Columns, Slenderness and Structural Walls

Learning purpose. Develop an explicit force, geometry, materials and evidence model before any final code compliance calculation. Numerical examples in this unit are educational and not an authorization to construct.

Guided investigation: a column is a system component

An axially loaded short column is an appealing introductory example because materials appear to share compressive force. But real columns see bending from beam-end moments, lateral drift, construction imperfections and unavoidable eccentricity. Reinforcement and concrete stresses cannot be assigned independently if both materials are assumed to deform compatibly. A P–M interaction analysis traces the combined capacity for a sequence of neutral-axis positions; this geometry explains why high compression and high bending demand interact rather than add linearly.

Construct a rectangular section diagram with symmetric layers of reinforcing steel. Assume a linear strain diagram and an extreme concrete compression strain appropriate to the illustrative model. For each neutral-axis depth, compute concrete compression and steel forces from their strain-compatible constitutive relationships. Sum P and M about the section centroid. Plot the resulting points and inspect where one layer yields in compression or tension. A nonrectangular section or a biaxial column requires a more advanced section analysis, not merely a reused rectangular curve.

The resulting nominal locus is not a code-approved capacity surface. Design reduction factors or partial material factors, tied versus spiral confinement, minimum eccentricity, reinforcement limits and tie spacing need separate application. Even these cross-section checks cannot represent member stability. A slender column under axial force will bend and thereby generate additional moment. A braced frame, where translational sway is restrained, differs from an unbraced frame where column stability interacts across the entire story.

Take a two-story frame and mark where a lateral horizontal force enters the diaphragm and travels into columns or walls. If the frame is intended to resist earthquakes, think about intended plastic hinges and the protection of joints and columns from brittle mechanisms. Inadequate confinement near a joint can invalidate assumptions about sustained plastic rotation, and poorly developed beam bars can prevent a frame from redistributing moment. A wall may reduce lateral drift but introduce transfer and foundation overturning demands.

Create two check records: one for the cross-section with strain compatibility, and another for the member and lateral system with end restraint, effective stiffness, second-order effects, axial stability and seismic requirements as applicable. An independent reviewer should be able to see where nominal resistance ends and the yet-to-be-performed code checks begin. A large P–M number does not certify a column or a building.

---

Lesson 5.1: Short tied columns under axial load

Physical meaning and governing ideas

A reinforced concrete column uses concrete compression and longitudinal steel to resist axial force, but imperfection and accidental eccentricity keep pure axial loading an idealization. Transverse ties restrain longitudinal bars and provide confinement and resistance to bar buckling; their spacing and anchorage are not optional finishing details.

Analysis and design reasoning

Begin with gross area, steel area and compatible strain response, then consider minimum eccentricities, load combinations and strength reduction requirements. Recognize that a nominal concentric compressive sum is not the permitted design capacity of a tied column. Column behavior is sensitive to restraint and detailing near beam-column joints.

Worked numerical or geometric illustration

For a 400×450 mm gross section with Ast=2400 mm², concrete area excluding steel is 177600 mm². A simple materials sum 0.85×32×177600+420×2400=5.84 MN is only a nominal upper-bound style thought experiment; it is not an allowable column load.

Interpretation and failure checks

Check column dimensions against the bar arrangement and tie bends: a theoretically adequate area may not be constructible with proper clearances.

Study task 5.1. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

---

Lesson 5.2: Axial force and bending interaction

Physical meaning and governing ideas

A column section under combined axial load and bending cannot be designed from independent axial and flexural capacities added together. A P–M interaction locus follows strain compatibility, concrete stress distribution and the stress reached in each reinforcement layer. The point describing demand must fall within a suitably reduced and code-checked capacity domain.

Analysis and design reasoning

Generate section points by varying neutral-axis position and solving steel strain from plane sections. Sum axial forces and moments about a consistent centroid; preserve the sign convention. For biaxial loading, use an appropriate surface or conservative approved interaction method rather than projecting a three-dimensional demand onto one arbitrary axis.

Worked numerical or geometric illustration

A column carrying Pu=900 kN with eccentricity e=80 mm has moment due solely to that eccentricity M=Pu e=72 kN·m. End moments, second-order effects and lateral loading may substantially increase the design moment and cannot be omitted.

Interpretation and failure checks

Use the interaction laboratory to find why a section can have substantial nominal compression resistance yet limited moment capacity at particular axial force.

