Strain Energy and Impact Loading

Learning Objectives

  • Define strain energy, strain energy density, modulus of resilience, and modulus of toughness.
  • Calculate the impact factor and analyze the effects of impact loading on structures.
  • Apply Castigliano's Theorem and the Maxwell-Betti Reciprocal Theorem to calculate structural displacements.

In structural engineering, not all loads are applied slowly (static loads). When a load is applied suddenly or dropped from a height, it causes significant dynamic effects. To understand these effects, we must analyze Strain Energy—the energy absorbed by a material when it deforms.

Strain Energy

Strain Energy

Strain Energy (UU) is the internal potential energy stored in a structural member as a result of elastic deformation. It is equal to the work done by the external forces acting on the member.

Strain Energy (U)

For a member subjected to a static axial load PP that causes an elongation δ\delta, the load-deformation curve in the elastic region is a straight line. The work done (and thus the strain energy) is the area under this curve.

Strain Energy (Work Done)

Calculates the strain energy from the load-deformation curve.

U=12PδU = \frac{1}{2} P \delta

By substituting Hooke's Law (δ=PLAE\delta = \frac{PL}{AE}), we can express strain energy purely in terms of the applied load and member properties:

Strain Energy (Axial Load)

Expresses strain energy purely in terms of the applied load and member properties.

U=P2L2AEU = \frac{P^2 L}{2AE}

Alternatively, expressing it in terms of normal stress (σ=PA\sigma = \frac{P}{A}):

Strain Energy (Normal Stress)

Expresses strain energy in terms of normal stress.

U=σ2AL2E=σ2V2EU = \frac{\sigma^2 AL}{2E} = \frac{\sigma^2 V}{2E}

where V=ALV = AL is the total volume of the member.

Strain Energy Density

Strain energy density is the strain energy stored per unit volume of the material.

Strain Energy Density

Calculates the strain energy stored per unit volume of the material.

u=UV=σ22Eu = \frac{U}{V} = \frac{\sigma^2}{2E}

Modulus of Resilience (U_r)

Modulus of Resilience: The maximum strain energy density a material can absorb without undergoing permanent (plastic) deformation. It is calculated using the proportional limit (or yield stress, σY\sigma_Y).

Modulus of Resilience

Calculates the maximum strain energy density a material can absorb without undergoing permanent deformation.

Ur=σY22EU_r = \frac{\sigma_Y^2}{2E}

Modulus of Toughness

Modulus of Toughness: The total amount of strain energy density a material can absorb before catastrophic rupture. It represents the total area under the entire stress-strain curve (both elastic and plastic regions).

Resilience vs. Toughness in Steel

A material like high-strength steel has a high modulus of resilience but might be brittle. Structural steel (mild steel) has a very high modulus of toughness because of its massive plastic yielding region, making it excellent for resisting sudden shocks (like earthquakes).

Impact Loading

Impact Loading

Impact loading occurs when a moving mass strikes a structure, transferring its kinetic energy into the structure as strain energy. Because the energy must be absorbed over a very short deformation distance, the resulting stresses and forces are vastly higher than if the same mass was placed gently on the structure.

Impact Factor

The ratio of the maximum dynamic stress (or deflection) caused by an impact to the static stress (or deflection) that would be caused by the same weight resting quietly on the member.

Impact Factor (IF)

Consider a mass WW dropped from a height hh onto a collar at the end of a vertical rod. The total potential energy lost by the falling mass must equal the strain energy absorbed by the rod at its maximum elongation δmax\delta_{\text{max}}.

Energy Conservation in Impact

Equates the total potential energy lost by the falling mass to the strain energy absorbed.

W(h+δmax)=12PmaxδmaxW(h + \delta_{\text{max}}) = \frac{1}{2} P_{\text{max}} \delta_{\text{max}}

By solving the quadratic energy equation, the maximum dynamic deformation δmax\delta_{\text{max}} and maximum dynamic stress σmax\sigma_{\text{max}} are given by:

Maximum Dynamic Deformation

Calculates the maximum dynamic deformation caused by an impact load.

