Deflection of Beams

Learning Objectives

  • Explain why beam deflection is a serviceability concern distinct from member strength.
  • Relate bending moment to elastic-curve curvature through flexural rigidity.
  • Apply double integration, moment-area, conjugate-beam, and superposition concepts appropriately.
  • Select a deflection method based on the loading, desired output, and available standard solutions.
  • Interpret the elastic curve without confusing exaggerated visualization with physical scale.

Elastic Curve

The elastic curve is the deflected shape of a beam's longitudinal centroidal axis while the member remains within the assumptions of elastic beam theory.

Strength versus Serviceability

A beam can remain safely below its material strength limits and still be unsatisfactory because of excessive deflection or rotation. Serviceability checks protect architectural finishes, partitions, glazing, drainage slopes, occupant comfort, and the intended appearance and function of the building.

Serviceability Interface Context

An intact floor assembly, partition, and glazing meet the same beam, showing why deformation can matter to building finishes and interfaces even when the structural member remains continuous.

Connected steel beam beneath an intact floor finish, partition, and glazing interface.

Interpreting Finish Sensitivity

Treat the floor, partition, and glazing as contextual nonstructural interfaces connected to the beam; the figure does not establish damage, tolerance, a code limit, or a measured displacement.

Elastic-Curve Differential Equation

Small-deflection relationship between bending moment and beam curvature under Euler-Bernoulli assumptions.

EId2vdx2=M(x)EI\frac{d^2v}{dx^2}=M(x)

Flexural Rigidity

The product EIEI is the flexural rigidity. Increasing either the material modulus EE or the section moment of inertia II reduces elastic curvature for a given bending moment. Because II depends strongly on section depth, increasing depth is often far more effective than increasing width.

Deep versus Shallow Beam Context

Matching connected bays make beam depth a visible geometric cue for qualitative flexural-rigidity and sag discussion; exact EI and deflection relationships remain in the formula and simulation.

Two matching structural floor bays compare a shallow steel beam with a deeper one beneath similar slabs.

Reading the Depth Comparison

Compare the member proportions and the calm qualitative profiles, not a measured stiffness or deflection ranking. Section depth is shown as physical context for the II term, while the exact relationship remains in the lesson's equation and interactive model.

Beam Deflection Context

The continuous bowing between the supports provides physical context for an elastic curve and serviceability response; use the simulation and formulas for exact curvature, slope, and deflection.

Text-free technical illustration of an intact steel beam with a modest smooth sag between pin and roller supports.

Reading the Deflected Beam

The beam and its support contacts remain intact while the shallow, continuous sag makes deformation visible. The curvature is deliberately qualitative and not to scale; it is not a measured displacement, a design limit, or a substitute for the normalized elastic curve in the simulation.

Interactive Exploration

Vary span, distributed load, modulus, and moment of inertia. Observe how the full elastic curve and maximum deflection change, and treat the plotted deformation as an instructional graph rather than a physically scaled bent beam.

Beam Deflection: Elastic Curve

Concept and model scope

Simply supported beam with a full-span uniform load. The plot uses separate graph scales; numerical deflection values are the engineering result.

Controls

Uniform load

Full-span uniformly distributed load. Numerically, 1 kN/m equals 1 N/mm.

Range: 1–25 kN/m. Step: 1 kN/m.

8 kN/m

Span

Simply supported beam span. For this load case, maximum deflection varies with the fourth power of span.

Range: 2.0–10.0 m. Step: 0.1 m.

5.0 m

Elastic modulus

Young's modulus E. Larger E increases flexural rigidity and reduces elastic deflection.

Range: 8–210 GPa. Step: 1 GPa.

200 GPa

Moment of inertia

Second moment of area I about the bending axis. Larger I increases flexural rigidity.

Range: 10–500 ×10⁶ mm⁴. Step: 5 ×10⁶ mm⁴.

100 ×10⁶ mm⁴
δ = 0δmax02.5 m5.0 mmidspan · δmax = 3.26 mmVertical graph scale is normalized to the current δmax; this is not a physical-scale deformation drawing.
Maximum deflection
3.26 mm

5wL⁴/(384EI), downward at midspan.

Flexural rigidity
20000 GPa·10⁶ mm⁴

Displayed as the product of the entered E and I scales.

Symmetry check
θ(midspan) = 0

The symmetric loading and supports require a horizontal tangent at midspan.

Double Integration Method

The double integration method obtains slope and deflection by integrating the bending-moment equation twice and evaluating constants from boundary and continuity conditions.

Cantilever Boundary Context

A continuous wall connection at one end and an unobstructed projecting end provide physical context for a cantilever boundary condition; exact slope and deflection remain in the worked methods.

Steel beam fixed into a concrete wall and extending to an unobstructed free end.

Reading the Fixed Boundary

The rigid wall connection is the defining boundary in this qualitative view, while the projecting end has no second support. Use the equations and examples for the boundary conditions, signs, and numerical response.

Slope and Deflection by Integration

Sequential integrations of the elastic-curve equation.

