Virtual Work, Equilibrium Checks and Energy Ideas

Virtual work offers another route to equilibrium when a compatible small displacement field is easier to construct than several moment equations. For rigid systems, ideal internal constraint reactions do no work under compatible virtual motions. This method becomes central in structural analysis, deflection calculations and finite-element formulations, but at the introductory statics level the goal is to see why displacement compatibility and external balance cannot be separated. Engineering consistency checks also draw on work and energy ideas.

Engineering boundary: These exercises analyze idealized statics models for education. They do not certify structural strength, earthquake/wind resistance, geotechnical bearing, connection design, construction safety, or compliance with any jurisdiction’s building code. Real works require local standards, authoritative loading data and qualified professional checking.

Learning outcomes

• Formulate an explicit free-body model for virtual work, equilibrium checks and energy ideas.

• Use signed forces, consistent units and geometric measurements to construct equilibrium relations.

• Check results with an independently chosen equilibrium statement or limiting case.

• Distinguish what idealized statics establishes from what structural design and field verification must establish.

Lesson 7.1: Virtual displacements and work balance

A virtual displacement is a mathematically admissible infinitesimal change in geometry consistent with the ideal constraints; it is not necessarily a motion that actually occurs. At equilibrium, total virtual work of applied forces and couples vanishes for every admissible virtual displacement.

Governing relationships

For a force F through displacement δr, δW=F·δr. For a couple M through small rotation δθ, δW=Mδθ. Work is expressed in force×length units.

Worked engineering example

If an ideal lever has effort arm 2 m and load arm 0.5 m, compatible rotations give δxeffort/δxload=4; force balance by virtual work gives effort = load/4.

Model audit and common error

A real mechanism loses energy through friction and deformation; ideal virtual work does not predict efficiency.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving virtual displacements and work balance. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For virtual work, equilibrium checks and energy ideas, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 7.2: Lever, pulley and linkage equilibrium

Mechanical advantage is not a free multiplication of energy. A large displacement at a low-force point balances a smaller displacement at a high-force point in an ideal lossless model.

Governing relationships

For an ideal pivoted lever at equilibrium, F1a=F2b from moments or F1δx1=F2δx2 from virtual work.

Worked engineering example

An ideal lever with a=1.2 m, b=0.3 m and a 400 N resisting load needs F=400×0.3/1.2=100 N. The effort point moves four times farther.

Model audit and common error

Bearing friction, lever bending and out-of-plane eccentricity can invalidate the ideal result.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving lever, pulley and linkage equilibrium. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For virtual work, equilibrium checks and energy ideas, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 7.3: Energy and compatibility introduction

Rigid equilibrium sets net force and moment to zero, whereas deformable structural analysis also needs displacement compatibility and material behavior. The energy viewpoint is valuable precisely because it connects internal deformation and external work.

Governing relationships

For a linearly elastic spring, U=½kδ² and force F=kδ. The derivative dU/dδ=kδ recovers elastic force in a one-degree-of-freedom system.

Worked engineering example

A linear spring with k=5000 N/m stretched δ=0.02 m stores U=0.5×5000×0.02²=1 J and produces F=100 N.

Model audit and common error

A static force calculation does not prove that the system stays in the small-deformation regime.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving energy and compatibility introduction. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For virtual work, equilibrium checks and energy ideas, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 7.4: Independent equilibrium audits and limiting cases

Reliable structural calculations use independent balance checks and physical limiting cases. A result should reduce correctly when a load vanishes, moves to a support, or geometry becomes symmetric.

Governing relationships

For a beam with total downward load W and reactions RA,RB, check RA+RB=W, then choose a second origin and verify the moment sum; merely reproducing the same equation is not independent confirmation.

Worked engineering example

A 10 kN point force moving from beam midspan to the left support must cause its right-support reaction to approach zero, while the left-support reaction approaches 10 kN.

Model audit and common error

Beware of a numerical solver returning a finite answer for near-singular geometry without any warning.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving independent equilibrium audits and limiting cases. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For virtual work, equilibrium checks and energy ideas, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Integrated civil case study

Temporary jack and lever exercise: model the jack mechanism, compute required input effort by both moments and virtual work, and identify why friction, movement and stability would change the real outcome.