δmax=δst[1+1+2hδst]\delta_{\text{max}} = \delta_{\text{st}} \left[ 1 + \sqrt{1 + \frac{2h}{\delta_{\text{st}}}} \right]

Maximum Dynamic Stress

Calculates the maximum dynamic stress caused by an impact load.

σmax=σst[1+1+2hδst]\sigma_{\text{max}} = \sigma_{\text{st}} \left[ 1 + \sqrt{1 + \frac{2h}{\delta_{\text{st}}}} \right]

Variables

SymbolDescriptionUnit
δst=WLAE\delta_{\text{st}} = \frac{WL}{AE}(Static deflection)-
σst=WA\sigma_{\text{st}} = \frac{W}{A}(Static stress)-

The term in the brackets is the Impact Factor (IF):

Impact Factor (Axial)

Calculates the impact factor for an axial load.

IF=1+1+2hδstIF = 1 + \sqrt{1 + \frac{2h}{\delta_{\text{st}}}}

Suddenly Applied Loads

If a load is "suddenly applied" (meaning it is released from h=0h=0 without being lowered slowly), the equation simplifies to IF=1+1+0=2IF = 1 + \sqrt{1 + 0} = 2. Therefore, a suddenly applied load causes exactly twice the stress and deformation of a statically applied load!

Transverse Impact on Beams

Impact is not limited to axial loads. When a weight WW is dropped from a height hh onto a beam (transverse impact), the same energy conservation principles apply. The kinetic energy of the falling mass is converted into the strain energy of bending.

The Impact Factor (IF) formula remains identical to the axial case:

Impact Factor (Transverse)

Calculates the impact factor for a transverse impact on a beam.

IF=1+1+2hδstIF = 1 + \sqrt{1 + \frac{2h}{\delta_{\text{st}}}}

However, the static deflection (δst\delta_{\text{st}}) and static stress (σst\sigma_{\text{st}}) used in the formula must be calculated using beam bending formulas at the point of impact.

For example, if a weight WW is dropped on the center of a simply supported beam:

  • δst=WL348EI\delta_{\text{st}} = \frac{W L^3}{48 EI}
  • Mst=WL4M_{\text{st}} = \frac{W L}{4}
  • σst=MstcI\sigma_{\text{st}} = \frac{M_{\text{st}} c}{I}

The dynamic maximums are simply these static values multiplied by the Impact Factor:

  • δmax=IF×δst\delta_{\text{max}} = IF \times \delta_{\text{st}}
  • σmax=IF×σst\sigma_{\text{max}} = IF \times \sigma_{\text{st}}
Strain Energy and Impact Analysis Workflow

Choose the energy balance for gradual, sudden, or falling loading, then check whether the peak response stays within the elastic model.

The sudden and falling-load paths assume an ideal elastic member and the stated energy balance. A peak state outside the elastic or small-displacement assumptions is a hold: continue with an appropriate nonlinear or dynamic model rather than reporting the linear-elastic result as valid.

Strain Energy and Impact Analysis WorkflowChoose the energy balance for gradual, sudden, or falling loading, then check whether the peak response stays within the elastic model.. The sudden and falling-load paths assume an ideal elastic member and the stated energy balance. A peak state outside the elastic or small-displacement assumptions is a hold: continue with an appropriate nonlinear or dynamic model rather than reporting the linear-elastic result as valid.. Define member stiffness and loading history → Which loading history?; Which loading history? — Gradual → Use linear elastic U=½Pδ; Which loading history? — Sudden → Equate constant-load work to peak strain energy; Which loading history? — Falling → Include gravitational work W(h+δmax); Use linear elastic U=½Pδ → Solve peak elastic deformation / stress; Equate constant-load work to peak strain energy → Solve peak elastic deformation / stress; Include gravitational work W(h+δmax) → Solve peak elastic deformation / stress; Solve peak elastic deformation / stress → Peak state remains in model's elastic range?; Peak state remains in model's elastic range? — Yes → Report energy-consistent response; Peak state remains in model's elastic range? — No → Hold: use an appropriate nonlinear / dynamic model

