Settlement, Thermal Actions and Imposed Strains

Conceptual overview

Applied loads are not the only source of internal force. A continuous beam may develop bending when a support moves, even when nobody places a load on the beam. A restrained bar may become compressive after heating or tensile after cooling. Construction stage, shrinkage, creep and bearing rotation can also introduce imposed changes. In analysis, a specified movement is a boundary condition; it belongs in the displacement vector and therefore changes the right-hand side of the reduced stiffness equations. Imposed strain has a different mathematical representation from mechanical force, although the two may be combined at the level of equivalent nodal actions. The engineering consequence is that reaction and moment signs may reverse as boundary movement changes.

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 — Bridge bearing replacement

During bearing replacement, the central support of a two-span steel beam is temporarily lowered by several millimeters. Model settlement as prescribed displacement, not an external force. Determine the change in support reactions and end moments, and contrast the result with the unchanged-support baseline. Identify which assumptions are violated if the bearing loses contact and the support reaction cannot become tensile. Such a unilateral contact problem requires updated constraints and is not solved by simply accepting a negative bearing reaction from a fixed-support linear model.

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.

4.1 Support settlement as prescribed movement

Learning target and physical interpretation

A settlement at a support is a known vertical displacement, not an unknown load. Solve unknown rotations and member actions while imposing the settlement in the constrained DOF.

Governing relation

K_ff u_f=F_f−K_fc u_c with u_c including the nonzero settlement.

Worked engineering interpretation

A 10 mm downward shift at a middle support of a two-span continuous beam creates end moments even with zero externally imposed UDL.

What the idealization omits

Forcing the settled support displacement to zero and adding an arbitrary point force does not enforce the correct kinematic condition.

Verification and interpretation of results

Compute reactions and confirm total vertical reaction remains zero when all external vertical loads are zero and only settlement is present.

Applied case and decision boundary

Use Lab 07 to evaluate load reversal during bearing replacement. 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: Forcing the settled support displacement to zero and adding an arbitrary point force does not enforce the correct kinematic condition. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 07 to evaluate load reversal during bearing replacement., 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.

4.2 Lack-of-fit in axial members

Learning target and physical interpretation

A fabricated bar may be longer or shorter than the gap between its supports. Assembly then requires either movement of supports or internal stress. Lack-of-fit can be modeled as an initial strain or equivalent displacement.

Governing relation

N=(Δ_actual−Δ_free)/(ΣL/(EA)) for members in series where Δ_free includes thermal and fabrication effects.

Worked engineering interpretation

If an immovable two-bar system would freely lengthen by 1 mm but is confined to a fixed 0 mm net length change, it carries a uniform compressive force proportional to 1 mm divided by axial flexibility.

What the idealization omits

Assuming the stress is the same in sections of unequal area is wrong; axial force is constant in ideal series bars but σ_i=N/A_i varies.

Verification and interpretation of results

Check the sum of mechanical plus thermal extensions equals the actual support-to-support displacement.

Applied case and decision boundary

Explain temperature-induced bearing forces at a restrained bridge expansion joint. 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: Assuming the stress is the same in sections of unequal area is wrong; axial force is constant in ideal series bars but σ_i=N/A_i varies. Explain which physical check exposes the error.

5. Professional judgment: Using the case Explain temperature-induced bearing forces at a restrained bridge expansion joint., 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.

4.3 Thermal strain and stiffness contrast

Learning target and physical interpretation

Thermal free strain is approximately αΔT for a uniform isotropic segment. E, A, α and segment length each influence force under restraint, and all four must be expressed in coherent units.

Governing relation

Δ_free=Σ α_i ΔT_i L_i; N=−Δ_free/(Σ L_i/(E_i A_i)) for fixed overall length.

Worked engineering interpretation

A 3 m steel segment with α=12×10⁻⁶/K warmed 50°C would lengthen 1.8 mm if unrestrained, before accounting for other segments.

What the idealization omits

Thermal force does not exist in an unconstrained prismatic bar under uniform temperature change, apart from secondary effects such as gradients or incompatible connections.

Verification and interpretation of results

Vary heating while comparing a fixed-fixed series bar against an unconstrained counterpart; ensure force vanishes when Δ_free=0.

Applied case and decision boundary

Use Lab 08 to contrast steel and aluminum segments in a restrained service 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: Thermal force does not exist in an unconstrained prismatic bar under uniform temperature change, apart from secondary effects such as gradients or incompatible connections. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 08 to contrast steel and aluminum segments in a restrained service 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.

