Deflection, Energy and Virtual Work

Unit purpose

Deformation is not an optional second-stage check. Structural serviceability, compatibility and stiffness-controlled load distribution depend on displacement prediction. This unit develops several independent methods so a single derivation is not trusted without cross-checking.

Field scenario

A slender cantilever canopy may pass an approximate force check while undergoing excessive tip movement. Compute deflections from a point load and a distributed action, then test sensitivity to length and second moment of area.

Topics and worked reasoning

Topic 5.1: Differential-equation integration

In Euler–Bernoulli bending, v double-prime is related to the moment through EI under an explicitly chosen curvature sign. Integrating the moment diagram twice introduces constants that are fixed by displacement or rotation boundary conditions. For a simply supported span without settlement the support deflections are zero, while rotations are generally not constrained. For a cantilever fixed at x=0, deflection and slope are zero at the built-in end. Boundary conditions belong to the displacement field, not just the applied-force diagram. Piecewise moments require continuity conditions at internal load locations.

Engineering practice: Work with one global sign convention from the start. For a simple-span central load, derive the left half displacement, impose zero end displacement and zero midspan rotation from symmetry, and obtain PL cubed/(48EI) at the center. Explain why symmetry conditions cannot be used for an off-center load.

Fully evaluated study example

For a linear elastic cantilever with length 6.0 m, end force 28 kN, full-span load 3.1 kN/m, stiffness E=200 GPa and I=2.30×10^-4 m^4, the free-tip downward deflection is 54.743 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Reasoning and interpretation

How many constants follow from integrating the curvature equation? Start by writing the physical reason, then use an equation or laboratory parameter sweep to support the claim. Compare the result against an extreme case: zero applied load, negligible rise, very long span, perfectly fixed restraint, or load at a support, whichever is physically appropriate. In a valid limiting case, the response should approach a predictable behavior. When a calculation fails an extreme-case check, investigate modeling assumptions before blaming numerical precision.

Practice tasks

1. Rework the given example with a 15% increase in the characteristic load and predict whether force, moment or displacement scales linearly. State all inputs kept fixed.
2. Investigate this potential failure of the model: How many constants follow from integrating the curvature equation? Document the assumption at risk and describe a more complete model.
3. Draw the free-body diagram or curvature/influence diagram appropriate to differential-equation integration. Label axes, internal forces, signs, physical units and boundary conditions.
4. Use a unit or equilibrium residual to check your solution. Describe an error that would pass the numerical residual check but still lead to an incorrect engineering interpretation.

Applied investigation

Choose a structural example relevant to differential-equation integration. List the information required from a drawing, material schedule or site measurement. Separate what can be concluded directly from equilibrium from what depends on stiffness, geometry, or an unverified boundary condition. Run two different reasonable parameter settings and write a short explanation of why the predicted response changes. Avoid treating either result as a certified design action.

Topic 5.2: Conjugate beam and area-moment ideas

The moment-area theorems connect the area of the M/EI diagram to the change in slope and the first moment of that area to tangential deviation. The conjugate-beam analogy interprets the original curvature diagram as loading on an auxiliary beam with transformed supports, enabling slope and deflection to be found from shear and moment in the conjugate system. This method is useful when the moment field can be integrated geometrically. It is also vulnerable to sign mistakes and incorrect support transformations when overhangs or internal hinges are present.

Engineering practice: Sketch the M/EI distribution from a known bending-moment diagram before performing area calculations. Compare an integration solution with a moment-area solution for a simply supported beam under one central force. Note that using a constant EI in a variable-section beam changes the curvature distribution and can lead to a misleading result.

Fully evaluated study example

For a linear elastic cantilever with length 9.0 m, end force 34 kN, full-span load 2.0 kN/m, stiffness E=200 GPa and I=2.60×10^-4 m^4, the free-tip downward deflection is 190.428 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Reasoning and interpretation

How are support conditions mapped to the conjugate beam? Start by writing the physical reason, then use an equation or laboratory parameter sweep to support the claim. Compare the result against an extreme case: zero applied load, negligible rise, very long span, perfectly fixed restraint, or load at a support, whichever is physically appropriate. In a valid limiting case, the response should approach a predictable behavior. When a calculation fails an extreme-case check, investigate modeling assumptions before blaming numerical precision.