Define member stiffness and loading history → Which loading history?; Which loading history? — Gradual → Use linear elastic U=½Pδ; Which loading history? — Sudden → Equate constant-load work to peak strain energy; Which loading history? — Falling → Include gravitational work W(h+δmax); Use linear elastic U=½Pδ → Solve peak elastic deformation / stress; Equate constant-load work to peak strain energy → Solve peak elastic deformation / stress; Include gravitational work W(h+δmax) → Solve peak elastic deformation / stress; Solve peak elastic deformation / stress → Peak state remains in model's elastic range?; Peak state remains in model's elastic range? — Yes → Report energy-consistent response; Peak state remains in model's elastic range? — No → Hold: use an appropriate nonlinear / dynamic model

  • Define member stiffness and loading history: terminator
  • Which loading history?: decision
  • Use linear elastic U=½Pδ: process
  • Equate constant-load work to peak strain energy: process
  • Include gravitational work W(h+δmax): process
  • Solve peak elastic deformation / stress: process
  • Peak state remains in model's elastic range?: decision
  • Hold: use an appropriate nonlinear / dynamic model: terminator
  • Report energy-consistent response: terminator

Energy Methods in Structural Analysis

Castigliano's Theorem

For a stable structure with small displacements and a linearly elastic response, Castigliano's second theorem gives the displacement or rotation associated with an applied generalized load. Differentiate the total strain energy with respect to that force or moment, with its direction defining the positive response direction.

Castigliano's Theorem (Displacement)

Calculates the displacement at any point in a linear elastic structure.

Δi=∂U∂Pi\Delta_i = \frac{\partial U}{\partial P_i}

Similarly, the partial derivative of the strain energy with respect to an applied moment MiM_i gives the rotation θi\theta_i at the point of application in the direction of MiM_i.

Castigliano's Theorem (Rotation)

Calculates the rotation at any point in a linear elastic structure.

θi=∂U∂Mi\theta_i = \frac{\partial U}{\partial M_i}

Strain Energy Formulations for Different Loadings:

  • Axial Loading: U=∫N22AEdxU = \int \frac{N^2}{2AE} dx
  • Bending: U=∫M22EIdxU = \int \frac{M^2}{2EI} dx
  • Torsion: U=∫T22GJdxU = \int \frac{T^2}{2GJ} dx
  • Transverse Shear: U=∫fsV22GAdxU = \int \frac{f_s V^2}{2GA} dx (where fsf_s is a form factor)

Dummy load method: If no real generalized load acts at the target point in the requested direction, introduce a dummy force QQ for translation or dummy couple MQM_Q for rotation. Keep the actual loads in the model, express the internal actions and total strain energy in terms of them and the dummy variable, differentiate first, then set only the dummy variable to zero. A real load in another direction does not replace the requested force or couple.

Castigliano Displacement with a Dummy Load When Needed

Use strain-energy differentiation to find a signed displacement or rotation in the requested direction. If no real generalized load acts at the target degree of freedom in that direction, introduce a dummy force or couple, differentiate first, and set the dummy load to zero afterward.

This lesson's method assumes a stable, linearly elastic structure with small displacements and a valid total strain-energy expression. Retain actual loads when adding a dummy load; an applied load in another direction does not remove the need for the requested direction's dummy. Resolve unknown reactions or redundants with the appropriate equilibrium and compatibility equations before evaluating U. Include axial, bending, torsional, and shear energy only where they are part of the adopted model. If the computed response leaves the stated linear-elastic range, stop and use an appropriate nonlinear/inelastic analysis.