4.4 Temperature gradients in beams

Learning target and physical interpretation

A through-depth temperature difference can cause curvature; a uniform temperature increase principally changes length. The modeling choice depends on whether axial and rotational restraint permit free motion.

Governing relation

For a linear thermal strain gradient through depth h, unconstrained thermal curvature κ≈αΔT_top-bottom/h, with sign depending on axis orientation.

Worked engineering interpretation

A concrete slab with warmer top and cooler bottom tends to curve when the depthwise strain field is unconstrained, but continuity can generate additional restraint moments.

What the idealization omits

Using a uniform-temperature axial formula to describe a curvature problem discards the gradient and can predict zero bending when the structure actually warps.

Verification and interpretation of results

Test the limiting case of equal top and bottom temperatures: pure gradient curvature should vanish.

Applied case and decision boundary

Discuss daytime temperature-gradient stresses on a continuous bridge deck. 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 uniform-temperature axial formula to describe a curvature problem discards the gradient and can predict zero bending when the structure actually warps. Explain which physical check exposes the error.

5. Professional judgment: Using the case Discuss daytime temperature-gradient stresses on a continuous bridge deck., 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.

4.5 Long-term deformation and stage analysis

Learning target and physical interpretation

Creep and shrinkage change structural response over time and can redistribute reactions in indeterminate systems. A single instantaneous-E stiffness computation cannot automatically predict years of movement.

Governing relation

An effective-modulus approximation may use E_eff≈E/(1+φ), where φ is a creep coefficient defined for the particular loading history.

Worked engineering interpretation

If the creep coefficient is 2, a simplified effective modulus may be one-third of the instantaneous modulus, but this is not a complete time-stepping material law.

What the idealization omits

Applying one creep factor to all loads irrespective of age at loading and moisture conditions can generate misleading long-term moment predictions.

Verification and interpretation of results

Run stage-by-stage sensitivity analyses for plausible φ values and explain what cannot be resolved without a calibrated material model.

Applied case and decision boundary

Outline temporary support removal after phased concrete bridge construction. 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: Applying one creep factor to all loads irrespective of age at loading and moisture conditions can generate misleading long-term moment predictions. Explain which physical check exposes the error.

5. Professional judgment: Using the case Outline temporary support removal after phased concrete bridge construction., 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 4 — Settlement, thermal actions and imposed strains

Conceptual overview

Applied loads are not the only source of internal force. A continuous beam may develop bending when a support moves, even when nobody places a load on the beam. A restrained bar may become compressive after heating or tensile after cooling. Construction stage, shrinkage, creep and bearing rotation can also introduce imposed changes. In analysis, a specified movement is a boundary condition; it belongs in the displacement vector and therefore changes the right-hand side of the reduced stiffness equations. Imposed strain has a different mathematical representation from mechanical force, although the two may be combined at the level of equivalent nodal actions. The engineering consequence is that reaction and moment signs may reverse as boundary movement changes.

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 — Bridge bearing replacement

During bearing replacement, the central support of a two-span steel beam is temporarily lowered by several millimeters. Model settlement as prescribed displacement, not an external force. Determine the change in support reactions and end moments, and contrast the result with the unchanged-support baseline. Identify which assumptions are violated if the bearing loses contact and the support reaction cannot become tensile. Such a unilateral contact problem requires updated constraints and is not solved by simply accepting a negative bearing reaction from a fixed-support linear model.

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.

4.1 Support settlement as prescribed movement

Learning target and physical interpretation

A settlement at a support is a known vertical displacement, not an unknown load. Solve unknown rotations and member actions while imposing the settlement in the constrained DOF.

Governing relation

K_ff u_f=F_f−K_fc u_c with u_c including the nonzero settlement.

Worked engineering interpretation

A 10 mm downward shift at a middle support of a two-span continuous beam creates end moments even with zero externally imposed UDL.

What the idealization omits

Forcing the settled support displacement to zero and adding an arbitrary point force does not enforce the correct kinematic condition.

Verification and interpretation of results

Compute reactions and confirm total vertical reaction remains zero when all external vertical loads are zero and only settlement is present.

Applied case and decision boundary

Use Lab 07 to evaluate load reversal during bearing replacement. 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: Forcing the settled support displacement to zero and adding an arbitrary point force does not enforce the correct kinematic condition. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 07 to evaluate load reversal during bearing replacement., 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.

4.2 Lack-of-fit in axial members

Learning target and physical interpretation

A fabricated bar may be longer or shorter than the gap between its supports. Assembly then requires either movement of supports or internal stress. Lack-of-fit can be modeled as an initial strain or equivalent displacement.

