Deflection of Structures

Learning Objectives

  • Relate curvature, slope, and deflection to bending moment for small-deflection beam theory.
  • Apply correct boundary and continuity conditions to elastic-curve calculations.
  • Select among double integration, moment-area, conjugate-beam, virtual-work, and Castigliano methods.
  • Calculate beam, frame, and truss displacements using geometric or energy approaches.
  • Distinguish tangential deviation from actual vertical deflection.
  • Verify displacement results using symmetry, boundary conditions, units, and expected deformation shape.

Elastic Curve

The elastic curve is the deflected shape of a member's reference axis under load while the response remains within the assumptions of the adopted elastic beam theory.

Slope

Slope θ\theta is the local rotation of the elastic curve. Under the small-slope approximation, θ≈dv/dx\theta\approx dv/dx.

Deflection

Deflection vv is the translational displacement of a point on the member relative to its undeformed reference position in the selected direction.

Moment-Curvature Relation

For an Euler-Bernoulli beam under small deformation and a consistent sign convention.

EI d2vdx2=M(x)EI\,\frac{d^2v}{dx^2}=M(x)

Variables

SymbolDescriptionUnit
EEYoung's modulus-
IISecond moment of area about the bending axis-
vvTransverse deflection-
M(x)M(x)Bending moment function under the adopted sign convention-

Sign Convention Must Remain Consistent

Some texts use the opposite sign in the moment-curvature equation because of different positive moment and deflection conventions. Do not mix formulas from different sign systems; boundary conditions and the final deformation direction must be interpreted consistently.

Essential Boundary Conditions

  • Fixed end: translation and rotation are restrained, so v=0v=0 and θ=0\theta=0 at the fixed point.
  • Pin or roller support: transverse deflection is zero in the restrained direction, while rotation is generally free.
  • Free end: translation and rotation are not prescribed; end shear or moment is zero only when no corresponding end load or couple is applied.
  • Internal hinge: bending moment is zero at the hinge and connected points share the hinge translation, while the ideal hinge permits relative rotation of the adjoining member segments.

Double-Integration Method

Write M(x)M(x), integrate EIv′′=M(x)EI v''=M(x) to obtain slope, integrate again to obtain deflection, and determine the constants from boundary and continuity conditions. This method is direct and transparent when the piecewise moment equations are manageable.

Integrated Elastic-Curve Relations

The two integrations introduce constants that are determined from support and continuity conditions.

EI θ(x)=∫M(x) dx+C1EI\,\theta(x)=\int M(x)\,dx+C_1EI v(x)=∫ ⁣ ⁣∫M(x) dx dx+C1x+C2EI\,v(x)=\int\!\!\int M(x)\,dx\,dx+C_1x+C_2

Variables

SymbolDescriptionUnit
θ(x)\theta(x)Slope-
v(x)v(x)Deflection-
C1,C2C_1,C_2Integration constants-

Moment-Area Theorem 1

The change in slope between points AA and BB equals the signed area under the M/EIM/EI diagram between those points:

θB−θA=∫ABMEI dx\theta_B-\theta_A=\int_A^B\frac{M}{EI}\,dx

Moment-Area Theorem 2

The tangential deviation of one point from the tangent drawn at another equals the first moment of the M/EIM/EI area between the points about the point where the deviation is measured. Tangential deviation is a geometric quantity relative to a tangent; it is not automatically the absolute deflection from the undeformed axis.

Conjugate-Beam Method

Create a conjugate beam loaded by the real beam's M/EIM/EI diagram. With the correct support transformation, shear in the conjugate beam corresponds to slope in the real beam and bending moment in the conjugate beam corresponds to deflection. The method converts a geometric integration problem into an equilibrium problem.

Interactive Exploration

Use the conjugate-beam simulation to connect the real beam, its M/EIM/EI loading, the conjugate support model, and the resulting slope/deflection quantities. Confirm support transformations and signs before reading numerical output.

Conjugate Beam Transformation

10 kN
5 m

Notice how the real beam's fixed support becomes free on the conjugate beam, and the free end becomes fixed. The M/EI diagram is applied as a distributed load to the conjugate beam.

Principle of Virtual Work

For a linear elastic structure, a desired displacement can be obtained by applying a compatible virtual unit action in the displacement direction and equating external virtual work to the corresponding internal virtual work.

Virtual Work for Truss Joint Deflection

Axial-deformation form for pin-jointed trusses.

