Displacement Methods and Moment Distribution

Conceptual overview

A displacement method chooses rotations and translations as unknowns and computes restoring member forces from these movements. Instead of releasing redundants, one imagines restraints added to prevent joint movement and then releases them by satisfying force and moment equilibrium. Slope–deflection, moment distribution and matrix stiffness all express the same linear mechanics using different bookkeeping. In a non-sway continuous beam, support vertical displacement may be prescribed, so joint rotations become the principal unknowns. In a sway frame, lateral translations must also be retained. The stiffness coefficient for an Euler–Bernoulli prismatic member is dependent on its far-end condition; a far-end fixed member exhibits rotational stiffness 4EI/L, while a far-end hinged member exhibits 3EI/L. These coefficients do not apply to a short deep beam when shear flexibility is significant.

Learning outcomes

After working through this unit you should be able to: (1) construct a physically consistent idealization, (2) express the relevant force, stiffness and displacement variables using compatible units, (3) solve a demonstrative problem by hand or a transparent numerical method, (4) interpret response patterns and their sensitivity to input parameters, and (5) identify what the simplified model cannot prove. In this unit the most important observation is not merely a numerical answer but the connection between structural behavior and the assumed support or constitutive conditions.

Unit engineering case — Two-span classroom girder

A classroom floor girder crosses one interior column and carries different distributed live loads on each span. Estimate support moments by slope–deflection and by the direct stiffness method. Explain any sign differences from end-action conventions. Test three load patterns: full-span dead load, live load on the left span only and live load on the right span only. For each pattern, record joint equilibrium, total support reaction and a deformation sketch. Use at least two stiffness values to illustrate that under linear, uniformly scaled EI, moment distribution depends more on relative EI than absolute EI.

Investigation sequence

Begin by preparing a diagram that names every member, support, load, imposed movement and unknown. Choose an independent coordinate system and assign a sign to every generalized force. Derive a minimal reference calculation before using a computational model. Then run a sensitivity study in which only one physical assumption changes at a time. Record both the predicted movement and the reaction or internal-force counterpart. Check conservation of force and moment where the physical model supports those checks. Finally compare the calculation with a plausible measurement or with a more complete independent model. Label the evidence as a verification benchmark, an assumption or a genuinely observed fact.

Suggested written report

A useful report has a one-paragraph engineering question, a table of parameter values with units, an explicit statement of boundary conditions, the governing equations, a readable figure of the system, a table of numerical results, at least one independent verification, and a conclusion that identifies unverified assumptions. State what happens when the governing input doubles, when stiffness increases, and when a support is relaxed or stiffened. Explain whether the observed changes follow from algebraic scaling or from a change in load path. Keep the original units in the file even if a graph displays a converted unit.

2.1 Slope–deflection equations

Learning target and physical interpretation

Relate a beam end moment to both end rotations, chord rotation from differential support displacement, and moments fixed by member loading. Use a clearly stated sign convention throughout the calculation.

Governing relation

For a prismatic beam without chord rotation: M_ab= (2EI/L)(2θa+θb)+FEM_ab.

Worked engineering interpretation

A joint between two fixed-ended spans is allowed to rotate. Its unknown θ enters both adjacent member end moments, and ΣMjoint=0 completes the analysis.

What the idealization omits

Copying fixed-end moment signs from a different convention without transforming them can make a physically symmetric loading condition produce asymmetric moments.

Verification and interpretation of results

Solve the joint balance, substitute θ into both member formulas and check equal-and-opposite member action at the joint.

Applied case and decision boundary

Interpret the negative support bending that governs reinforcement detailing at an interior concrete beam support. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Copying fixed-end moment signs from a different convention without transforming them can make a physically symmetric loading condition produce asymmetric moments. Explain which physical check exposes the error.

5. Professional judgment: Using the case Interpret the negative support bending that governs reinforcement detailing at an interior concrete beam support., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

2.2 Moment distribution factors

Learning target and physical interpretation

Distribute an unbalanced joint moment in proportion to the rotational stiffnesses of connected members. Carry-over transmits part of the correction to a restrained far end.

Governing relation

Distribution factor DF_i=k_i/Σ_j k_j, with ΣDF=1; fixed-far-end k=4EI/L.

Worked engineering interpretation

If members entering a joint have stiffnesses 4 and 6 MN·m/rad, a 50 kN·m balancing action distributes 20 and 30 kN·m with the corrective sign.

What the idealization omits

Using a fixed-end carry-over factor of one-half for a member whose far end is released can corrupt the balancing sequence.

Verification and interpretation of results

At the end of each sweep, record joint residual moments; repeated sweeps should reduce the residual for a stable linear structure.

