Verification, Model Comparison and Reporting

Unit purpose

Verification is the bridge between doing algebra and producing a usable structural calculation. Checking more digits does not cure the wrong model. Reliable work combines conservation laws, compatibility, benchmark comparisons, numerical convergence and transparent documentation.

Field scenario

Build a mini verification plan for a local footbridge model: decide which assumptions need photographs or drawings, which numbers need independent calculations, and which uncertainties cannot be resolved by a spreadsheet alone.

Topics and worked reasoning

Topic 8.1: Equilibrium and compatibility residuals

Numerical answers must satisfy all equations that define the selected model, not merely look plausible. For a determinate beam compute vertical force and moment residuals after solving reactions. For a truss compute the vector residual at each joint, ideally scaled by a representative applied load. For a beam with imposed displacement compute the difference between predicted and prescribed support deflection. Large residuals may signal unit inconsistency, a sign mistake, an unconstrained degree of freedom or an implementation error.

Engineering practice: Create a verification table showing each residual, its normalization and an acceptance tolerance. Distinguish floating-point rounding, discretization error and modeling error. A zero residual cannot prove the assumed support behavior is physically correct; it proves only internal consistency for the specified equations.

Fully evaluated study example

To verify a beam of span 8.0 m carrying P=24 kN at x=5.5 m and w=0.9 kN/m, compute RA=11.100 kN and RB=20.100 kN independently from opposite moment balances. The applied vertical force total is 31.200 kN; the reaction total is 31.200 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

Reasoning and interpretation

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

Practice tasks

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

Applied investigation

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

Topic 8.2: Sensitivity and idealization limits

A structural output can be highly sensitive to certain model choices. In a simple beam, flexural deflection varies inversely with EI, but reaction distribution does not. In a propped cantilever with settlement, reaction distribution depends on stiffness and support movement. In a shallow arch, horizontal thrust varies inversely with rise. These parameter sensitivities are useful diagnostic tests and can reveal unrealistic designs or data errors before detailed analysis. Sensitivity to material stiffness and sensitivity to geometry are distinct questions.

Engineering practice: Perturb at least three independent inputs by a small percentage and report the relative response change. Identify a case where the response is nonlinear in the parameter and explain why a linear extrapolation is misleading. Make sure a sensitivity study retains consistent units, boundary conditions and load directions.

Fully evaluated study example

To verify a beam of span 10.0 m carrying P=45 kN at x=5.0 m and w=2.4 kN/m, compute RA=34.500 kN and RB=34.500 kN independently from opposite moment balances. The applied vertical force total is 69.000 kN; the reaction total is 69.000 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

Reasoning and interpretation

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

Practice tasks

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

Applied investigation

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

Topic 8.3: Closed-form versus numerical models

A numerical beam model approximates a continuous structure by discrete degrees of freedom and elements. For a simple linear beam with constant stiffness, exact element stiffness relations can reproduce certain nodal responses with very few elements, while recovering distributed-load curvature and local field response may require careful consistent loading and interpolation. A converged numerical result should be compared with a known closed-form benchmark where possible. Using a finer mesh without checking the model equations, boundary conditions or input units is not sufficient verification.

Engineering practice: Compare an integration-based deflection result with a numerical quadrature approximation or a stiffness-method calculation. Refine the numerical representation and plot error against number of divisions. State the reference solution, input values, convergence criterion and remaining discrepancies. A visually smooth plot should not be taken as evidence of numerical accuracy.

Fully evaluated study example

To verify a beam of span 7.5 m carrying P=38 kN at x=4.5 m and w=3.8 kN/m, compute RA=29.450 kN and RB=37.050 kN independently from opposite moment balances. The applied vertical force total is 66.500 kN; the reaction total is 66.500 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

Reasoning and interpretation

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

Practice tasks

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

Applied investigation

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

Topic 8.4: Engineering communication and reproducibility

A defensible structural calculation communicates geometry, load cases, sign conventions, units, support idealizations, material properties, method selection, checks and limitations. It separates model inputs from computed responses and design acceptance criteria. Versioned analytical scripts, traceable parameter values and explicit test cases make independent reproduction possible. This volume teaches preliminary elastic structural analysis rather than jurisdiction-specific design checking. Load factors, material resistance factors and detailing rules depend on authoritative codes and cannot be invented from the formulas in a teaching lab.

Engineering practice: Write a one-page calculation note for a hypothetical pedestrian bridge: define the question, assumptions, load transfer, beam or truss model, critical response, verification check and next required investigations. Add a limitation statement about omitted vibration, foundation behavior, fatigue or connection stiffness. A useful report presents uncertainty and scope rather than attaching false precision to a single number.

Fully evaluated study example

To verify a beam of span 5.5 m carrying P=19 kN at x=3.5 m and w=2.1 kN/m, compute RA=12.684 kN and RB=17.866 kN independently from opposite moment balances. The applied vertical force total is 30.550 kN; the reaction total is 30.550 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

Reasoning and interpretation

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

Practice tasks

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

Applied investigation

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

Deep dive and misconception repair

Verification clinic: use independent equations and independent models

Verification begins with a question about what could be wrong. A spreadsheet that gives a smooth graph may contain a sign reversal, a unit conversion mistake or an incorrect boundary condition. Global equilibrium residuals detect some of these defects, but not all. A simple-beam calculation with an incorrect point-load location can still satisfy equilibrium with respect to the incorrect input. Therefore verification has layers: first validate the geometry and load schedule, then check the analytical equations, then test the numerical implementation and finally compare response against observed physical plausibility.

