Matrix Stiffness of Trusses and Rigid Frames

Conceptual overview

Computers do not understand a structural diagram unless the analyst supplies nodes, connectivity, supports, member properties and load vectors. A stiffness formulation converts those data into equations. For a truss, each node has two in-plane translational degrees of freedom and member axial stiffness acts along its axis. For a frame, axial displacement, lateral displacement and in-plane rotation must all be present. Transformations from local member axes to global coordinates preserve the distinction between a horizontal structural direction and a member's own axial direction. A global stiffness matrix is ordinarily symmetric if the underlying linear elastic mechanics are conservative and the model is assembled consistently. Singular global stiffness may indicate an unconstrained rigid-body motion or a true structural mechanism rather than a numerical software fault.

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 — Industrial portal roof sway

A rectangular portal supports a lateral wind action and roof gravity action. First build a two-column and one-beam stiffness model with fixed bases. Then examine the effects of connection release, frame height and member EI on roof drift. Draw the local member end moments using a consistent sign convention. Comment on why a correct global reaction sum does not guarantee that bending and axial force recovery are correct. In a professional analysis, wind pressure distribution, second-order effects, bracing, diaphragm action and load combinations must be considered separately.

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.

3.1 Bar-element geometry and transformation

Learning target and physical interpretation

Compute element length and direction cosines from node coordinates. The bar resists only change of length; forces transverse to its axis enter through the coordinate transformation rather than an invented bending stiffness.

Governing relation

k_global=(EA/L)[[c²,cs,−c²,−cs],[cs,s²,−cs,−s²],[−c²,−cs,c²,cs],[−cs,−s²,cs,s²]].

Worked engineering interpretation

A diagonal bar connecting (0,0) to (3,4) m has L=5 m, c=0.6 and s=0.8, which determines all 16 entries of its 4×4 matrix.

What the idealization omits

Using a positive direction cosine in all quadrants gives wrong signs in off-diagonal stiffness contributions and can rotate the predicted deformation field.

Verification and interpretation of results

Check that a uniform rigid translation creates zero bar force and that axial extension EAΔ/L agrees with matrix force recovery.

Applied case and decision boundary

Use a steel bracing example to interpret member axial forces after imposing a lateral floor movement. 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 positive direction cosine in all quadrants gives wrong signs in off-diagonal stiffness contributions and can rotate the predicted deformation field. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use a steel bracing example to interpret member axial forces after imposing a lateral floor movement., 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.

3.2 Global assembly and constraints

Learning target and physical interpretation

Scatter every element stiffness contribution into the global matrix at indices corresponding to connected nodal DOFs. Enforce prescribed displacements and solve only for unknown DOFs.

Governing relation

K_ff u_f = F_f − K_fc u_c; R_c = K_cf u_f + K_cc u_c − F_c.

Worked engineering interpretation

A two-bay frame may have 18 free and restrained DOFs before simplifying symmetry; indexing is the primary practical risk in manual assembly.

What the idealization omits

Deleting a support row without deleting the matching column breaks symmetry; setting a diagonal entry to a giant number can obscure reaction recovery.

Verification and interpretation of results

Reinsert the solved displacement vector into the original complete system and check the constrained residual components as support reactions.

Applied case and decision boundary

Discuss why settlement and imposed temperature strain must enter the same equilibrium bookkeeping. 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: Deleting a support row without deleting the matching column breaks symmetry; setting a diagonal entry to a giant number can obscure reaction recovery. Explain which physical check exposes the error.

5. Professional judgment: Using the case Discuss why settlement and imposed temperature strain must enter the same equilibrium bookkeeping., 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.

3.3 Three-bar truss numerical model

Learning target and physical interpretation

A top joint joined to three fixed lower joints produces a two-unknown stiffness system. Different bar lengths and orientations contribute different off-diagonal coupling and effective vertical stiffness.

Governing relation

For top displacements (ux,uy), K = Σ(EA/L)[[cx²,cxcy],[cxcy,cy²]] and bar force N_i=(EA/L)(cx ux+cy uy).

Worked engineering interpretation

For a symmetric support arrangement and vertical force, ux is zero by symmetry, but the center vertical member still changes the partition of axial force.

What the idealization omits

Adding a bar while reusing forces from the former two-bar solution violates equilibrium and compatibility: the new stiffness must be assembled and solved again.

Verification and interpretation of results

Check horizontal and vertical nodal equilibrium from recovered bar forces and document the chosen sign for tension.

Applied case and decision boundary

Use Lab 05 to measure how a vertical hanger relieves two diagonal members. 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: Adding a bar while reusing forces from the former two-bar solution violates equilibrium and compatibility: the new stiffness must be assembled and solved again. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 05 to measure how a vertical hanger relieves two diagonal members., 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.

