Slabs, Stairs and Floor Systems

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: slab actions in two dimensions

The simplest slab calculation takes a unit-width strip carrying gravity area pressure. For a simple one-way strip, this converts directly to a line load in kN/m. Its moment is then obtained from beam equilibrium. That calculation provides a good learning baseline because the units are transparent and the moment changes predictably with span. However, it does not show the negative moments near continuous support, torsional corner restraint, concentrated column reactions or the plate action of a two-way system.

Sketch one rectangular floor bay three ways. In the first sketch, support it primarily along two opposing edges and annotate the one-way span. In the second, add substantial support on all four edges and draw conceptual bending in both orthogonal directions. In the third, remove the perimeter beams and carry the slab directly on columns. The panel now demands concentrated slab-to-column force transfer and possible punching near supports. Without a correct analysis method, dividing total load between x and y directions based on a guessed ratio is not acceptable. In an initial model, show candidate column and middle strips but do not present them as code-prescribed widths until those rules have been checked.

Add a rectangular opening near an interior column. The opening removes concrete from potential shear-perimeter paths and changes local bending and reinforcement continuity. Sketch how the critical punching perimeter might be truncated, using the governing code’s treatment rather than the unperforated square formula. If the opening interrupts top steel near a column, new bars may be required around it and must develop their forces. A flat slab that is numerically strong at mid-panel can still have a critical slab–column connection.

Deflection deserves a separate investigation. With gross elastic stiffness, a simple one-way strip has displacement proportional to L⁴/h³ under a fixed uniform load and material modulus. If span grows by 20%, the fourth-power effect is about a factor of 2.07 even before cracking. If concrete cracks significantly, reduced effective stiffness can amplify this response; sustained loads can add creep deflection. Evaluate the relative timing of partition erection and slab support removal, since damage to finishes may arise from differential deflection after completion.

Finally trace diaphragm actions. An ordinary floor slab can also transmit lateral inertia to structural walls and frames. Large openings can interrupt collectors and chords, and slab-to-wall connectors require detail. Create a clear division between the gravity design, slab-column connections, serviceability, fire resistance and diaphragm force paths. The project should not be called complete because a single slab bending check passes.

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Lesson 4.1: One-way slabs and supported strips

Physical meaning and governing ideas

An ideal one-way slab delivers most load to a pair of opposing supports. A one-metre-wide strip provides a useful bending and shear model, but continuity, openings, edge conditions and slab aspect ratio can alter action distribution. Serviceability often sets a minimum viable depth long before a nominal bending calculation becomes difficult.

Analysis and design reasoning

Compute load per metre width correctly: an area pressure of 6 kN/m² on a 1 m strip gives 6 kN/m. Locate positive moment steel near the bottom in the span, negative moment steel near the top above continuous supports, and distribution or temperature reinforcement as applicable. Reinforcing orientation and lap locations must follow the actual moment field.

Worked numerical or geometric illustration

For a 4.0 m simple span with 6 kN/m² area load, a 1 m strip sees 6 kN/m, maximum positive moment 12 kN·m/m and support reaction 12 kN/m. If an uncracked elastic calculation yields very small deflection, test how cracking and creep may multiply that prediction.

Interpretation and failure checks

Do not import beam support continuity rules blindly into slab strips that also participate in diaphragm, punching or transfer actions.

Study task 4.1. 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.

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Lesson 4.2: Two-way slabs and column strips

Physical meaning and governing ideas

A slab supported on four sides can resist load by two-way plate bending. Moment patterns depend on panel shape, support stiffness, edge continuity and column geometry. In flat slabs, transfer into columns produces concentrated shear and unbalanced moment demands. Panel-strip methods are simplified routes only within their defined applicability ranges.

Analysis and design reasoning

Identify panel dimensions, geometry irregularities, drop panels, beams and openings near columns. Distinguish middle strips from column strips, and positive from negative flexural regions. Use a code-permitted analysis such as a finite-element plate model or an eligible simplified method; do not assume a simple Lx/Ly interpolation is a design procedure.

Worked numerical or geometric illustration

A 5×6 m panel under 7 kN/m² total area load carries 210 kN in aggregate before considering self-weight separately. This total load does not determine the reinforcement for both directions without estimating bending distribution and transfer into the supports.

Interpretation and failure checks

Inspect punching and moment transfer in the neighborhood of columns before selecting reinforcement solely from mid-panel flexural estimates.

Study task 4.2. 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.

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Lesson 4.3: Flat slabs and punching shear

Physical meaning and governing ideas

Punching failure can develop around a concentrated support without a large warning flexural deflection. The critical perimeter geometry changes at edge and corner columns and near slab openings. Unbalanced column moments create nonuniform shear stress around the perimeter, so a concentric uniform-stress calculation is only a baseline.