Study task 5.2. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

---

Lesson 5.3: Slenderness and second-order effects

Physical meaning and governing ideas

Slender compression members bend under initial eccentricity and lateral drift, causing additional P–Δ and P–δ moments. Slenderness cannot be judged from physical height alone: end restraint, effective stiffness, axial load and braced or unbraced sway conditions matter. Long-term creep further decreases effective stiffness and raises second-order sensitivity.

Analysis and design reasoning

Identify whether a frame can sway, estimate an appropriate effective length or use a second-order analysis, and include cracked stiffness consistent with the chosen analysis method. Verify stability under governing combinations, with geometric and material nonlinearities as required. A first-order axial–moment check may be unconservative when deflection magnifies moment.

Worked numerical or geometric illustration

For an ideal pin-ended elastic strut EI=40 MN·m² and length L=5 m, Euler load Pe=π² EI/L²≈15.79 MN. That is an elastic bifurcation reference, not reinforced concrete code axial capacity. Cracking, creep, imperfections and inelasticity can govern far earlier.

Interpretation and failure checks

Increasing reinforcement does not automatically solve a sway problem. Bracing, geometry and system stiffness may provide more effective remedies.

Study task 5.3. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

---

Lesson 5.4: Shear walls, coupling beams and boundary regions

Physical meaning and governing ideas

Walls provide lateral stiffness and gravity capacity, but their behavior depends on aspect ratio, flexural axial interaction, shear cracking, confinement and boundary details. A coupled-wall system also develops axial action through coupling beams, redistributing overturning demand. Connections to foundations and diaphragms complete the load path.

Analysis and design reasoning

Sketch shear, axial force, overturning moment and displacement along the wall height. Determine whether flexure, shear, sliding, wall instability or boundary crushing controls. Under seismic demands, detailed confinement and capacity design can be essential even when simple elastic section calculations predict excess strength.

Worked numerical or geometric illustration

For an elastic 9 m cantilever wall subjected to a 140 kN top lateral force, base moment is 1260 kN·m. If idealized flexural rigidity is EI=110 MN·m², roof deflection from bending alone is F H³/(3 EI)=309 mm? Check units: 140×729/(3×110)=309 mm. Such a large result should prompt review of realistic EI and load levels.

Interpretation and failure checks

Do not transfer a beam stirrup design formula to a wall without examining its in-plane shear fields and code wall detailing requirements.

Study task 5.4. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

---

Numerical practice — solved and interpreted

The eight problems below vary geometry and actions within a consistent instructional model. They develop reproducible calculation habits, but do not substitute for independent code checks.

Practice scenario 5.1

Problem. A rectangular gross column has b=330 mm, h=365 mm, concentric component P=510 kN and nominal eccentricity e=23 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=11.73 kN·m. The average compressive stress is P/A=4.234 MPa. The flexural extreme stress is 6Pe/(bh²)=1.601 MPa, giving edges 5.835 MPa and 2.633 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.2

Problem. A rectangular gross column has b=340 mm, h=380 mm, concentric component P=620 kN and nominal eccentricity e=31 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=19.22 kN·m. The average compressive stress is P/A=4.799 MPa. The flexural extreme stress is 6Pe/(bh²)=2.349 MPa, giving edges 7.148 MPa and 2.450 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.3

Problem. A rectangular gross column has b=350 mm, h=395 mm, concentric component P=730 kN and nominal eccentricity e=39 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=28.47 kN·m. The average compressive stress is P/A=5.280 MPa. The flexural extreme stress is 6Pe/(bh²)=3.128 MPa, giving edges 8.408 MPa and 2.152 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.4

Problem. A rectangular gross column has b=360 mm, h=410 mm, concentric component P=840 kN and nominal eccentricity e=47 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=39.48 kN·m. The average compressive stress is P/A=5.691 MPa. The flexural extreme stress is 6Pe/(bh²)=3.914 MPa, giving edges 9.605 MPa and 1.777 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.5

Problem. A rectangular gross column has b=370 mm, h=425 mm, concentric component P=950 kN and nominal eccentricity e=55 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=52.25 kN·m. The average compressive stress is P/A=6.041 MPa. The flexural extreme stress is 6Pe/(bh²)=4.691 MPa, giving edges 10.732 MPa and 1.350 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.6