3.4 Plane-frame local stiffness

Learning target and physical interpretation

The plane-frame element couples two translational DOFs and one rotational DOF at each end; its local stiffness is the sum of axial EA/L and Euler–Bernoulli bending terms.

Governing relation

Local bending terms: 12EI/L³, 6EI/L², 4EI/L and 2EI/L; axial term EA/L.

Worked engineering interpretation

For a prismatic 6 m steel beam with EI=30 MN·m², its local rotational diagonal is 4EI/L=20 MN·m/rad.

What the idealization omits

A released member end is not the same as a rigid joint with small EI. To model an actual hinge, release the member rotational DOF correctly.

Verification and interpretation of results

Check that a member oriented vertically produces the same physical stiffness after correct transformation as a rotated horizontal equivalent.

Applied case and decision boundary

Compare a sway portal and a non-sway portal under the same horizontal action. 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: A released member end is not the same as a rigid joint with small EI. To model an actual hinge, release the member rotational DOF correctly. Explain which physical check exposes the error.

5. Professional judgment: Using the case Compare a sway portal and a non-sway portal under the same horizontal action., 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.

3.5 Portal-frame sway and load recovery

Learning target and physical interpretation

A lateral force on a rigid portal bends columns and beam, producing horizontal drift and joint rotation. Recover member end forces from solved displacements in local coordinates, then transform and sum reactions.

Governing relation

Member end actions q_local = k_local T u_global for the no-member-load case.

Worked engineering interpretation

A horizontal roof force of 80 kN acting on a frame with two fixed bases must be balanced by −80 kN total horizontal support reactions, independent of drift magnitude.

What the idealization omits

Displaying only frame drift hides dangerous local member moments; output both joint translations and near/far member moments for a meaningful check.

Verification and interpretation of results

Verify both horizontal equilibrium and equal-and-opposite shears at rigid joints, then vary EI and EA separately.

Applied case and decision boundary

Use Lab 06 to compare flexural-frame stiffness with a hypothetical braced alternative. 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: Displaying only frame drift hides dangerous local member moments; output both joint translations and near/far member moments for a meaningful check. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 06 to compare flexural-frame stiffness with a hypothetical braced alternative., 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 3 — Matrix stiffness of trusses and rigid frames

Conceptual overview

Computers do not understand a structural diagram unless the analyst supplies nodes, connectivity, supports, member properties and load vectors. A stiffness formulation converts those data into equations. For a truss, each node has two in-plane translational degrees of freedom and member axial stiffness acts along its axis. For a frame, axial displacement, lateral displacement and in-plane rotation must all be present. Transformations from local member axes to global coordinates preserve the distinction between a horizontal structural direction and a member's own axial direction. A global stiffness matrix is ordinarily symmetric if the underlying linear elastic mechanics are conservative and the model is assembled consistently. Singular global stiffness may indicate an unconstrained rigid-body motion or a true structural mechanism rather than a numerical software fault.

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 — Industrial portal roof sway

A rectangular portal supports a lateral wind action and roof gravity action. First build a two-column and one-beam stiffness model with fixed bases. Then examine the effects of connection release, frame height and member EI on roof drift. Draw the local member end moments using a consistent sign convention. Comment on why a correct global reaction sum does not guarantee that bending and axial force recovery are correct. In a professional analysis, wind pressure distribution, second-order effects, bracing, diaphragm action and load combinations must be considered separately.

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.

3.1 Bar-element geometry and transformation

Learning target and physical interpretation

Compute element length and direction cosines from node coordinates. The bar resists only change of length; forces transverse to its axis enter through the coordinate transformation rather than an invented bending stiffness.

Governing relation

k_global=(EA/L)[[c²,cs,−c²,−cs],[cs,s²,−cs,−s²],[−c²,−cs,c²,cs],[−cs,−s²,cs,s²]].

Worked engineering interpretation

A diagonal bar connecting (0,0) to (3,4) m has L=5 m, c=0.6 and s=0.8, which determines all 16 entries of its 4×4 matrix.

What the idealization omits

Using a positive direction cosine in all quadrants gives wrong signs in off-diagonal stiffness contributions and can rotate the predicted deformation field.

Verification and interpretation of results

Check that a uniform rigid translation creates zero bar force and that axial extension EAΔ/L agrees with matrix force recovery.

Applied case and decision boundary

Use a steel bracing example to interpret member axial forces after imposing a lateral floor movement. 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 positive direction cosine in all quadrants gives wrong signs in off-diagonal stiffness contributions and can rotate the predicted deformation field. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use a steel bracing example to interpret member axial forces after imposing a lateral floor movement., 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.