Applied case and decision boundary

Model an unequal-span elevated walkway subjected to gravity loading. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Using a fixed-end carry-over factor of one-half for a member whose far end is released can corrupt the balancing sequence. Explain which physical check exposes the error.

5. Professional judgment: Using the case Model an unequal-span elevated walkway subjected to gravity loading., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

2.3 Continuous beam stiffness assembly

Learning target and physical interpretation

A two-node Euler–Bernoulli beam carries transverse translation and rotation at each end. Assemble shared DOFs by summing local matrices, then restrain the supported vertical DOFs while retaining joint rotation.

Governing relation

k_e=(EI/L³)[[12,6L,−12,6L],[6L,4L²,−6L,2L²],[−12,−6L,12,−6L],[6L,2L²,−6L,4L²]].

Worked engineering interpretation

For two spans of 5 m and 7 m, shared vertical and rotation DOFs at the interior support make a six-DOF system before boundary conditions.

What the idealization omits

Treating the interior support as a hinge by accidentally duplicating its rotation eliminates continuity and produces incorrect negative moments.

Verification and interpretation of results

Check matrix symmetry, reaction sum against total UDL, and that interior end moments satisfy joint equilibrium.

Applied case and decision boundary

Use Lab 03 to compare unequal UDLs on a bridge deck strip. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Treating the interior support as a hinge by accidentally duplicating its rotation eliminates continuity and produces incorrect negative moments. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 03 to compare unequal UDLs on a bridge deck strip., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

2.4 Fixed-end loads and consistent nodal forces

Learning target and physical interpretation

A distributed member load cannot be replaced by arbitrary nodal forces without changing equilibrium or deformation response. Consistent nodal loads are obtained by the same interpolation functions used for the displacement field.

Governing relation

For a downward constant UDL w on an upward-positive beam DOF system, f_e=[−wL/2,−wL²/12,−wL/2,+wL²/12].

Worked engineering interpretation

A 10 kN/m load over 6 m contributes −30 kN at each end and fixed-end generalized nodal moments of magnitude 30 kN·m.

What the idealization omits

Forgetting the equivalent nodal moments causes the rotations to be wrong, even if the total applied vertical force remains correct.

Verification and interpretation of results

Sum all equivalent nodal forces and moments about an arbitrary datum and confirm that they reproduce the distributed load.

Applied case and decision boundary

Explain why a mesh convergence check is necessary for pointwise internal-force plots. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Forgetting the equivalent nodal moments causes the rotations to be wrong, even if the total applied vertical force remains correct. Explain which physical check exposes the error.

5. Professional judgment: Using the case Explain why a mesh convergence check is necessary for pointwise internal-force plots., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

2.5 Comparison of hand and matrix methods

Learning target and physical interpretation

Hand calculations expose mechanics while matrix assembly scales to numerous joints. Compare moment distribution with direct stiffness on the same ideal model, not merely similarly named structures.

Governing relation

Kθ θ=−M_fixed for a rotationally restrained two-span beam with one free interior rotation.

Worked engineering interpretation

For left and right spans fixed at outer ends, the unknown central rotation is θB=−(FEM_B,left+FEM_B,right)/(4EI/L1+4EI/L2).

What the idealization omits

Different end-moment sign conventions can produce numbers with opposite signs that are physically equivalent; draw the action pairs before concluding a disagreement.

Verification and interpretation of results

Check that the moment distribution correction and the matrix θB give the same final joint end moments to the selected numerical tolerance.

Applied case and decision boundary

Use Lab 04 as an auditable demonstration before adopting a larger frame model. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Different end-moment sign conventions can produce numbers with opposite signs that are physically equivalent; draw the action pairs before concluding a disagreement. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 04 as an auditable demonstration before adopting a larger frame model., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

End-of-unit self-check

1. Which unknown in the five lesson models is governed principally by equilibrium, and which requires a deformation condition? Explain for two different examples.
2. Identify a boundary-condition assumption in this unit that could change if a real bearing loses contact or a joint begins to slip.
3. Form a dimensionless or unit-consistent ratio from a model response and a characteristic load or span, then explain how that ratio changes with structural stiffness.
4. Explain how a model could satisfy overall force balance and nevertheless predict the wrong deformation field.
5. Design a two-point numerical verification plan that does not reuse the same solver for both checks.
6. Identify the limitation of treating code design or material yielding as if it were included in these ideal elastic analyses.

Short answer guidance: A complete answer should use a named force path or deformation coordinate rather than generic phrases. A verification answer should specify the numerical quantity being compared and the expected reference; a validation answer should identify a relevant observation or tested physical condition.