Worked problem studio

Each problem below is a separate numerical case. Work it first without looking at the answer, then compare the force and moment checks. Changing dimensions can reveal near-singular geometry or support reactions that switch direction.

Worked problem 7.1

Given: An ideal lever lifts a resisting 12.0 kN at short arm 0.25 m using an effort at 1.20 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.500 kN. Under the same tiny angular change, effort displacement is a/b=4.800 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.2

Given: An ideal lever lifts a resisting 14.0 kN at short arm 0.29 m using an effort at 1.40 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.900 kN. Under the same tiny angular change, effort displacement is a/b=4.828 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.3

Given: An ideal lever lifts a resisting 16.0 kN at short arm 0.33 m using an effort at 1.60 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=3.300 kN. Under the same tiny angular change, effort displacement is a/b=4.848 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.4

Given: An ideal lever lifts a resisting 18.0 kN at short arm 0.37 m using an effort at 1.80 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=3.700 kN. Under the same tiny angular change, effort displacement is a/b=4.865 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.5

Given: An ideal lever lifts a resisting 20.0 kN at short arm 0.41 m using an effort at 2.00 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=4.100 kN. Under the same tiny angular change, effort displacement is a/b=4.878 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.6

Given: An ideal lever lifts a resisting 22.0 kN at short arm 0.25 m using an effort at 2.20 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.500 kN. Under the same tiny angular change, effort displacement is a/b=8.800 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.7

Given: An ideal lever lifts a resisting 24.0 kN at short arm 0.29 m using an effort at 2.40 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.900 kN. Under the same tiny angular change, effort displacement is a/b=8.276 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.8

Given: An ideal lever lifts a resisting 26.0 kN at short arm 0.33 m using an effort at 2.60 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=3.300 kN. Under the same tiny angular change, effort displacement is a/b=7.879 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.9

Given: An ideal lever lifts a resisting 28.0 kN at short arm 0.37 m using an effort at 2.80 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=3.700 kN. Under the same tiny angular change, effort displacement is a/b=7.568 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.10

Given: An ideal lever lifts a resisting 30.0 kN at short arm 0.41 m using an effort at 3.00 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=4.100 kN. Under the same tiny angular change, effort displacement is a/b=7.317 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.11

Given: An ideal lever lifts a resisting 32.0 kN at short arm 0.25 m using an effort at 3.20 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.500 kN. Under the same tiny angular change, effort displacement is a/b=12.800 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.12

Given: An ideal lever lifts a resisting 34.0 kN at short arm 0.29 m using an effort at 3.40 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.900 kN. Under the same tiny angular change, effort displacement is a/b=11.724 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Unit assessment (with guidance)

Q1. State the virtual-work condition for a rigid system with ideal constraints.

Guidance: Use the governing equation and a labeled diagram from Unit 7; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q2. A lever has a 1.5 m effort arm and a 0.3 m load arm. What effort holds a 2 kN load?

Guidance: Use the governing equation and a labeled diagram from Unit 7; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q3. Explain how a lever can reduce force demand without providing free energy.

Guidance: Use the governing equation and a labeled diagram from Unit 7; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q4. Which additional relationships become necessary when a rigid statics model is replaced by an elastic structural model?

Guidance: Use the governing equation and a labeled diagram from Unit 7; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Interactive experiment

Open [simulation 7](../simulations/simulation-07.html) and vary at least three parameters. Create an observation table from the CSV. Describe one change for which the response is nonlinear, ill-conditioned or physically infeasible.

End-of-unit checklist

• External forces and reactions identified on isolated body.

• Dimensions, datums, load units and sign convention documented.

• Vertical/horizontal resultant and moment balance checked separately.

• Real-world limits and uncertainty recorded before any design interpretation.