Δ=∑iniNiLiAiEi\Delta=\sum_i\frac{n_iN_iL_i}{A_iE_i}

Variables

SymbolDescriptionUnit
Δ\DeltaDesired displacement-
NiN_iReal axial force in member i-
nin_iVirtual axial force caused by a unit virtual load-
LiL_iMember length-
AiA_iMember area-
EiE_iYoung's modulus-

Virtual Work for Beam or Frame Deflection

Bending-dominated form when axial and shear deformation are neglected.

Δ=∫mMEI dx\Delta=\int\frac{mM}{EI}\,dx

Variables

SymbolDescriptionUnit
MMReal bending moment-
mmVirtual bending moment caused by a unit virtual action-
EEYoung's modulus-
IISecond moment of area-

Castigliano's Second Theorem

For a linearly elastic structure whose strain energy UU is expressed in terms of applied generalized loads, the displacement associated with load PP is ∂U/∂P\partial U/\partial P.

Castigliano Displacement Relation

Generalized displacement associated with an applied generalized load.

ΔP=∂U∂P\Delta_P=\frac{\partial U}{\partial P}

Variables

SymbolDescriptionUnit
ΔP\Delta_PDisplacement in the direction of generalized load P-
UUElastic strain energy-
PPGeneralized force or moment-

Maxwell-Betti Reciprocity

For a stable linear elastic structure, the displacement at coordinate ii caused by a unit load at coordinate jj equals the displacement at coordinate jj caused by the same unit load at coordinate ii, provided the same compatible generalized coordinates and sign conventions are used.

Deflection Method Selection

Choose the method based on the structural type, the number of displacements required, and the complexity of the moment or member-force expressions. The shortest method is the one that exposes the required quantity with the fewest unnecessary unknowns.

Deflection Method SelectionDefine displacement or rotation required → Is a continuous deflection curve required?; Is a continuous deflection curve required? — Yes → Use double integration method; Is a continuous deflection curve required? — No → Is the structure a truss or flexural frame?; Is the structure a truss or flexural frame? — Truss → Truss: use virtual work with member axial forces; Is the structure a truss or flexural frame? — Beam / frame → Is M/EI geometry simple or piecewise standard?; Is M/EI geometry simple or piecewise standard? — Yes → Use moment-area or conjugate-beam method; Is M/EI geometry simple or piecewise standard? — No → Use virtual work (unit-load) or Castigliano; Use double integration method → Check boundary conditions, symmetry, units, and direction; Truss: use virtual work with member axial forces → Check boundary conditions, symmetry, units, and direction; Use moment-area or conjugate-beam method → Check boundary conditions, symmetry, units, and direction; Use virtual work (unit-load) or Castigliano → Check boundary conditions, symmetry, units, and direction; Check boundary conditions, symmetry, units, and direction → Verified displacement result

Define displacement or rotation required → Is a continuous deflection curve required?; Is a continuous deflection curve required? — Yes → Use double integration method; Is a continuous deflection curve required? — No → Is the structure a truss or flexural frame?; Is the structure a truss or flexural frame? — Truss → Truss: use virtual work with member axial forces; Is the structure a truss or flexural frame? — Beam / frame → Is M/EI geometry simple or piecewise standard?; Is M/EI geometry simple or piecewise standard? — Yes → Use moment-area or conjugate-beam method; Is M/EI geometry simple or piecewise standard? — No → Use virtual work (unit-load) or Castigliano; Use double integration method → Check boundary conditions, symmetry, units, and direction; Truss: use virtual work with member axial forces → Check boundary conditions, symmetry, units, and direction; Use moment-area or conjugate-beam method → Check boundary conditions, symmetry, units, and direction; Use virtual work (unit-load) or Castigliano → Check boundary conditions, symmetry, units, and direction; Check boundary conditions, symmetry, units, and direction → Verified displacement result

  • Define displacement or rotation required: terminator
  • Is a continuous deflection curve required?: decision
  • Use double integration method: process
  • Is the structure a truss or flexural frame?: decision
  • Truss: use virtual work with member axial forces: process
  • Is M/EI geometry simple or piecewise standard?: decision
  • Use moment-area or conjugate-beam method: process
  • Use virtual work (unit-load) or Castigliano: process
  • Check boundary conditions, symmetry, units, and direction: process
  • Verified displacement result: terminator
Key Takeaways
  • The elastic curve connects bending moment to curvature, slope, and deflection under small-deformation beam assumptions.
  • Correct support, hinge, and continuity conditions are essential to every deflection method.
  • Moment-area and conjugate-beam methods exploit the geometry of the M/EIM/EI diagram.
  • Virtual work is efficient for selected displacements in beams, frames, and trusses.
  • Castigliano relates displacement to derivatives of elastic strain energy.
  • Always verify deformation direction, boundary values, symmetry, and dimensional consistency.

References