Castigliano Displacement with a Dummy Load When NeededUse strain-energy differentiation to find a signed displacement or rotation in the requested direction. If no real generalized load acts at the target degree of freedom in that direction, introduce a dummy force or couple, differentiate first, and set the dummy load to zero afterward.. This lesson's method assumes a stable, linearly elastic structure with small displacements and a valid total strain-energy expression. Retain actual loads when adding a dummy load; an applied load in another direction does not remove the need for the requested direction's dummy. Resolve unknown reactions or redundants with the appropriate equilibrium and compatibility equations before evaluating U. Include axial, bending, torsional, and shear energy only where they are part of the adopted model. If the computed response leaves the stated linear-elastic range, stop and use an appropriate nonlinear/inelastic analysis.. Specify target point and signed displacement or rotation → Stable, linear-elastic small-displacement model?; Stable, linear-elastic small-displacement model? — No → Hold: select a method valid for this response regime; Stable, linear-elastic small-displacement model? — Yes → Are loads, reactions, properties and energy terms available?; Are loads, reactions, properties and energy terms available? — No → Hold: obtain data and resolve reactions or redundants; Are loads, reactions, properties and energy terms available? — Yes → Does a real generalized load act at target in that direction?; Does a real generalized load act at target in that direction? — Yes → Use that applied force or moment as the variable; Does a real generalized load act at target in that direction? — No → Add dummy force Q or couple M_Q in the target direction; Use that applied force or moment as the variable → Express internal actions in the real and dummy loads; Add dummy force Q or couple M_Q in the target direction → Express internal actions in the real and dummy loads; Express internal actions in the real and dummy loads → Assemble total U from applicable strain-energy terms; Assemble total U from applicable strain-energy terms → Differentiate U with respect to the target load variable; Differentiate U with respect to the target load variable → Was a dummy force or couple introduced?; Was a dummy force or couple introduced? — Yes → After differentiation, set dummy to zero; use actual real loads; Was a dummy force or couple introduced? — No → Evaluate derivative at the actual applied-load value; After differentiation, set dummy to zero; use actual real loads → Do stresses and displacements satisfy the model assumptions?; Evaluate derivative at the actual applied-load value → Do stresses and displacements satisfy the model assumptions?; Do stresses and displacements satisfy the model assumptions? — Yes → Report signed displacement or rotation with assumptions; Do stresses and displacements satisfy the model assumptions? — No → Hold: use a model that represents the material response

Specify target point and signed displacement or rotation → Stable, linear-elastic small-displacement model?; Stable, linear-elastic small-displacement model? — No → Hold: select a method valid for this response regime; Stable, linear-elastic small-displacement model? — Yes → Are loads, reactions, properties and energy terms available?; Are loads, reactions, properties and energy terms available? — No → Hold: obtain data and resolve reactions or redundants; Are loads, reactions, properties and energy terms available? — Yes → Does a real generalized load act at target in that direction?; Does a real generalized load act at target in that direction? — Yes → Use that applied force or moment as the variable; Does a real generalized load act at target in that direction? — No → Add dummy force Q or couple M_Q in the target direction; Use that applied force or moment as the variable → Express internal actions in the real and dummy loads; Add dummy force Q or couple M_Q in the target direction → Express internal actions in the real and dummy loads; Express internal actions in the real and dummy loads → Assemble total U from applicable strain-energy terms; Assemble total U from applicable strain-energy terms → Differentiate U with respect to the target load variable; Differentiate U with respect to the target load variable → Was a dummy force or couple introduced?; Was a dummy force or couple introduced? — Yes → After differentiation, set dummy to zero; use actual real loads; Was a dummy force or couple introduced? — No → Evaluate derivative at the actual applied-load value; After differentiation, set dummy to zero; use actual real loads → Do stresses and displacements satisfy the model assumptions?; Evaluate derivative at the actual applied-load value → Do stresses and displacements satisfy the model assumptions?; Do stresses and displacements satisfy the model assumptions? — Yes → Report signed displacement or rotation with assumptions; Do stresses and displacements satisfy the model assumptions? — No → Hold: use a model that represents the material response