Unit 2 — Displacement methods and moment distribution

Conceptual overview

A displacement method chooses rotations and translations as unknowns and computes restoring member forces from these movements. Instead of releasing redundants, one imagines restraints added to prevent joint movement and then releases them by satisfying force and moment equilibrium. Slope–deflection, moment distribution and matrix stiffness all express the same linear mechanics using different bookkeeping. In a non-sway continuous beam, support vertical displacement may be prescribed, so joint rotations become the principal unknowns. In a sway frame, lateral translations must also be retained. The stiffness coefficient for an Euler–Bernoulli prismatic member is dependent on its far-end condition; a far-end fixed member exhibits rotational stiffness 4EI/L, while a far-end hinged member exhibits 3EI/L. These coefficients do not apply to a short deep beam when shear flexibility is significant.

Learning outcomes

After working through this unit you should be able to: (1) construct a physically consistent idealization, (2) express the relevant force, stiffness and displacement variables using compatible units, (3) solve a demonstrative problem by hand or a transparent numerical method, (4) interpret response patterns and their sensitivity to input parameters, and (5) identify what the simplified model cannot prove. In this unit the most important observation is not merely a numerical answer but the connection between structural behavior and the assumed support or constitutive conditions.

Unit engineering case — Two-span classroom girder

A classroom floor girder crosses one interior column and carries different distributed live loads on each span. Estimate support moments by slope–deflection and by the direct stiffness method. Explain any sign differences from end-action conventions. Test three load patterns: full-span dead load, live load on the left span only and live load on the right span only. For each pattern, record joint equilibrium, total support reaction and a deformation sketch. Use at least two stiffness values to illustrate that under linear, uniformly scaled EI, moment distribution depends more on relative EI than absolute EI.

Investigation sequence

Begin by preparing a diagram that names every member, support, load, imposed movement and unknown. Choose an independent coordinate system and assign a sign to every generalized force. Derive a minimal reference calculation before using a computational model. Then run a sensitivity study in which only one physical assumption changes at a time. Record both the predicted movement and the reaction or internal-force counterpart. Check conservation of force and moment where the physical model supports those checks. Finally compare the calculation with a plausible measurement or with a more complete independent model. Label the evidence as a verification benchmark, an assumption or a genuinely observed fact.

Suggested written report

A useful report has a one-paragraph engineering question, a table of parameter values with units, an explicit statement of boundary conditions, the governing equations, a readable figure of the system, a table of numerical results, at least one independent verification, and a conclusion that identifies unverified assumptions. State what happens when the governing input doubles, when stiffness increases, and when a support is relaxed or stiffened. Explain whether the observed changes follow from algebraic scaling or from a change in load path. Keep the original units in the file even if a graph displays a converted unit.

2.1 Slope–deflection equations

Learning target and physical interpretation

Relate a beam end moment to both end rotations, chord rotation from differential support displacement, and moments fixed by member loading. Use a clearly stated sign convention throughout the calculation.

Governing relation

For a prismatic beam without chord rotation: M_ab= (2EI/L)(2θa+θb)+FEM_ab.

Worked engineering interpretation

A joint between two fixed-ended spans is allowed to rotate. Its unknown θ enters both adjacent member end moments, and ΣMjoint=0 completes the analysis.

What the idealization omits

Copying fixed-end moment signs from a different convention without transforming them can make a physically symmetric loading condition produce asymmetric moments.

Verification and interpretation of results

Solve the joint balance, substitute θ into both member formulas and check equal-and-opposite member action at the joint.

Applied case and decision boundary

Interpret the negative support bending that governs reinforcement detailing at an interior concrete beam support. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Copying fixed-end moment signs from a different convention without transforming them can make a physically symmetric loading condition produce asymmetric moments. Explain which physical check exposes the error.

5. Professional judgment: Using the case Interpret the negative support bending that governs reinforcement detailing at an interior concrete beam support., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

2.2 Moment distribution factors

Learning target and physical interpretation

Distribute an unbalanced joint moment in proportion to the rotational stiffnesses of connected members. Carry-over transmits part of the correction to a restrained far end.

Governing relation

Distribution factor DF_i=k_i/Σ_j k_j, with ΣDF=1; fixed-far-end k=4EI/L.

Worked engineering interpretation

If members entering a joint have stiffnesses 4 and 6 MN·m/rad, a 50 kN·m balancing action distributes 20 and 30 kN·m with the corrective sign.

What the idealization omits

Using a fixed-end carry-over factor of one-half for a member whose far end is released can corrupt the balancing sequence.

Verification and interpretation of results

At the end of each sweep, record joint residual moments; repeated sweeps should reduce the residual for a stable linear structure.