3.2 Global assembly and constraints

Learning target and physical interpretation

Scatter every element stiffness contribution into the global matrix at indices corresponding to connected nodal DOFs. Enforce prescribed displacements and solve only for unknown DOFs.

Governing relation

K_ff u_f = F_f − K_fc u_c; R_c = K_cf u_f + K_cc u_c − F_c.

Worked engineering interpretation

A two-bay frame may have 18 free and restrained DOFs before simplifying symmetry; indexing is the primary practical risk in manual assembly.

What the idealization omits

Deleting a support row without deleting the matching column breaks symmetry; setting a diagonal entry to a giant number can obscure reaction recovery.

Verification and interpretation of results

Reinsert the solved displacement vector into the original complete system and check the constrained residual components as support reactions.

Applied case and decision boundary

Discuss why settlement and imposed temperature strain must enter the same equilibrium bookkeeping. 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: Deleting a support row without deleting the matching column breaks symmetry; setting a diagonal entry to a giant number can obscure reaction recovery. Explain which physical check exposes the error.

5. Professional judgment: Using the case Discuss why settlement and imposed temperature strain must enter the same equilibrium bookkeeping., 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.

3.3 Three-bar truss numerical model

Learning target and physical interpretation

A top joint joined to three fixed lower joints produces a two-unknown stiffness system. Different bar lengths and orientations contribute different off-diagonal coupling and effective vertical stiffness.

Governing relation

For top displacements (ux,uy), K = Σ(EA/L)[[cx²,cxcy],[cxcy,cy²]] and bar force N_i=(EA/L)(cx ux+cy uy).

Worked engineering interpretation

For a symmetric support arrangement and vertical force, ux is zero by symmetry, but the center vertical member still changes the partition of axial force.

What the idealization omits

Adding a bar while reusing forces from the former two-bar solution violates equilibrium and compatibility: the new stiffness must be assembled and solved again.

Verification and interpretation of results

Check horizontal and vertical nodal equilibrium from recovered bar forces and document the chosen sign for tension.

Applied case and decision boundary

Use Lab 05 to measure how a vertical hanger relieves two diagonal members. 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: Adding a bar while reusing forces from the former two-bar solution violates equilibrium and compatibility: the new stiffness must be assembled and solved again. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 05 to measure how a vertical hanger relieves two diagonal members., 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.

3.4 Plane-frame local stiffness

Learning target and physical interpretation

The plane-frame element couples two translational DOFs and one rotational DOF at each end; its local stiffness is the sum of axial EA/L and Euler–Bernoulli bending terms.

Governing relation

Local bending terms: 12EI/L³, 6EI/L², 4EI/L and 2EI/L; axial term EA/L.

Worked engineering interpretation

For a prismatic 6 m steel beam with EI=30 MN·m², its local rotational diagonal is 4EI/L=20 MN·m/rad.

What the idealization omits

A released member end is not the same as a rigid joint with small EI. To model an actual hinge, release the member rotational DOF correctly.

Verification and interpretation of results

Check that a member oriented vertically produces the same physical stiffness after correct transformation as a rotated horizontal equivalent.

Applied case and decision boundary

Compare a sway portal and a non-sway portal under the same horizontal action. 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: A released member end is not the same as a rigid joint with small EI. To model an actual hinge, release the member rotational DOF correctly. Explain which physical check exposes the error.

5. Professional judgment: Using the case Compare a sway portal and a non-sway portal under the same horizontal action., 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.

3.5 Portal-frame sway and load recovery

Learning target and physical interpretation

A lateral force on a rigid portal bends columns and beam, producing horizontal drift and joint rotation. Recover member end forces from solved displacements in local coordinates, then transform and sum reactions.

Governing relation

Member end actions q_local = k_local T u_global for the no-member-load case.

Worked engineering interpretation

A horizontal roof force of 80 kN acting on a frame with two fixed bases must be balanced by −80 kN total horizontal support reactions, independent of drift magnitude.

What the idealization omits

Displaying only frame drift hides dangerous local member moments; output both joint translations and near/far member moments for a meaningful check.

Verification and interpretation of results

Verify both horizontal equilibrium and equal-and-opposite shears at rigid joints, then vary EI and EA separately.

Applied case and decision boundary

Use Lab 06 to compare flexural-frame stiffness with a hypothetical braced alternative. 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: Displaying only frame drift hides dangerous local member moments; output both joint translations and near/far member moments for a meaningful check. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 06 to compare flexural-frame stiffness with a hypothetical braced alternative., 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.