  • Specify target point and signed displacement or rotation: terminator
  • Stable, linear-elastic small-displacement model?: decision
  • Hold: select a method valid for this response regime: terminator
  • Are loads, reactions, properties and energy terms available?: decision
  • Hold: obtain data and resolve reactions or redundants: terminator
  • Does a real generalized load act at target in that direction?: decision
  • Use that applied force or moment as the variable: process
  • Add dummy force Q or couple M_Q in the target direction: process
  • Express internal actions in the real and dummy loads: process
  • Assemble total U from applicable strain-energy terms: process
  • Differentiate U with respect to the target load variable: process
  • Was a dummy force or couple introduced?: decision
  • After differentiation, set dummy to zero; use actual real loads: process
  • Evaluate derivative at the actual applied-load value: process
  • Do stresses and displacements satisfy the model assumptions?: decision
  • Hold: use a model that represents the material response: terminator
  • Report signed displacement or rotation with assumptions: terminator

Maxwell-Betti Reciprocal Theorem

The Maxwell-Betti Reciprocal Theorem is one of the most elegant and fundamental principles derived from strain energy. It forms the basis of many structural analysis methods, including the Flexibility Method (Method of Consistent Deformations) and the formulation of stiffness matrices in Finite Element Analysis.

The Theorem: For any linear elastic structure subjected to two independent load sets (let's say a force PP applied at point A, and a force QQ applied at point B), the theorem states that the work done by the first set of loads acting through the displacements caused by the second set of loads is equal to the work done by the second set of loads acting through the displacements caused by the first set of loads.

In its simplest form for two point loads:

Maxwell-Betti Reciprocal Theorem

Relates the work done by two independent load sets on a linear elastic structure.

P⋅ΔAB=Q⋅ΔBAP \cdot \Delta_{AB} = Q \cdot \Delta_{BA}

Variables

SymbolDescriptionUnit
ΔAB\Delta_{AB}the displacement at point A caused *only* by the load QQ applied at point B.-
ΔBA\Delta_{BA}the displacement at point B caused *only* by the load PP applied at point A.-

Key Consequence: If you apply a unit load (P=1P=1, Q=1Q=1), the theorem proves that the influence coefficients are symmetric. The deflection at A due to a load at B is exactly equal to the deflection at B due to a load at A (δAB=δBA\delta_{AB} = \delta_{BA}).

Interactive Tool: Strain Energy Simulator

Interact with the Strain Energy Simulator below to visualize these concepts in action.

Strain Energy and Impact Loading

Concept and model scope

Compare a gradually applied load, a suddenly applied weight, and a weight dropped through a gap onto an axial elastic member. Stiffness is AE/L. Gradual work is one-half Wδ; sudden/drop work is W(h+δ). A suddenly applied load is the zero-drop-height dynamic case, not gradual loading. The model neglects bar mass, damping, contact losses, stress waves and plasticity; the weight stays engaged through maximum displacement. All plotted values are ideal elastic trial responses. The 250 MPa yield reference is a teaching assumption.

Controls

Loading history

Controls external work and peak response. Sudden loading produces twice the static displacement in this ideal undamped model.

Applied load or weight

Static force W, also the falling weight in the dynamic cases. It is a force, not a mass in kilograms.

Range: 0.0–10.0 kN. Step: 0.1 kN.

2.0 kN

Member length

Original axial member length. Increasing it reduces stiffness AE/L.

Range: 0.5–5.0 m. Step: 0.1 m.

2.0 m

Cross-sectional area

Uniform area resisting axial force; affects stiffness and stress.

Range: 100–2000 mm². Step: 50 mm².

500 mm²

Elastic modulus

Linear elastic axial stiffness. Yield behavior is excluded.