Analysis and design reasoning

Draw the column, critical perimeter, opening truncations and applied slab gravity and lateral reactions. Distinguish design punching demand from allowable concrete and shear reinforcement capacity under the selected code. Slab-column connections in seismic regions may also need drift compatibility and integrity reinforcement even when gravity punching checks pass.

Worked numerical or geometric illustration

For an interior 400 mm square column with effective slab depth d=200 mm, a perimeter at d/2 from each face gives a square side c+d=600 mm and length b0=4×600=2400 mm. A trial 600 kN transferred shear gives average v=600000/(2400×200)=1.25 MPa, without correction for moment transfer or code resistance.

Interpretation and failure checks

A number below a guessed shear strength is not proof of safety. Edge geometry, flexural cracks, slab rotation and reinforcement layout can change punching behavior.

Study task 4.3. 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.

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Lesson 4.4: Stairs, openings and diaphragm roles

Physical meaning and governing ideas

Stair flights may span longitudinally between landings or transversely between side beams. The concrete waist slab geometry, incline and live-load projection determine reactions. Openings remove slab strips, concentrate forces and change local reinforcement requirements. Floor diaphragms also transmit lateral loads, which can govern collectors and connections.

Analysis and design reasoning

Choose the actual flight support arrangement and establish loads on the relevant horizontal or inclined projection consistently. Draw reinforcement continuity through landings and around openings; avoid terminating principal bars at re-entrant corners. If the slab is a diaphragm, model collectors, chords and load transfer separately from gravity flexure.

Worked numerical or geometric illustration

A staircase inclined at 30° to horizontal with a 3 m horizontal run has an inclined length 3/cos30°≈3.464 m. Loads expressed per unit inclined area must be carefully converted if the structural analysis uses horizontal projection; otherwise member demands are misestimated.

Interpretation and failure checks

A structurally credible stair must have complete support and anchorage details, not only a waist-thickness calculation.

Study task 4.4. 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.

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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 4.1

Problem. A simply supported one-way slab strip 1 m wide is subjected to 3.55 kN/m² over a span 3.04 m and has thickness 135 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 3.55 kN/m for a 1 m strip. A simple span gives M=qL²/8=4.101 kN·m/m and V=qL/2=5.396 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.2

Problem. A simply supported one-way slab strip 1 m wide is subjected to 4.10 kN/m² over a span 3.28 m and has thickness 140 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 4.10 kN/m for a 1 m strip. A simple span gives M=qL²/8=5.514 kN·m/m and V=qL/2=6.724 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.3

Problem. A simply supported one-way slab strip 1 m wide is subjected to 4.65 kN/m² over a span 3.52 m and has thickness 145 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 4.65 kN/m for a 1 m strip. A simple span gives M=qL²/8=7.202 kN·m/m and V=qL/2=8.184 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.4

Problem. A simply supported one-way slab strip 1 m wide is subjected to 5.20 kN/m² over a span 3.76 m and has thickness 150 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 5.20 kN/m for a 1 m strip. A simple span gives M=qL²/8=9.189 kN·m/m and V=qL/2=9.776 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.5

Problem. A simply supported one-way slab strip 1 m wide is subjected to 5.75 kN/m² over a span 4.00 m and has thickness 155 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 5.75 kN/m for a 1 m strip. A simple span gives M=qL²/8=11.500 kN·m/m and V=qL/2=11.500 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.6

Problem. A simply supported one-way slab strip 1 m wide is subjected to 6.30 kN/m² over a span 4.24 m and has thickness 160 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 6.30 kN/m for a 1 m strip. A simple span gives M=qL²/8=14.157 kN·m/m and V=qL/2=13.356 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.7

Problem. A simply supported one-way slab strip 1 m wide is subjected to 6.85 kN/m² over a span 4.48 m and has thickness 165 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 6.85 kN/m for a 1 m strip. A simple span gives M=qL²/8=17.185 kN·m/m and V=qL/2=15.344 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.8

Problem. A simply supported one-way slab strip 1 m wide is subjected to 7.40 kN/m² over a span 4.72 m and has thickness 170 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 7.40 kN/m for a 1 m strip. A simple span gives M=qL²/8=20.608 kN·m/m and V=qL/2=17.464 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.1

How can a student recognize that a slab is not appropriately modeled as one-way?