Governing relation

N=(Δ_actual−Δ_free)/(ΣL/(EA)) for members in series where Δ_free includes thermal and fabrication effects.

Worked engineering interpretation

If an immovable two-bar system would freely lengthen by 1 mm but is confined to a fixed 0 mm net length change, it carries a uniform compressive force proportional to 1 mm divided by axial flexibility.

What the idealization omits

Assuming the stress is the same in sections of unequal area is wrong; axial force is constant in ideal series bars but σ_i=N/A_i varies.

Verification and interpretation of results

Check the sum of mechanical plus thermal extensions equals the actual support-to-support displacement.

Applied case and decision boundary

Explain temperature-induced bearing forces at a restrained bridge expansion joint. 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: Assuming the stress is the same in sections of unequal area is wrong; axial force is constant in ideal series bars but σ_i=N/A_i varies. Explain which physical check exposes the error.

5. Professional judgment: Using the case Explain temperature-induced bearing forces at a restrained bridge expansion joint., 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.

4.3 Thermal strain and stiffness contrast

Learning target and physical interpretation

Thermal free strain is approximately αΔT for a uniform isotropic segment. E, A, α and segment length each influence force under restraint, and all four must be expressed in coherent units.

Governing relation

Δ_free=Σ α_i ΔT_i L_i; N=−Δ_free/(Σ L_i/(E_i A_i)) for fixed overall length.

Worked engineering interpretation

A 3 m steel segment with α=12×10⁻⁶/K warmed 50°C would lengthen 1.8 mm if unrestrained, before accounting for other segments.

What the idealization omits

Thermal force does not exist in an unconstrained prismatic bar under uniform temperature change, apart from secondary effects such as gradients or incompatible connections.

Verification and interpretation of results

Vary heating while comparing a fixed-fixed series bar against an unconstrained counterpart; ensure force vanishes when Δ_free=0.

Applied case and decision boundary

Use Lab 08 to contrast steel and aluminum segments in a restrained service 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: Thermal force does not exist in an unconstrained prismatic bar under uniform temperature change, apart from secondary effects such as gradients or incompatible connections. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 08 to contrast steel and aluminum segments in a restrained service 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.

4.4 Temperature gradients in beams

Learning target and physical interpretation

A through-depth temperature difference can cause curvature; a uniform temperature increase principally changes length. The modeling choice depends on whether axial and rotational restraint permit free motion.

Governing relation

For a linear thermal strain gradient through depth h, unconstrained thermal curvature κ≈αΔT_top-bottom/h, with sign depending on axis orientation.

Worked engineering interpretation

A concrete slab with warmer top and cooler bottom tends to curve when the depthwise strain field is unconstrained, but continuity can generate additional restraint moments.

What the idealization omits

Using a uniform-temperature axial formula to describe a curvature problem discards the gradient and can predict zero bending when the structure actually warps.

Verification and interpretation of results

Test the limiting case of equal top and bottom temperatures: pure gradient curvature should vanish.

Applied case and decision boundary

Discuss daytime temperature-gradient stresses on a continuous bridge deck. 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 uniform-temperature axial formula to describe a curvature problem discards the gradient and can predict zero bending when the structure actually warps. Explain which physical check exposes the error.

5. Professional judgment: Using the case Discuss daytime temperature-gradient stresses on a continuous bridge deck., 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.

4.5 Long-term deformation and stage analysis

Learning target and physical interpretation

Creep and shrinkage change structural response over time and can redistribute reactions in indeterminate systems. A single instantaneous-E stiffness computation cannot automatically predict years of movement.

Governing relation

An effective-modulus approximation may use E_eff≈E/(1+φ), where φ is a creep coefficient defined for the particular loading history.

Worked engineering interpretation

If the creep coefficient is 2, a simplified effective modulus may be one-third of the instantaneous modulus, but this is not a complete time-stepping material law.

What the idealization omits

Applying one creep factor to all loads irrespective of age at loading and moisture conditions can generate misleading long-term moment predictions.

Verification and interpretation of results

Run stage-by-stage sensitivity analyses for plausible φ values and explain what cannot be resolved without a calibrated material model.

Applied case and decision boundary

Outline temporary support removal after phased concrete bridge construction. 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: Applying one creep factor to all loads irrespective of age at loading and moisture conditions can generate misleading long-term moment predictions. Explain which physical check exposes the error.

5. Professional judgment: Using the case Outline temporary support removal after phased concrete bridge construction., 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.