Unit 7: Virtual work, equilibrium checks and energy ideas

Virtual work offers another route to equilibrium when a compatible small displacement field is easier to construct than several moment equations. For rigid systems, ideal internal constraint reactions do no work under compatible virtual motions. This method becomes central in structural analysis, deflection calculations and finite-element formulations, but at the introductory statics level the goal is to see why displacement compatibility and external balance cannot be separated. Engineering consistency checks also draw on work and energy ideas.

Engineering boundary: These exercises analyze idealized statics models for education. They do not certify structural strength, earthquake/wind resistance, geotechnical bearing, connection design, construction safety, or compliance with any jurisdiction’s building code. Real works require local standards, authoritative loading data and qualified professional checking.

Learning outcomes

• Formulate an explicit free-body model for virtual work, equilibrium checks and energy ideas.

• Use signed forces, consistent units and geometric measurements to construct equilibrium relations.

• Check results with an independently chosen equilibrium statement or limiting case.

• Distinguish what idealized statics establishes from what structural design and field verification must establish.

Lesson 7.1: Virtual displacements and work balance

A virtual displacement is a mathematically admissible infinitesimal change in geometry consistent with the ideal constraints; it is not necessarily a motion that actually occurs. At equilibrium, total virtual work of applied forces and couples vanishes for every admissible virtual displacement.

Governing relationships

For a force F through displacement δr, δW=F·δr. For a couple M through small rotation δθ, δW=Mδθ. Work is expressed in force×length units.

Worked engineering example

If an ideal lever has effort arm 2 m and load arm 0.5 m, compatible rotations give δxeffort/δxload=4; force balance by virtual work gives effort = load/4.

Model audit and common error

A real mechanism loses energy through friction and deformation; ideal virtual work does not predict efficiency.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving virtual displacements and work balance. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For virtual work, equilibrium checks and energy ideas, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 7.2: Lever, pulley and linkage equilibrium

Mechanical advantage is not a free multiplication of energy. A large displacement at a low-force point balances a smaller displacement at a high-force point in an ideal lossless model.

Governing relationships

For an ideal pivoted lever at equilibrium, F1a=F2b from moments or F1δx1=F2δx2 from virtual work.

Worked engineering example

An ideal lever with a=1.2 m, b=0.3 m and a 400 N resisting load needs F=400×0.3/1.2=100 N. The effort point moves four times farther.

Model audit and common error

Bearing friction, lever bending and out-of-plane eccentricity can invalidate the ideal result.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving lever, pulley and linkage equilibrium. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For virtual work, equilibrium checks and energy ideas, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 7.3: Energy and compatibility introduction

Rigid equilibrium sets net force and moment to zero, whereas deformable structural analysis also needs displacement compatibility and material behavior. The energy viewpoint is valuable precisely because it connects internal deformation and external work.

Governing relationships

For a linearly elastic spring, U=½kδ² and force F=kδ. The derivative dU/dδ=kδ recovers elastic force in a one-degree-of-freedom system.

Worked engineering example

A linear spring with k=5000 N/m stretched δ=0.02 m stores U=0.5×5000×0.02²=1 J and produces F=100 N.

Model audit and common error

A static force calculation does not prove that the system stays in the small-deformation regime.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving energy and compatibility introduction. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For virtual work, equilibrium checks and energy ideas, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 7.4: Independent equilibrium audits and limiting cases

Reliable structural calculations use independent balance checks and physical limiting cases. A result should reduce correctly when a load vanishes, moves to a support, or geometry becomes symmetric.

Governing relationships

For a beam with total downward load W and reactions RA,RB, check RA+RB=W, then choose a second origin and verify the moment sum; merely reproducing the same equation is not independent confirmation.

Worked engineering example

A 10 kN point force moving from beam midspan to the left support must cause its right-support reaction to approach zero, while the left-support reaction approaches 10 kN.

Model audit and common error

Beware of a numerical solver returning a finite answer for near-singular geometry without any warning.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving independent equilibrium audits and limiting cases. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For virtual work, equilibrium checks and energy ideas, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Integrated civil case study

Temporary jack and lever exercise: model the jack mechanism, compute required input effort by both moments and virtual work, and identify why friction, movement and stability would change the real outcome.