Problem. A rectangular gross column has b=380 mm, h=440 mm, concentric component P=1060 kN and nominal eccentricity e=63 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=66.78 kN·m. The average compressive stress is P/A=6.340 MPa. The flexural extreme stress is 6Pe/(bh²)=5.446 MPa, giving edges 11.786 MPa and 0.893 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.7

Problem. A rectangular gross column has b=390 mm, h=455 mm, concentric component P=1170 kN and nominal eccentricity e=71 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=83.07 kN·m. The average compressive stress is P/A=6.593 MPa. The flexural extreme stress is 6Pe/(bh²)=6.173 MPa, giving edges 12.767 MPa and 0.420 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.8

Problem. A rectangular gross column has b=400 mm, h=470 mm, concentric component P=1280 kN and nominal eccentricity e=79 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=101.12 kN·m. The average compressive stress is P/A=6.809 MPa. The flexural extreme stress is 6Pe/(bh²)=6.866 MPa, giving edges 13.675 MPa and -0.058 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Knowledge check with explanatory answers

Question 5.1

Why is a gross material sum not an allowable column load?

Discussion answer. An axial sum of concrete and steel stress capacities ignores minimum eccentricity, interaction with bending, reduction factors, confinement differences, reinforcement limits and stability. Real columns carry moments and may be slender. A nominal sum is useful for scale, but it must not be described as design capacity or permitted service load.

Question 5.2

What is the physical meaning of a point on a P–M interaction curve?

Discussion answer. It represents axial force and moment resultants from one assumed compatible section strain distribution and material stress state. The locus changes when the neutral axis moves, steel layers change stress sign or reach yield, and concrete compression changes. A project demand pair must be checked against the correctly reduced, code-defined resistance region with member effects included.

Question 5.3

How do first-order and second-order moments differ?

Discussion answer. First-order moments come from equilibrium of the original geometry. Second-order moments include the additional effects of axial forces acting through displacements, such as P–Δ and P–δ. They can amplify bending in slender or sway systems. Effective member stiffness, restraint and long-term behavior influence the magnitude; a cross-section capacity check cannot by itself determine the amplification.

Question 5.4

What must a shear wall transfer into its foundation?

Discussion answer. The wall may transmit vertical gravity and seismic axial actions, in-plane shear, overturning moment, uplift and concentrated boundary forces. A foundation must resist them with compatible soil behavior, reinforcement and connection details. The wall also needs a diaphragm connection to receive lateral load. A diagram of wall base shear alone is incomplete.

Question 5.5

Why do tied and spiral columns require different treatment?

Discussion answer. Transverse reinforcement arrangement changes confinement, longitudinal-bar stability and postpeak response. A governing code may prescribe different nominal strength limitations, factors and detailing for tied versus spirally reinforced columns. Counting the same total transverse steel area does not ensure equivalent confinement geometry or seismic performance.

Unit project and experiment

For a 400×450 mm column with symmetric reinforcement, generate a nominal axial–moment interaction using strain compatibility. Select at least three applied P–M demand pairs; explain whether they appear inside or outside the nominal curve and why nominal comparison is insufficient. Compute the moment from a given eccentricity and list other sources of design moment. Estimate an elastic Euler load with a declared effective length, then discuss why it is not a reinforced-concrete axial design limit. Compare sway and nonsway structural scenarios. Prepare a detailed tie cage with realistic dimensions, check bar fit, and specify that actual strength factors, slenderness and seismic detailing require the governing code.

Deliverable: A transparent engineering memo with assumptions, diagrams, units, hand-checks, source references and a list of issues that need a governing-code and professional review.

Related interactive simulations

• [P–M interaction construction](../labs/lab-column.html) — adjust inputs, inspect curves, export data and explain limits of the model.

• [Gross-section eccentricity thought experiment](../labs/lab-eccentric.html) — adjust inputs, inspect curves, export data and explain limits of the model.

• [Wall height, lateral force and stiffness](../labs/lab-wall.html) — adjust inputs, inspect curves, export data and explain limits of the model.

Unit 5: Columns and walls

Learning purpose. Develop an explicit force, geometry, materials and evidence model before any final code compliance calculation. Numerical examples in this unit are educational and not an authorization to construct.