Discussion answer. A slab with meaningful support on all four sides can develop two-dimensional moment patterns, and a slab directly supported on columns introduces concentrated slab-column transfer. Panel aspect ratio alone is not enough to establish the governing behavior when support stiffness, discontinuities and openings are present. Sketch boundaries and reinforcement direction before selecting a code-permitted plate or strip analysis.

Question 4.2

Why does an interior opening near a column require a revised punching check?

Discussion answer. The opening can remove parts of the load-transfer perimeter and change the distribution of shear and unbalanced moment. It may also interrupt reinforcement needed to transmit tension around the column. The critical perimeter and applicable reduction for nearby openings must follow the governing method. A punching check performed as if the slab were solid may be unconservative.

Question 4.3

What does a gross-inertia deflection result fail to include?

Discussion answer. Gross inertia assumes an uncracked concrete section and does not represent reduced stiffness from flexural cracking or nonlinear time effects. Under sustained loading creep and shrinkage may cause added deformation. Support restraint and slab two-way action also change the response. Use it as an initial bound or comparison, not as a reliable long-term serviceability decision by itself.

Question 4.4

Why must staircase loads be related to a consistent projection?

Discussion answer. A stair flight is inclined, but design loads may be given on horizontal projected area or on sloping surface area. Confusing the two changes the resulting load per unit flight length. The structural support direction and landing reactions must also be established. Show geometry, conversion factors and free-body diagrams before calculating flight moments.

Question 4.5

Can the sum of loads on a two-way slab uniquely determine its reinforcement?

Discussion answer. No. A total panel gravity load sets overall equilibrium but not the distribution of positive and negative bending or slab-column shear. Support stiffness, panel geometry and analysis method determine moment fields. Reinforcement placement additionally follows crack, development, punching and detailing requirements. A complete model must connect actions to specific strips and locations.

Unit project and experiment

Analyze a 6×6 m flat slab panel supported at four columns, with an opening near one column and a specified imposed load. Prepare a plan with column strip and middle strip candidates. Discuss alternative two-way analysis methods and why unsupported bending-strip allocations are not trustworthy. Calculate gravity load totals and a first-pass concentric punching stress for an unperforated interior-column case, then sketch the truncated perimeter caused by the opening and discuss altered demand. Indicate top steel areas near column regions and bottom steel zones only after selecting an appropriate code model. Add serviceability, fire, progressive-collapse integrity and construction-stage considerations. Present a drawing with noted uncertainties.

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

• [One-way slab span and deflection laboratory](../labs/lab-slab.html) — adjust inputs, inspect curves, export data and explain limits of the model.

Unit 4: Slabs, stairs and floor systems

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: slab actions in two dimensions

The simplest slab calculation takes a unit-width strip carrying gravity area pressure. For a simple one-way strip, this converts directly to a line load in kN/m. Its moment is then obtained from beam equilibrium. That calculation provides a good learning baseline because the units are transparent and the moment changes predictably with span. However, it does not show the negative moments near continuous support, torsional corner restraint, concentrated column reactions or the plate action of a two-way system.

Sketch one rectangular floor bay three ways. In the first sketch, support it primarily along two opposing edges and annotate the one-way span. In the second, add substantial support on all four edges and draw conceptual bending in both orthogonal directions. In the third, remove the perimeter beams and carry the slab directly on columns. The panel now demands concentrated slab-to-column force transfer and possible punching near supports. Without a correct analysis method, dividing total load between x and y directions based on a guessed ratio is not acceptable. In an initial model, show candidate column and middle strips but do not present them as code-prescribed widths until those rules have been checked.

Add a rectangular opening near an interior column. The opening removes concrete from potential shear-perimeter paths and changes local bending and reinforcement continuity. Sketch how the critical punching perimeter might be truncated, using the governing code’s treatment rather than the unperforated square formula. If the opening interrupts top steel near a column, new bars may be required around it and must develop their forces. A flat slab that is numerically strong at mid-panel can still have a critical slab–column connection.

Deflection deserves a separate investigation. With gross elastic stiffness, a simple one-way strip has displacement proportional to L⁴/h³ under a fixed uniform load and material modulus. If span grows by 20%, the fourth-power effect is about a factor of 2.07 even before cracking. If concrete cracks significantly, reduced effective stiffness can amplify this response; sustained loads can add creep deflection. Evaluate the relative timing of partition erection and slab support removal, since damage to finishes may arise from differential deflection after completion.

Finally trace diaphragm actions. An ordinary floor slab can also transmit lateral inertia to structural walls and frames. Large openings can interrupt collectors and chords, and slab-to-wall connectors require detail. Create a clear division between the gravity design, slab-column connections, serviceability, fire resistance and diaphragm force paths. The project should not be called complete because a single slab bending check passes.