Worked problem studio

Each problem below is a separate numerical case. Work it first without looking at the answer, then compare the force and moment checks. Changing dimensions can reveal near-singular geometry or support reactions that switch direction.

Worked problem 7.1

Given: An ideal lever lifts a resisting 12.0 kN at short arm 0.25 m using an effort at 1.20 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.500 kN. Under the same tiny angular change, effort displacement is a/b=4.800 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.2

Given: An ideal lever lifts a resisting 14.0 kN at short arm 0.29 m using an effort at 1.40 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.900 kN. Under the same tiny angular change, effort displacement is a/b=4.828 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.3

Given: An ideal lever lifts a resisting 16.0 kN at short arm 0.33 m using an effort at 1.60 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=3.300 kN. Under the same tiny angular change, effort displacement is a/b=4.848 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.4

Given: An ideal lever lifts a resisting 18.0 kN at short arm 0.37 m using an effort at 1.80 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=3.700 kN. Under the same tiny angular change, effort displacement is a/b=4.865 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.5

Given: An ideal lever lifts a resisting 20.0 kN at short arm 0.41 m using an effort at 2.00 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=4.100 kN. Under the same tiny angular change, effort displacement is a/b=4.878 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.6

Given: An ideal lever lifts a resisting 22.0 kN at short arm 0.25 m using an effort at 2.20 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.500 kN. Under the same tiny angular change, effort displacement is a/b=8.800 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.7

Given: An ideal lever lifts a resisting 24.0 kN at short arm 0.29 m using an effort at 2.40 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.900 kN. Under the same tiny angular change, effort displacement is a/b=8.276 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.8

Given: An ideal lever lifts a resisting 26.0 kN at short arm 0.33 m using an effort at 2.60 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=3.300 kN. Under the same tiny angular change, effort displacement is a/b=7.879 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.9

Given: An ideal lever lifts a resisting 28.0 kN at short arm 0.37 m using an effort at 2.80 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=3.700 kN. Under the same tiny angular change, effort displacement is a/b=7.568 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.10

Given: An ideal lever lifts a resisting 30.0 kN at short arm 0.41 m using an effort at 3.00 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=4.100 kN. Under the same tiny angular change, effort displacement is a/b=7.317 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.11

Given: An ideal lever lifts a resisting 32.0 kN at short arm 0.25 m using an effort at 3.20 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.500 kN. Under the same tiny angular change, effort displacement is a/b=12.800 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 7.12

Given: An ideal lever lifts a resisting 34.0 kN at short arm 0.29 m using an effort at 3.40 m from the pivot. Determine effort and displacement ratio.

Solution: Moment or virtual-work balance gives E a=Q b, so E=2.900 kN. Under the same tiny angular change, effort displacement is a/b=11.724 times the load displacement. This assumes a rigid lever, ideal pivot and compatible motion.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Unit assessment (with guidance)

Q1. State the virtual-work condition for a rigid system with ideal constraints.

Guidance: Use the governing equation and a labeled diagram from Unit 7; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q2. A lever has a 1.5 m effort arm and a 0.3 m load arm. What effort holds a 2 kN load?

Guidance: Use the governing equation and a labeled diagram from Unit 7; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q3. Explain how a lever can reduce force demand without providing free energy.

Guidance: Use the governing equation and a labeled diagram from Unit 7; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q4. Which additional relationships become necessary when a rigid statics model is replaced by an elastic structural model?

Guidance: Use the governing equation and a labeled diagram from Unit 7; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Interactive experiment

Open [simulation 7](../simulations/simulation-07.html) and vary at least three parameters. Create an observation table from the CSV. Describe one change for which the response is nonlinear, ill-conditioned or physically infeasible.

End-of-unit checklist

• External forces and reactions identified on isolated body.

• Dimensions, datums, load units and sign convention documented.

• Vertical/horizontal resultant and moment balance checked separately.

• Real-world limits and uncertainty recorded before any design interpretation.

1. [Civil loading cases, construction stages and modeling uncertainty](units/unit-08.md)