Guided investigation: a column is a system component

An axially loaded short column is an appealing introductory example because materials appear to share compressive force. But real columns see bending from beam-end moments, lateral drift, construction imperfections and unavoidable eccentricity. Reinforcement and concrete stresses cannot be assigned independently if both materials are assumed to deform compatibly. A P–M interaction analysis traces the combined capacity for a sequence of neutral-axis positions; this geometry explains why high compression and high bending demand interact rather than add linearly.

Construct a rectangular section diagram with symmetric layers of reinforcing steel. Assume a linear strain diagram and an extreme concrete compression strain appropriate to the illustrative model. For each neutral-axis depth, compute concrete compression and steel forces from their strain-compatible constitutive relationships. Sum P and M about the section centroid. Plot the resulting points and inspect where one layer yields in compression or tension. A nonrectangular section or a biaxial column requires a more advanced section analysis, not merely a reused rectangular curve.

The resulting nominal locus is not a code-approved capacity surface. Design reduction factors or partial material factors, tied versus spiral confinement, minimum eccentricity, reinforcement limits and tie spacing need separate application. Even these cross-section checks cannot represent member stability. A slender column under axial force will bend and thereby generate additional moment. A braced frame, where translational sway is restrained, differs from an unbraced frame where column stability interacts across the entire story.

Take a two-story frame and mark where a lateral horizontal force enters the diaphragm and travels into columns or walls. If the frame is intended to resist earthquakes, think about intended plastic hinges and the protection of joints and columns from brittle mechanisms. Inadequate confinement near a joint can invalidate assumptions about sustained plastic rotation, and poorly developed beam bars can prevent a frame from redistributing moment. A wall may reduce lateral drift but introduce transfer and foundation overturning demands.

Create two check records: one for the cross-section with strain compatibility, and another for the member and lateral system with end restraint, effective stiffness, second-order effects, axial stability and seismic requirements as applicable. An independent reviewer should be able to see where nominal resistance ends and the yet-to-be-performed code checks begin. A large P–M number does not certify a column or a building.

---

Lesson 5.1: Short tied columns under axial load

Physical meaning and governing ideas

A reinforced concrete column uses concrete compression and longitudinal steel to resist axial force, but imperfection and accidental eccentricity keep pure axial loading an idealization. Transverse ties restrain longitudinal bars and provide confinement and resistance to bar buckling; their spacing and anchorage are not optional finishing details.

Analysis and design reasoning

Begin with gross area, steel area and compatible strain response, then consider minimum eccentricities, load combinations and strength reduction requirements. Recognize that a nominal concentric compressive sum is not the permitted design capacity of a tied column. Column behavior is sensitive to restraint and detailing near beam-column joints.

Worked numerical or geometric illustration

For a 400×450 mm gross section with Ast=2400 mm², concrete area excluding steel is 177600 mm². A simple materials sum 0.85×32×177600+420×2400=5.84 MN is only a nominal upper-bound style thought experiment; it is not an allowable column load.

Interpretation and failure checks

Check column dimensions against the bar arrangement and tie bends: a theoretically adequate area may not be constructible with proper clearances.

Study task 5.1. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

---

Lesson 5.2: Axial force and bending interaction

Physical meaning and governing ideas

A column section under combined axial load and bending cannot be designed from independent axial and flexural capacities added together. A P–M interaction locus follows strain compatibility, concrete stress distribution and the stress reached in each reinforcement layer. The point describing demand must fall within a suitably reduced and code-checked capacity domain.

Analysis and design reasoning

Generate section points by varying neutral-axis position and solving steel strain from plane sections. Sum axial forces and moments about a consistent centroid; preserve the sign convention. For biaxial loading, use an appropriate surface or conservative approved interaction method rather than projecting a three-dimensional demand onto one arbitrary axis.

Worked numerical or geometric illustration

A column carrying Pu=900 kN with eccentricity e=80 mm has moment due solely to that eccentricity M=Pu e=72 kN·m. End moments, second-order effects and lateral loading may substantially increase the design moment and cannot be omitted.

Interpretation and failure checks

Use the interaction laboratory to find why a section can have substantial nominal compression resistance yet limited moment capacity at particular axial force.

Study task 5.2. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

---

Lesson 5.3: Slenderness and second-order effects

Physical meaning and governing ideas

Slender compression members bend under initial eccentricity and lateral drift, causing additional P–Δ and P–δ moments. Slenderness cannot be judged from physical height alone: end restraint, effective stiffness, axial load and braced or unbraced sway conditions matter. Long-term creep further decreases effective stiffness and raises second-order sensitivity.