Practice tasks

1. Rework the given example with a 15% increase in the characteristic load and predict whether force, moment or displacement scales linearly. State all inputs kept fixed.
2. Investigate this potential failure of the model: How are support conditions mapped to the conjugate beam? Document the assumption at risk and describe a more complete model.
3. Draw the free-body diagram or curvature/influence diagram appropriate to conjugate beam and area-moment ideas. Label axes, internal forces, signs, physical units and boundary conditions.
4. Use a unit or equilibrium residual to check your solution. Describe an error that would pass the numerical residual check but still lead to an incorrect engineering interpretation.

Applied investigation

Choose a structural example relevant to conjugate beam and area-moment ideas. List the information required from a drawing, material schedule or site measurement. Separate what can be concluded directly from equilibrium from what depends on stiffness, geometry, or an unverified boundary condition. Run two different reasonable parameter settings and write a short explanation of why the predicted response changes. Avoid treating either result as a certified design action.

Topic 5.3: Virtual work and the unit-load method

The unit-load method evaluates a displacement by multiplying the real internal-force field with a compatible virtual internal-force field. In a beam where flexure dominates, Delta=int M(x)m(x)/(EI) dx, with m produced by a unit load or moment at the target degree of freedom. Contributions from axial and shear deformation can be added where important. The unit virtual action must be placed in the exact direction of the desired response. A downward unit load produces the corresponding downward displacement under compatible signed work conventions.

Engineering practice: For a cantilever with a tip load P and uniform line load w, M(x) in magnitude equals P(L-x)+w(L-x) squared/2 and m(x)=L-x. Integration yields tip displacement PL cubed/(3EI)+wL to the fourth/(8EI). Independently compute the integral numerically and compare. Explain how error changes when the numerical integration is coarsened.

Fully evaluated study example

For a linear elastic cantilever with length 12.0 m, end force 41 kN, full-span load 4.8 kN/m, stiffness E=200 GPa and I=2.90×10^-4 m^4, the free-tip downward deflection is 621.683 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Reasoning and interpretation

Where is the virtual unit force applied and what is its orientation? Start by writing the physical reason, then use an equation or laboratory parameter sweep to support the claim. Compare the result against an extreme case: zero applied load, negligible rise, very long span, perfectly fixed restraint, or load at a support, whichever is physically appropriate. In a valid limiting case, the response should approach a predictable behavior. When a calculation fails an extreme-case check, investigate modeling assumptions before blaming numerical precision.

Practice tasks

1. Rework the given example with a 15% increase in the characteristic load and predict whether force, moment or displacement scales linearly. State all inputs kept fixed.
2. Investigate this potential failure of the model: Where is the virtual unit force applied and what is its orientation? Document the assumption at risk and describe a more complete model.
3. Draw the free-body diagram or curvature/influence diagram appropriate to virtual work and the unit-load method. Label axes, internal forces, signs, physical units and boundary conditions.
4. Use a unit or equilibrium residual to check your solution. Describe an error that would pass the numerical residual check but still lead to an incorrect engineering interpretation.

Applied investigation

Choose a structural example relevant to virtual work and the unit-load method. List the information required from a drawing, material schedule or site measurement. Separate what can be concluded directly from equilibrium from what depends on stiffness, geometry, or an unverified boundary condition. Run two different reasonable parameter settings and write a short explanation of why the predicted response changes. Avoid treating either result as a certified design action.

Topic 5.4: Strain energy and reciprocal response

Linear elastic strain energy stored by bending is U=int M²/(2EI) dx, with analogous axial and torsional terms. Castigliano differentiation with respect to an applied generalized force produces a displacement under the assumptions of the theorem. Maxwell–Betti reciprocity states symmetry of corresponding displacement influence coefficients for a linear elastic conservative system with consistent boundary conditions. These results are powerful cross-checks for displacement formulas, but reciprocity can fail or require care when nonconservative actions or nonlinear contact is involved.