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Lesson 4.1: One-way slabs and supported strips

Physical meaning and governing ideas

An ideal one-way slab delivers most load to a pair of opposing supports. A one-metre-wide strip provides a useful bending and shear model, but continuity, openings, edge conditions and slab aspect ratio can alter action distribution. Serviceability often sets a minimum viable depth long before a nominal bending calculation becomes difficult.

Analysis and design reasoning

Compute load per metre width correctly: an area pressure of 6 kN/m² on a 1 m strip gives 6 kN/m. Locate positive moment steel near the bottom in the span, negative moment steel near the top above continuous supports, and distribution or temperature reinforcement as applicable. Reinforcing orientation and lap locations must follow the actual moment field.

Worked numerical or geometric illustration

For a 4.0 m simple span with 6 kN/m² area load, a 1 m strip sees 6 kN/m, maximum positive moment 12 kN·m/m and support reaction 12 kN/m. If an uncracked elastic calculation yields very small deflection, test how cracking and creep may multiply that prediction.

Interpretation and failure checks

Do not import beam support continuity rules blindly into slab strips that also participate in diaphragm, punching or transfer actions.

Study task 4.1. 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.

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Lesson 4.2: Two-way slabs and column strips

Physical meaning and governing ideas

A slab supported on four sides can resist load by two-way plate bending. Moment patterns depend on panel shape, support stiffness, edge continuity and column geometry. In flat slabs, transfer into columns produces concentrated shear and unbalanced moment demands. Panel-strip methods are simplified routes only within their defined applicability ranges.

Analysis and design reasoning

Identify panel dimensions, geometry irregularities, drop panels, beams and openings near columns. Distinguish middle strips from column strips, and positive from negative flexural regions. Use a code-permitted analysis such as a finite-element plate model or an eligible simplified method; do not assume a simple Lx/Ly interpolation is a design procedure.

Worked numerical or geometric illustration

A 5×6 m panel under 7 kN/m² total area load carries 210 kN in aggregate before considering self-weight separately. This total load does not determine the reinforcement for both directions without estimating bending distribution and transfer into the supports.

Interpretation and failure checks

Inspect punching and moment transfer in the neighborhood of columns before selecting reinforcement solely from mid-panel flexural estimates.

Study task 4.2. 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.

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Lesson 4.3: Flat slabs and punching shear

Physical meaning and governing ideas

Punching failure can develop around a concentrated support without a large warning flexural deflection. The critical perimeter geometry changes at edge and corner columns and near slab openings. Unbalanced column moments create nonuniform shear stress around the perimeter, so a concentric uniform-stress calculation is only a baseline.

Analysis and design reasoning

Draw the column, critical perimeter, opening truncations and applied slab gravity and lateral reactions. Distinguish design punching demand from allowable concrete and shear reinforcement capacity under the selected code. Slab-column connections in seismic regions may also need drift compatibility and integrity reinforcement even when gravity punching checks pass.

Worked numerical or geometric illustration

For an interior 400 mm square column with effective slab depth d=200 mm, a perimeter at d/2 from each face gives a square side c+d=600 mm and length b0=4×600=2400 mm. A trial 600 kN transferred shear gives average v=600000/(2400×200)=1.25 MPa, without correction for moment transfer or code resistance.

Interpretation and failure checks

A number below a guessed shear strength is not proof of safety. Edge geometry, flexural cracks, slab rotation and reinforcement layout can change punching behavior.

Study task 4.3. 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.

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Lesson 4.4: Stairs, openings and diaphragm roles

Physical meaning and governing ideas

Stair flights may span longitudinally between landings or transversely between side beams. The concrete waist slab geometry, incline and live-load projection determine reactions. Openings remove slab strips, concentrate forces and change local reinforcement requirements. Floor diaphragms also transmit lateral loads, which can govern collectors and connections.

Analysis and design reasoning

Choose the actual flight support arrangement and establish loads on the relevant horizontal or inclined projection consistently. Draw reinforcement continuity through landings and around openings; avoid terminating principal bars at re-entrant corners. If the slab is a diaphragm, model collectors, chords and load transfer separately from gravity flexure.

Worked numerical or geometric illustration

A staircase inclined at 30° to horizontal with a 3 m horizontal run has an inclined length 3/cos30°≈3.464 m. Loads expressed per unit inclined area must be carefully converted if the structural analysis uses horizontal projection; otherwise member demands are misestimated.

Interpretation and failure checks

A structurally credible stair must have complete support and anchorage details, not only a waist-thickness calculation.

Study task 4.4. 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.