Residuals should be normalized where useful. A vertical equilibrium difference of 0.001 kN is negligible for an example with 100 kN total applied load but may be significant for a highly sensitive micro-scale instrument. A relative residual divides the absolute imbalance by a meaningful characteristic load, while an absolute residual provides the physical units. Report both if decisions depend on tolerances. In joint-equilibrium analysis of a truss, check horizontal and vertical closure at every node to locate the specific joint where a sign or direction-cosine error appears.

Numerical convergence is a separate issue. A midpoint quadrature of a smooth virtual-work integrand should approach the analytic result as the segment count increases. An incorrect integrand may also converge, but to the wrong value. That is why a known exact benchmark is valuable. Conversely, if a model uses physically meaningful irregular geometry and no closed form is available, compare against a simpler limiting case and independently check global conservation.

Sensitivity without invented precision

A sensitivity study should identify whether uncertainty in geometry, stiffness, loading or supports dominates the answer. Small span errors can have an amplified effect on distributed-load deflection because span enters to the fourth power. The same stiffness uncertainty leaves a determinate beam reaction unchanged while altering displacement. Support settlement changes an indeterminate reaction in proportion to EI. These are mechanistic comparisons, not statistical uncertainty estimates; to quantify probability one would need justified distributions for uncertain inputs.

Engineering documentation is part of the calculation. A reproducible note states what coordinate system is used, what direction counts as positive, what force and length units are used, which loads act simultaneously, what restraints are applied and what idealizations are excluded. It provides the governing equations, numerical values and at least one independent closure check. If a program produces a plot and downloadable CSV data, record its version and input settings so the figures can be regenerated later.

The calculations in this book are for educational elastic analysis. They are not code design checks. A bridge, building or foundation requires an appropriate local standard, load combinations, material resistance models, serviceability criteria and competent engineering review. A report that omits this scope boundary risks inviting misuse. The strongest report is not the one that hides limitations; it is the one that states exactly what has and has not been established.

Multi-case calculations and validation

Case 8.1: Reaction shift

To verify a beam of span 6.0 m carrying P=28 kN at x=2.0 m and w=3.1 kN/m, compute RA=27.967 kN and RB=18.633 kN independently from opposite moment balances. The applied vertical force total is 46.600 kN; the reaction total is 46.600 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

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

Case 8.2: Longer girder

To verify a beam of span 9.0 m carrying P=34 kN at x=6.0 m and w=2.0 kN/m, compute RA=20.333 kN and RB=31.667 kN independently from opposite moment balances. The applied vertical force total is 52.000 kN; the reaction total is 52.000 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

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

Case 8.3: Footbridge loading

To verify a beam of span 12.0 m carrying P=41 kN at x=4.0 m and w=4.8 kN/m, compute RA=56.133 kN and RB=42.467 kN independently from opposite moment balances. The applied vertical force total is 98.600 kN; the reaction total is 98.600 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

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

Case 8.4: Short lintel

To verify a beam of span 4.5 m carrying P=15 kN at x=1.5 m and w=1.2 kN/m, compute RA=12.700 kN and RB=7.700 kN independently from opposite moment balances. The applied vertical force total is 20.400 kN; the reaction total is 20.400 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

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

Case 8.5: Beam with equipment

To verify a beam of span 8.0 m carrying P=24 kN at x=5.5 m and w=0.9 kN/m, compute RA=11.100 kN and RB=20.100 kN independently from opposite moment balances. The applied vertical force total is 31.200 kN; the reaction total is 31.200 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

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

Case 8.6: Symmetric midspan

To verify a beam of span 10.0 m carrying P=45 kN at x=5.0 m and w=2.4 kN/m, compute RA=34.500 kN and RB=34.500 kN independently from opposite moment balances. The applied vertical force total is 69.000 kN; the reaction total is 69.000 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

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

Case 8.7: Warehouse girder

To verify a beam of span 7.5 m carrying P=38 kN at x=4.5 m and w=3.8 kN/m, compute RA=29.450 kN and RB=37.050 kN independently from opposite moment balances. The applied vertical force total is 66.500 kN; the reaction total is 66.500 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

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

Case 8.8: Roof beam

To verify a beam of span 5.5 m carrying P=19 kN at x=3.5 m and w=2.1 kN/m, compute RA=12.684 kN and RB=17.866 kN independently from opposite moment balances. The applied vertical force total is 30.550 kN; the reaction total is 30.550 kN. Both totals agree. A second solver that uses the same incorrect load input may still pass this check, so geometry validation and an external load schedule remain essential.

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

Interactive experiments

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

• [Simulation 13: Quadrature accuracy and equilibrium](../labs/13-verification-convergence.html)

Unit assessment

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

Project brief

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