Analysis and design reasoning

Identify whether a frame can sway, estimate an appropriate effective length or use a second-order analysis, and include cracked stiffness consistent with the chosen analysis method. Verify stability under governing combinations, with geometric and material nonlinearities as required. A first-order axial–moment check may be unconservative when deflection magnifies moment.

Worked numerical or geometric illustration

For an ideal pin-ended elastic strut EI=40 MN·m² and length L=5 m, Euler load Pe=π² EI/L²≈15.79 MN. That is an elastic bifurcation reference, not reinforced concrete code axial capacity. Cracking, creep, imperfections and inelasticity can govern far earlier.

Interpretation and failure checks

Increasing reinforcement does not automatically solve a sway problem. Bracing, geometry and system stiffness may provide more effective remedies.

Study task 5.3. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

---

Lesson 5.4: Shear walls, coupling beams and boundary regions

Physical meaning and governing ideas

Walls provide lateral stiffness and gravity capacity, but their behavior depends on aspect ratio, flexural axial interaction, shear cracking, confinement and boundary details. A coupled-wall system also develops axial action through coupling beams, redistributing overturning demand. Connections to foundations and diaphragms complete the load path.

Analysis and design reasoning

Sketch shear, axial force, overturning moment and displacement along the wall height. Determine whether flexure, shear, sliding, wall instability or boundary crushing controls. Under seismic demands, detailed confinement and capacity design can be essential even when simple elastic section calculations predict excess strength.

Worked numerical or geometric illustration

For an elastic 9 m cantilever wall subjected to a 140 kN top lateral force, base moment is 1260 kN·m. If idealized flexural rigidity is EI=110 MN·m², roof deflection from bending alone is F H³/(3 EI)=309 mm? Check units: 140×729/(3×110)=309 mm. Such a large result should prompt review of realistic EI and load levels.

Interpretation and failure checks

Do not transfer a beam stirrup design formula to a wall without examining its in-plane shear fields and code wall detailing requirements.

Study task 5.4. Reconstruct the main diagram using your own sign convention, then identify one missing input, one model assumption that could fail, one relevant limit state and one construction detail that must be verified. Explain how you would obtain evidence for the missing input.

---

Numerical practice — solved and interpreted

The eight problems below vary geometry and actions within a consistent instructional model. They develop reproducible calculation habits, but do not substitute for independent code checks.

Practice scenario 5.1

Problem. A rectangular gross column has b=330 mm, h=365 mm, concentric component P=510 kN and nominal eccentricity e=23 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=11.73 kN·m. The average compressive stress is P/A=4.234 MPa. The flexural extreme stress is 6Pe/(bh²)=1.601 MPa, giving edges 5.835 MPa and 2.633 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.2

Problem. A rectangular gross column has b=340 mm, h=380 mm, concentric component P=620 kN and nominal eccentricity e=31 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=19.22 kN·m. The average compressive stress is P/A=4.799 MPa. The flexural extreme stress is 6Pe/(bh²)=2.349 MPa, giving edges 7.148 MPa and 2.450 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.3

Problem. A rectangular gross column has b=350 mm, h=395 mm, concentric component P=730 kN and nominal eccentricity e=39 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=28.47 kN·m. The average compressive stress is P/A=5.280 MPa. The flexural extreme stress is 6Pe/(bh²)=3.128 MPa, giving edges 8.408 MPa and 2.152 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.4

Problem. A rectangular gross column has b=360 mm, h=410 mm, concentric component P=840 kN and nominal eccentricity e=47 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=39.48 kN·m. The average compressive stress is P/A=5.691 MPa. The flexural extreme stress is 6Pe/(bh²)=3.914 MPa, giving edges 9.605 MPa and 1.777 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.5

Problem. A rectangular gross column has b=370 mm, h=425 mm, concentric component P=950 kN and nominal eccentricity e=55 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=52.25 kN·m. The average compressive stress is P/A=6.041 MPa. The flexural extreme stress is 6Pe/(bh²)=4.691 MPa, giving edges 10.732 MPa and 1.350 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.6

Problem. A rectangular gross column has b=380 mm, h=440 mm, concentric component P=1060 kN and nominal eccentricity e=63 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=66.78 kN·m. The average compressive stress is P/A=6.340 MPa. The flexural extreme stress is 6Pe/(bh²)=5.446 MPa, giving edges 11.786 MPa and 0.893 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

---

Practice scenario 5.7

Problem. A rectangular gross column has b=390 mm, h=455 mm, concentric component P=1170 kN and nominal eccentricity e=71 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=83.07 kN·m. The average compressive stress is P/A=6.593 MPa. The flexural extreme stress is 6Pe/(bh²)=6.173 MPa, giving edges 12.767 MPa and 0.420 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Practice scenario 5.8

Problem. A rectangular gross column has b=400 mm, h=470 mm, concentric component P=1280 kN and nominal eccentricity e=79 mm. Compute gross elastic edge stresses and identify whether the uncracked model predicts tension.

