CELE Strength of Materials — Beam DeflectionsSummary

The elastic curve y(x) is the deflected shape of the beam's neutral axis. For a linearly elastic beam, the governing differential equation is EI·d²y/dx² = M(x), where EI is the flexural rigidity (product of Young's modulus E and second moment of area I). This equation is the foundation of all deflection methods. Small-deflection theory assumes that slopes are small (dy/dx ≪ 1), so sin(θ) ≈ θ and cos(θ) ≈ 1, making the curvature approximation κ ≈ d²y/dx² valid.

The most direct approach: (1) Write the bending-moment function M(x) using free-body diagrams and equilibrium. (2) Substitute into EI·d²y/dx² = M(x) and integrate once to obtain EI·dy/dx = ∫M dx + C₁ (slope equation). (3) Integrate again to obtain EI·y = ∫∫M dx + C₁·x + C₂ (deflection equation). (4) Apply boundary conditions to find C₁ and C₂. For a simply supported beam: y = 0 at x = 0 and x = L. For a cantilever fixed at x = 0: y = 0 and dy/dx = 0 at x = 0. Macaulay/singularity brackets ⟨x − a⟩ⁿ simplify multi-segment loading by switching terms on and off at discontinuities.

Two powerful geometric theorems based on the M/EI diagram: Theorem 1: The change in slope between points A and B equals the area of the M/EI diagram between them: θ_B − θ_A = ∫ₐᵇ (M/EI) dx. Theorem 2: The tangential deviation (vertical distance from point B to the tangent line at A) equals the first moment of the M/EI area about B: t_{B/A} = ∫ₐᵇ (M/EI)·x_B dx = (Area_{AB})·x̄_B. For a cantilever with a horizontal tangent at the fixed end, the deviation directly gives the deflection.

An elegant reframing of the moment-area theorems: (1) Create a fictitious 'conjugate beam' with the same length and boundary conditions related to the real beam. (2) Load the conjugate beam with the real beam's M/EI diagram as a distributed load. (3) In the conjugate beam, the shear force at any section equals the slope in the real beam at that section; the bending moment in the conjugate equals the deflection in the real beam. Support conversions: a real simple support becomes a conjugate simple support; a real fixed end becomes a conjugate free end (and vice versa). This converts a deflection problem into a familiar statics problem of finding shear and moment.

For standard loading cases on common support conditions, deflection formulas are pre-derived and tabulated. For a simply supported beam with span L and flexural rigidity EI: central point load P gives δ_max = PL³/(48EI) at midspan; full-span UDL w gives δ_max = 5wL⁴/(384EI) at midspan. For a cantilever of length L: point load P at the free end gives δ = PL³/(3EI); full UDL w gives δ = wL⁴/(8EI); moment M at free end gives δ = ML²/(2EI). Complex loadings are handled by superposing simpler cases (e.g., a cantilever with both a point load and a UDL: δ_total = δ_from_point + δ_from_UDL).

The Philippine National Building Code (NSCP 2015) Section 5.2.4 mandates deflection checks as a distinct serviceability limit state. Common limits: δ ≤ L/360 for live loads on members supporting fragile finishes (plaster, suspended ceilings); δ ≤ L/240 for live loads on other members. For floor vibration control, δ ≤ L/480 may apply. A beam may satisfy bending and shear stress criteria yet fail the deflection limit, requiring an increase in section size (larger I, smaller L, or stiffer material—higher E). This is why deflection calculations are performed in serviceability checks, separate from strength checks.

In a statically indeterminate structure (e.g., a propped cantilever or a continuous beam), the number of unknown reactions exceeds the equilibrium equations. The additional equations come from **geometric compatibility**: the actual deflection at a support must match the support condition. For a propped cantilever under load, the free-end deflection due to the applied load plus the deflection due to the prop reaction must sum to zero (the prop prevents vertical movement). This 'consistent deformation' principle is the entry point to indeterminate analysis, using the Flexibility or Stiffness methods. Deflection formulas are the raw material.

A critical practical skill: always convert lengths to a single unit (mm preferred) and forces to N before calculation. For example, if L = 6 m, w = 10 kN/m, E = 200 GPa, I = 80 × 10⁶ mm⁴: convert to L = 6000 mm, w = 10 N/mm, E = 200,000 N/mm², I = 80 × 10⁶ mm⁴, then substitute. Failure to convert causes 'unit explosions'—answers off by factors of 10³ to 10¹². The formula δ = 5wL⁴/(384EI) with L⁴ is particularly prone to this error.

The area-moment theorems are mathematical consequences of the double-integration method. Integrating EI·d²y/dx² = M(x) from A to B gives the slope change θ_B − θ_A = ∫(M/EI)dx, which is the area of the M/EI diagram. Further manipulation of the integrated form yields the tangential deviation formula. Thus, area-moment is not an independent method but a geometric shortcut that avoids explicit integration.

The conjugate-beam method is a reformulation of area-moment geometry using statics. In the conjugate beam, the M/EI diagram becomes a load, and the statics (finding shear and moment) directly reproduce the slope and deflection of the real beam. The two approaches give identical answers but use different conceptual tools—geometry for one, statics for the other. The conjugate beam is faster when moment/shear are easy to visualize.

Every standard formula (e.g., δ = PL³/(48EI) for a simple beam with central point load) is derived once by double integration or area-moment and then tabulated. In practice, these formulas are verified against the full derivation but applied directly. Superposition extends their use: a cantilever under both a point load and UDL is solved by δ_total = δ_point + δ_UDL, avoiding re-derivation of a custom loading case.

Indeterminate structures have more unknowns than equilibrium equations provide. The missing equations come from deflection compatibility: the actual deflection at a support must match the support condition (usually zero). For a propped cantilever, the deflection at the prop (from applied loads plus prop reaction) must be zero. This compatibility equation is solved using the deflection formulas derived in this chapter, establishing the direct link between elastic analysis and structural theory.

Serviceability is a distinct limit state (along with strength). The Philippine National Building Code NSCP 2015 Section 5.2.4 mandates deflection checks: δ ≤ L/360 for fragile finishes, δ ≤ L/240 for general members. These limits are derived from user comfort and functional requirements (doors jamming at ~L/200, floors bouncing at ~L/300). A section designed for strength alone may violate serviceability, requiring iteration: increase I, decrease L, or upgrade material to raise E.