Engineering practice: Calculate the energy of a cantilever under a tip load, differentiate with respect to P, and recover PL cubed/(3EI). Compare the result to direct integration. Propose an experimental test where two small forces are applied at distinct points and the cross-deflections are exchanged, then identify practical sources of measurement error.

Fully evaluated study example

For a linear elastic cantilever with length 4.5 m, end force 15 kN, full-span load 1.2 kN/m, stiffness E=200 GPa and I=3.20×10^-4 m^4, the free-tip downward deflection is 8.080 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Reasoning and interpretation

What energy terms are omitted if shear deformation is ignored? Start by writing the physical reason, then use an equation or laboratory parameter sweep to support the claim. Compare the result against an extreme case: zero applied load, negligible rise, very long span, perfectly fixed restraint, or load at a support, whichever is physically appropriate. In a valid limiting case, the response should approach a predictable behavior. When a calculation fails an extreme-case check, investigate modeling assumptions before blaming numerical precision.

Practice tasks

1. Rework the given example with a 15% increase in the characteristic load and predict whether force, moment or displacement scales linearly. State all inputs kept fixed.
2. Investigate this potential failure of the model: What energy terms are omitted if shear deformation is ignored? Document the assumption at risk and describe a more complete model.
3. Draw the free-body diagram or curvature/influence diagram appropriate to strain energy and reciprocal response. Label axes, internal forces, signs, physical units and boundary conditions.
4. Use a unit or equilibrium residual to check your solution. Describe an error that would pass the numerical residual check but still lead to an incorrect engineering interpretation.

Applied investigation

Choose a structural example relevant to strain energy and reciprocal response. List the information required from a drawing, material schedule or site measurement. Separate what can be concluded directly from equilibrium from what depends on stiffness, geometry, or an unverified boundary condition. Run two different reasonable parameter settings and write a short explanation of why the predicted response changes. Avoid treating either result as a certified design action.

Deep dive and misconception repair

Deflection clinic: when energy methods are especially useful

Direct integration works well when the moment field is simple and boundary conditions are easy to impose. It becomes cumbersome when the member geometry is complex, the desired displacement is at an internal point, or several bar and beam contributions are required. The virtual-work method instead evaluates the displacement of a chosen degree of freedom through products of actual and virtual internal-force fields. The virtual field is generated by applying a unit force in the desired displacement direction to the same linear elastic geometry.

A cantilever of length L under a downward tip force P has an internal bending moment magnitude P(L-x) at coordinate x measured from the fixed end. An imagined unit downward tip force produces virtual moment magnitude L-x. Their product divided by EI integrates to PL³/(3EI). A uniform downward line load w adds moment w(L-x)²/2, giving wL⁴/(8EI) after integration against L-x. These two contributions superpose because the material and geometry are linear in the model.

The integral can be evaluated numerically without changing the physics. With a midpoint rule, divide the span into n equal segments, evaluate M(x)m(x)/(EI) at each segment midpoint and multiply their sum by segment width. As n increases, a smooth integrand is sampled more accurately and the numerical result approaches the analytical integral. A convergence plot that flattens near the benchmark provides evidence of numerical accuracy for this particular formula, not evidence that the cantilever's supports or material assumptions match a real canopy.

Sign and virtual-load direction

Virtual work requires compatible signs. If the desired displacement is downward and the unit virtual force points downward, the resulting displacement is positive downward. If the virtual force is applied in the opposite direction, the corresponding signed displacement changes sign. A check with an obviously downward-deflecting cantilever can reveal inconsistent conventions quickly. Another powerful check is dimensional: actual moment times virtual moment times dx divided by EI must produce a displacement when the virtual generalized force is treated consistently as a unit action.

The strain energy U for pure flexure is the integral of M²/(2EI). With one varying applied force P, differentiating U with respect to P yields the corresponding displacement under Castigliano's theorem for a linear elastic conservative system. This creates a second analytical route to the cantilever expression. It is especially useful when multiple actions are present, because the cross terms in M² capture interactions between load cases.