Worked resolution. M=Pe=101.12 kN·m. The average compressive stress is P/A=6.809 MPa. The flexural extreme stress is 6Pe/(bh²)=6.866 MPa, giving edges 13.675 MPa and -0.058 MPa. Negative opposite-edge stress would indicate that gross uncracked compression assumptions are no longer self-consistent.

Engineering interpretation. The calculation is a stress visualization, not a column ultimate resistance. A real check requires compatible section strength, reduction factors, minimum eccentricity, slenderness and detailing.

Extension. Select one of the listed inputs and perturb it by approximately 10%. Calculate the new response, identify the governing mathematical proportionality where possible, and write a short conclusion about whether the result is sensitive to dimensions, material properties or boundary idealization.

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Knowledge check with explanatory answers

Question 5.1

Why is a gross material sum not an allowable column load?

Discussion answer. An axial sum of concrete and steel stress capacities ignores minimum eccentricity, interaction with bending, reduction factors, confinement differences, reinforcement limits and stability. Real columns carry moments and may be slender. A nominal sum is useful for scale, but it must not be described as design capacity or permitted service load.

Question 5.2

What is the physical meaning of a point on a P–M interaction curve?

Discussion answer. It represents axial force and moment resultants from one assumed compatible section strain distribution and material stress state. The locus changes when the neutral axis moves, steel layers change stress sign or reach yield, and concrete compression changes. A project demand pair must be checked against the correctly reduced, code-defined resistance region with member effects included.

Question 5.3

How do first-order and second-order moments differ?

Discussion answer. First-order moments come from equilibrium of the original geometry. Second-order moments include the additional effects of axial forces acting through displacements, such as P–Δ and P–δ. They can amplify bending in slender or sway systems. Effective member stiffness, restraint and long-term behavior influence the magnitude; a cross-section capacity check cannot by itself determine the amplification.

Question 5.4

What must a shear wall transfer into its foundation?

Discussion answer. The wall may transmit vertical gravity and seismic axial actions, in-plane shear, overturning moment, uplift and concentrated boundary forces. A foundation must resist them with compatible soil behavior, reinforcement and connection details. The wall also needs a diaphragm connection to receive lateral load. A diagram of wall base shear alone is incomplete.

Question 5.5

Why do tied and spiral columns require different treatment?

Discussion answer. Transverse reinforcement arrangement changes confinement, longitudinal-bar stability and postpeak response. A governing code may prescribe different nominal strength limitations, factors and detailing for tied versus spirally reinforced columns. Counting the same total transverse steel area does not ensure equivalent confinement geometry or seismic performance.

Unit project and experiment

For a 400×450 mm column with symmetric reinforcement, generate a nominal axial–moment interaction using strain compatibility. Select at least three applied P–M demand pairs; explain whether they appear inside or outside the nominal curve and why nominal comparison is insufficient. Compute the moment from a given eccentricity and list other sources of design moment. Estimate an elastic Euler load with a declared effective length, then discuss why it is not a reinforced-concrete axial design limit. Compare sway and nonsway structural scenarios. Prepare a detailed tie cage with realistic dimensions, check bar fit, and specify that actual strength factors, slenderness and seismic detailing require the governing code.

Deliverable: A transparent engineering memo with assumptions, diagrams, units, hand-checks, source references and a list of issues that need a governing-code and professional review.

Related interactive simulations

• [P–M interaction construction](labs/lab-column.html) — adjust inputs, inspect curves, export data and explain limits of the model.

• [Gross-section eccentricity thought experiment](labs/lab-eccentric.html) — adjust inputs, inspect curves, export data and explain limits of the model.

• [Wall height, lateral force and stiffness](labs/lab-wall.html) — adjust inputs, inspect curves, export data and explain limits of the model.