies and damping. The stiffness (inverse of flexibility) from the force method feeds into dynamic analysis. For instance, knowing the static flexibility δ₁₁ allows rapid calculation of natural frequency f = (1/2π)√(k/m) = (1/2π)√(1/(m·δ₁₁)). This connects the static method to dynamics, essential for seismic design under NSCP 2015 earthquake requirements.

In composite girders (steel beam with concrete slab), shear connectors ensure composite behavior, making the section behave as one. The indeterminate interaction between steel and concrete, as well as the interaction with pier supports, requires the force method to distribute loads correctly between the steel and concrete phases of construction. This affects the design moments and shear connector spacing, critical for bridges and parking structures.