Range: 20–210 GPa. Step: 5 GPa.

200 GPa
0.000.00.02401.20.04802.4Elastic resisting force (kN) · shaded area = stored energyExtension δ (mm)
Static extension
0.0400 mm
Peak elastic trial extension
0.0400 mm
Peak elastic trial force
2.000 kN
Peak elastic trial stress
4.00 MPa
Stored strain energy
0.040 J
External work
0.040 J
Dynamic / static response
1.000 ×

Within the stated teaching yield reference; ideal elastic energy balance only.

Compare loading histories

The tool distinguishes gradual loading from sudden loading and a dropped weight. At zero drop height, the dynamic response is the sudden-load case, with twice the static extension in the ideal undamped model. Compare stored strain energy with external work and check the displayed elastic-limit warning before interpreting a trial result.

Mechanics Visual Synthesis

Use these deterministic figures to connect the lesson equations to the corresponding free-body, section, transformation, or response geometry. Each visual addresses a separate mechanics question and is intentionally static so it can be compared directly with the interactive model above.

Strain energy as area under load–deformation response

For a linear elastic member, stored strain energy is the triangular area under the force–deformation line and equals one half P times delta.

Strain energy as area under load–deformation responseFor a linear elastic member, stored strain energy is the triangular area under the force–deformation line and equals one half P times delta.PδU = ½Pδ

Gradual and suddenly applied load energy balance

The ideal undamped sudden-load case must store the work of a constant applied load, producing a peak deformation twice the static deformation.

Gradual and suddenly applied load energy balanceThe ideal undamped sudden-load case must store the work of a constant applied load, producing a peak deformation twice the static deformation.gradual: δstsudden: 2δstenergy balance

Falling-weight impact energy

A dropped weight contributes gravitational work through the drop height plus the member deformation, which must equal the peak elastic strain energy in the ideal model.

Falling-weight impact energyA dropped weight contributes gravitational work through the drop height plus the member deformation, which must equal the peak elastic strain energy in the ideal model.W(h+δmax)U(δmax)solve peak response

Castigliano displacement from strain-energy sensitivity

For a linearly elastic structure, the displacement associated with an applied generalized load follows from the partial derivative of total strain energy with respect to that load.

Castigliano displacement from strain-energy sensitivityFor a linearly elastic structure, the displacement associated with an applied generalized load follows from the partial derivative of total strain energy with respect to that load.U(P)∂U/∂PδP = ∂U/∂P
Key Takeaways
  • Strain Energy (UU) is the potential energy stored internally in a member due to elastic deformation. It equals the area under the load-deformation curve.
  • Modulus of Resilience measures the energy absorption capacity up to the elastic limit, while Modulus of Toughness measures the total energy absorption up to rupture.
  • Impact Factor (IF) quantifies the severe multiplying effect of dynamic loads. A suddenly applied load (zero drop height) creates exactly double the static stress and deformation.
  • Transverse Impact: Works identically to axial impact, provided the static deflection (δst\delta_{\text{st}}) and stress (σst\sigma_{\text{st}}) are calculated using beam bending theory.
  • At fixed static stress and drop height in the ideal impact model, a larger static deflection lowers the impact factor; changes in section or material can also change the static stress, so compare the full response expression.
  • Castigliano's Theorem: For the stated stable, small-displacement, linear-elastic model, differentiate total strain energy with respect to the generalized force or moment acting in the target direction.
  • Dummy Load Method: When no real generalized load acts in the requested direction, retain real loads, add a force or couple dummy, differentiate, and only then set that dummy to zero.
  • Maxwell-Betti Reciprocal Theorem: A foundational principle stating that the work done by Load A moving through the displacement caused by Load B equals the work done by Load B moving through the displacement caused by Load A (P⋅ΔAB=Q⋅ΔBAP \cdot \Delta_{AB} = Q \cdot \Delta_{BA}).

References