Applied case and decision boundary

Model an unequal-span elevated walkway subjected to gravity loading. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Using a fixed-end carry-over factor of one-half for a member whose far end is released can corrupt the balancing sequence. Explain which physical check exposes the error.

5. Professional judgment: Using the case Model an unequal-span elevated walkway subjected to gravity loading., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

2.3 Continuous beam stiffness assembly

Learning target and physical interpretation

A two-node Euler–Bernoulli beam carries transverse translation and rotation at each end. Assemble shared DOFs by summing local matrices, then restrain the supported vertical DOFs while retaining joint rotation.

Governing relation

k_e=(EI/L³)[[12,6L,−12,6L],[6L,4L²,−6L,2L²],[−12,−6L,12,−6L],[6L,2L²,−6L,4L²]].

Worked engineering interpretation

For two spans of 5 m and 7 m, shared vertical and rotation DOFs at the interior support make a six-DOF system before boundary conditions.

What the idealization omits

Treating the interior support as a hinge by accidentally duplicating its rotation eliminates continuity and produces incorrect negative moments.

Verification and interpretation of results

Check matrix symmetry, reaction sum against total UDL, and that interior end moments satisfy joint equilibrium.

Applied case and decision boundary

Use Lab 03 to compare unequal UDLs on a bridge deck strip. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Treating the interior support as a hinge by accidentally duplicating its rotation eliminates continuity and produces incorrect negative moments. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 03 to compare unequal UDLs on a bridge deck strip., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

2.4 Fixed-end loads and consistent nodal forces

Learning target and physical interpretation

A distributed member load cannot be replaced by arbitrary nodal forces without changing equilibrium or deformation response. Consistent nodal loads are obtained by the same interpolation functions used for the displacement field.

Governing relation

For a downward constant UDL w on an upward-positive beam DOF system, f_e=[−wL/2,−wL²/12,−wL/2,+wL²/12].

Worked engineering interpretation

A 10 kN/m load over 6 m contributes −30 kN at each end and fixed-end generalized nodal moments of magnitude 30 kN·m.

What the idealization omits

Forgetting the equivalent nodal moments causes the rotations to be wrong, even if the total applied vertical force remains correct.

Verification and interpretation of results

Sum all equivalent nodal forces and moments about an arbitrary datum and confirm that they reproduce the distributed load.

Applied case and decision boundary

Explain why a mesh convergence check is necessary for pointwise internal-force plots. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Forgetting the equivalent nodal moments causes the rotations to be wrong, even if the total applied vertical force remains correct. Explain which physical check exposes the error.

5. Professional judgment: Using the case Explain why a mesh convergence check is necessary for pointwise internal-force plots., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

2.5 Comparison of hand and matrix methods

Learning target and physical interpretation

Hand calculations expose mechanics while matrix assembly scales to numerous joints. Compare moment distribution with direct stiffness on the same ideal model, not merely similarly named structures.

Governing relation

Kθ θ=−M_fixed for a rotationally restrained two-span beam with one free interior rotation.

Worked engineering interpretation

For left and right spans fixed at outer ends, the unknown central rotation is θB=−(FEM_B,left+FEM_B,right)/(4EI/L1+4EI/L2).

What the idealization omits

Different end-moment sign conventions can produce numbers with opposite signs that are physically equivalent; draw the action pairs before concluding a disagreement.

Verification and interpretation of results

Check that the moment distribution correction and the matrix θB give the same final joint end moments to the selected numerical tolerance.

Applied case and decision boundary

Use Lab 04 as an auditable demonstration before adopting a larger frame model. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Different end-moment sign conventions can produce numbers with opposite signs that are physically equivalent; draw the action pairs before concluding a disagreement. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 04 as an auditable demonstration before adopting a larger frame model., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

End-of-unit self-check

1. Which unknown in the five lesson models is governed principally by equilibrium, and which requires a deformation condition? Explain for two different examples.
2. Identify a boundary-condition assumption in this unit that could change if a real bearing loses contact or a joint begins to slip.
3. Form a dimensionless or unit-consistent ratio from a model response and a characteristic load or span, then explain how that ratio changes with structural stiffness.
4. Explain how a model could satisfy overall force balance and nevertheless predict the wrong deformation field.
5. Design a two-point numerical verification plan that does not reuse the same solver for both checks.
6. Identify the limitation of treating code design or material yielding as if it were included in these ideal elastic analyses.

Short answer guidance: A complete answer should use a named force path or deformation coordinate rather than generic phrases. A verification answer should specify the numerical quantity being compared and the expected reference; a validation answer should identify a relevant observation or tested physical condition.