Shear deformation can be neglected for a slender Euler–Bernoulli beam but may be important for a deep or short beam. Axial deformation often dominates a truss displacement even if bending is ignored in its members. The proper energy expression must reflect the intended model: bending-only virtual work for a beam, axial sum for an ideal truss, and potentially bending, shear and axial terms for a frame. Omitting a significant term gives an internally tidy but physically incomplete displacement prediction.

Multi-case calculations and validation

Case 5.1: Reaction shift

For a linear elastic cantilever with length 6.0 m, end force 28 kN, full-span load 3.1 kN/m, stiffness E=200 GPa and I=2.00×10^-4 m^4, the free-tip downward deflection is 62.955 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For deflection, energy and virtual work, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 5.2: Longer girder

For a linear elastic cantilever with length 9.0 m, end force 34 kN, full-span load 2.0 kN/m, stiffness E=200 GPa and I=2.20×10^-4 m^4, the free-tip downward deflection is 225.051 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For deflection, energy and virtual work, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 5.3: Footbridge loading

For a linear elastic cantilever with length 12.0 m, end force 41 kN, full-span load 4.8 kN/m, stiffness E=200 GPa and I=2.40×10^-4 m^4, the free-tip downward deflection is 751.200 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For deflection, energy and virtual work, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 5.4: Short lintel

For a linear elastic cantilever with length 4.5 m, end force 15 kN, full-span load 1.2 kN/m, stiffness E=200 GPa and I=2.60×10^-4 m^4, the free-tip downward deflection is 9.945 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For deflection, energy and virtual work, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 5.5: Beam with equipment

For a linear elastic cantilever with length 8.0 m, end force 24 kN, full-span load 0.9 kN/m, stiffness E=200 GPa and I=2.80×10^-4 m^4, the free-tip downward deflection is 81.371 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For deflection, energy and virtual work, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 5.6: Symmetric midspan

For a linear elastic cantilever with length 10.0 m, end force 45 kN, full-span load 2.4 kN/m, stiffness E=200 GPa and I=3.00×10^-4 m^4, the free-tip downward deflection is 300.000 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For deflection, energy and virtual work, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 5.7: Warehouse girder

For a linear elastic cantilever with length 7.5 m, end force 38 kN, full-span load 3.8 kN/m, stiffness E=200 GPa and I=3.20×10^-4 m^4, the free-tip downward deflection is 106.979 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For deflection, energy and virtual work, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Case 5.8: Roof beam

For a linear elastic cantilever with length 5.5 m, end force 19 kN, full-span load 2.1 kN/m, stiffness E=200 GPa and I=3.40×10^-4 m^4, the free-tip downward deflection is 19.028 mm. The point-force contribution scales as L cubed and the line-load contribution as L to the fourth. When the cross-section is changed, only the stiffness term changes in this idealized linear model. No allowance for shear deformation, cracking, creep or support rotation is included.

Verification challenge: Check the key result above against a limiting case in which one action is reduced to zero, or one geometric parameter is increased. For deflection, energy and virtual work, identify the response that changes most and state the governing physical mechanism. Do not use this case as a final capacity evaluation; it omits detailed component resistance and code-required factors.

Interactive experiments

Use these standalone HTML laboratories. They load without a web connection and export response curves in CSV format. Each lab shows a calculation formula and limitations.

• [Simulation 03: Elastic beam deflection laboratory](../labs/03-beam-deflection.html)
• [Simulation 08: Cantilever unit-load integration](../labs/08-cantilever-virtual.html)

Unit assessment

1. Explain the key idealization governing deflection, energy and virtual work, and describe one condition that would invalidate it.
2. Explain what mathematical or physical equilibrium, compatibility, or limiting-case checks are appropriate for this unit.
3. Compare the results of two parameter settings from a linked lab. Name the inputs, calculate one numerical difference and interpret its sign.
4. Provide one real-world detail that the model does not represent and specify how an engineer should investigate it.

Project brief

Prepare a two-page calculation note using the methods of deflection, energy and virtual work. The note must include a problem definition, a labeled free-body diagram, a parameter and unit table, equations, at least two worked numerical results, an independently evaluated check, a figure or CSV trace from a linked simulation, and a discussion of three limitations. The note is not a structural design certification.