EIdvdx=∫M(x) dx+C1EI\frac{dv}{dx}=\int M(x)\,dx+C_1EIv=∫ ⁣∫M(x) dx dx+C1x+C2EIv=\int\!\int M(x)\,dx\,dx+C_1x+C_2

Moment-Area Method

The moment-area method relates changes in slope and tangential deviation to the area and first moment of the M/EIM/EI diagram.

Moment-Area Theorems

Slope change and tangential deviation obtained from the M over EI diagram.

θB−θA=∫ABMEI dx\theta_B-\theta_A=\int_A^B\frac{M}{EI}\,dxtB/A=∫ABxBMEI dxt_{B/A}=\int_A^B x_B\frac{M}{EI}\,dx

Conjugate Beam Method

The conjugate-beam method converts an elastic-curve problem into an equivalent statics problem. The real beam's M/EIM/EI diagram becomes the loading on the conjugate beam; conjugate-beam shear corresponds to real-beam slope, and conjugate-beam moment corresponds to real-beam deflection. Correct support transformation is essential.

Superposition

For a linear-elastic system with small deflections, the response from several load cases is the algebraic sum of the responses produced by each load separately. This makes verified beam-formula tables highly efficient for standard loading patterns.

Beam Superposition Load-Case Context

A single beam shares one architectural assembly with several independent service contexts, illustrating why separate load cases can contribute to one overall response; the raster contains no load values or response sum.

One connected beam carrying floor, partition, ceiling, and service contexts along its span.

Interpreting Multiple Service Conditions

Read the floor, partition, ceiling, and service-support elements as distinct qualitative contexts on one connected beam. The algebraic combination and the linearity limits are defined by the adjacent concept and caution, not by the raster.

Superposition Limits

Do not use linear superposition after substantial yielding, large geometric change, support-condition change, or another nonlinearity that causes one load case to alter the structural response to another.

Method Selection Workflow

Choose a deflection method based on what must be found and how the loading is represented. The workflow below is a guide, not a replacement for engineering judgment.

Beam Deflection Method Selection
Beam Deflection Method SelectionDefine beam, EI, loading, and required response → Can the loading be decomposed into verified standard cases?; Can the loading be decomposed into verified standard cases? — Yes → Use superposition and beam-formula solutions; Can the loading be decomposed into verified standard cases? — No → Is a continuous slope/deflection equation required?; Use superposition and beam-formula solutions → Apply boundary, sign, unit, and symmetry checks; Is a continuous slope/deflection equation required? — Yes → Use double integration or singularity functions; Is a continuous slope/deflection equation required? — No → Is slope/deflection needed mainly at selected points?; Use double integration or singularity functions → Apply boundary, sign, unit, and symmetry checks; Is slope/deflection needed mainly at selected points? — Yes → Use moment-area where M/EI geometry is convenient; Is slope/deflection needed mainly at selected points? — No / alternate → Use conjugate beam when a statics transformation is advantageous; Use moment-area where M/EI geometry is convenient → Apply boundary, sign, unit, and symmetry checks; Use conjugate beam when a statics transformation is advantageous → Apply boundary, sign, unit, and symmetry checks; Apply boundary, sign, unit, and symmetry checks → Report slope/deflection and assumptions

Define beam, EI, loading, and required response → Can the loading be decomposed into verified standard cases?; Can the loading be decomposed into verified standard cases? — Yes → Use superposition and beam-formula solutions; Can the loading be decomposed into verified standard cases? — No → Is a continuous slope/deflection equation required?; Use superposition and beam-formula solutions → Apply boundary, sign, unit, and symmetry checks; Is a continuous slope/deflection equation required? — Yes → Use double integration or singularity functions; Is a continuous slope/deflection equation required? — No → Is slope/deflection needed mainly at selected points?; Use double integration or singularity functions → Apply boundary, sign, unit, and symmetry checks; Is slope/deflection needed mainly at selected points? — Yes → Use moment-area where M/EI geometry is convenient; Is slope/deflection needed mainly at selected points? — No / alternate → Use conjugate beam when a statics transformation is advantageous; Use moment-area where M/EI geometry is convenient → Apply boundary, sign, unit, and symmetry checks; Use conjugate beam when a statics transformation is advantageous → Apply boundary, sign, unit, and symmetry checks; Apply boundary, sign, unit, and symmetry checks → Report slope/deflection and assumptions

  • Define beam, EI, loading, and required response: terminator
  • Can the loading be decomposed into verified standard cases?: decision
  • Use superposition and beam-formula solutions: process
  • Is a continuous slope/deflection equation required?: decision
  • Use double integration or singularity functions: process
  • Is slope/deflection needed mainly at selected points?: decision
  • Use moment-area where M/EI geometry is convenient: process
  • Use conjugate beam when a statics transformation is advantageous: process
  • Apply boundary, sign, unit, and symmetry checks: process
  • Report slope/deflection and assumptions: terminator
Key Takeaways
  • Beam deflection is primarily a serviceability response and must be checked separately from strength.
  • Flexural rigidity EIEI controls elastic curvature and deflection.
  • Double integration gives a continuous response; moment-area and conjugate-beam methods can simplify selected-point calculations.
  • Superposition is efficient only when linearity and small-deflection assumptions remain valid.
  • Every method should satisfy support conditions, sign conventions, dimensions, and symmetry or limiting-case checks.

References