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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 4.1

Problem. A simply supported one-way slab strip 1 m wide is subjected to 3.55 kN/m² over a span 3.04 m and has thickness 135 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 3.55 kN/m for a 1 m strip. A simple span gives M=qL²/8=4.101 kN·m/m and V=qL/2=5.396 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.2

Problem. A simply supported one-way slab strip 1 m wide is subjected to 4.10 kN/m² over a span 3.28 m and has thickness 140 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 4.10 kN/m for a 1 m strip. A simple span gives M=qL²/8=5.514 kN·m/m and V=qL/2=6.724 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.3

Problem. A simply supported one-way slab strip 1 m wide is subjected to 4.65 kN/m² over a span 3.52 m and has thickness 145 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 4.65 kN/m for a 1 m strip. A simple span gives M=qL²/8=7.202 kN·m/m and V=qL/2=8.184 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.4

Problem. A simply supported one-way slab strip 1 m wide is subjected to 5.20 kN/m² over a span 3.76 m and has thickness 150 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 5.20 kN/m for a 1 m strip. A simple span gives M=qL²/8=9.189 kN·m/m and V=qL/2=9.776 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.5

Problem. A simply supported one-way slab strip 1 m wide is subjected to 5.75 kN/m² over a span 4.00 m and has thickness 155 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 5.75 kN/m for a 1 m strip. A simple span gives M=qL²/8=11.500 kN·m/m and V=qL/2=11.500 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.6

Problem. A simply supported one-way slab strip 1 m wide is subjected to 6.30 kN/m² over a span 4.24 m and has thickness 160 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 6.30 kN/m for a 1 m strip. A simple span gives M=qL²/8=14.157 kN·m/m and V=qL/2=13.356 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.7

Problem. A simply supported one-way slab strip 1 m wide is subjected to 6.85 kN/m² over a span 4.48 m and has thickness 165 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 6.85 kN/m for a 1 m strip. A simple span gives M=qL²/8=17.185 kN·m/m and V=qL/2=15.344 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.8

Problem. A simply supported one-way slab strip 1 m wide is subjected to 7.40 kN/m² over a span 4.72 m and has thickness 170 mm. Compute illustrative shear and moment per unit strip and explain why continuity changes reinforcement locations.

Worked resolution. The line load is 7.40 kN/m for a 1 m strip. A simple span gives M=qL²/8=20.608 kN·m/m and V=qL/2=17.464 kN/m. If the span increases 10% with pressure fixed, the simple-span moment increases by 1.1²−1=21%. Thickness influences stiffness roughly with its cube under uncracked assumptions.

Engineering interpretation. Separate gravity strength from crack control and long-term deflection. A two-way panel supported on four edges cannot be certified by duplicating this one-dimensional strip method in two directions.

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 4.1

How can a student recognize that a slab is not appropriately modeled as one-way?

Discussion answer. A slab with meaningful support on all four sides can develop two-dimensional moment patterns, and a slab directly supported on columns introduces concentrated slab-column transfer. Panel aspect ratio alone is not enough to establish the governing behavior when support stiffness, discontinuities and openings are present. Sketch boundaries and reinforcement direction before selecting a code-permitted plate or strip analysis.

Question 4.2

Why does an interior opening near a column require a revised punching check?

Discussion answer. The opening can remove parts of the load-transfer perimeter and change the distribution of shear and unbalanced moment. It may also interrupt reinforcement needed to transmit tension around the column. The critical perimeter and applicable reduction for nearby openings must follow the governing method. A punching check performed as if the slab were solid may be unconservative.

Question 4.3

What does a gross-inertia deflection result fail to include?

Discussion answer. Gross inertia assumes an uncracked concrete section and does not represent reduced stiffness from flexural cracking or nonlinear time effects. Under sustained loading creep and shrinkage may cause added deformation. Support restraint and slab two-way action also change the response. Use it as an initial bound or comparison, not as a reliable long-term serviceability decision by itself.

Question 4.4

Why must staircase loads be related to a consistent projection?

Discussion answer. A stair flight is inclined, but design loads may be given on horizontal projected area or on sloping surface area. Confusing the two changes the resulting load per unit flight length. The structural support direction and landing reactions must also be established. Show geometry, conversion factors and free-body diagrams before calculating flight moments.

Question 4.5

Can the sum of loads on a two-way slab uniquely determine its reinforcement?

Discussion answer. No. A total panel gravity load sets overall equilibrium but not the distribution of positive and negative bending or slab-column shear. Support stiffness, panel geometry and analysis method determine moment fields. Reinforcement placement additionally follows crack, development, punching and detailing requirements. A complete model must connect actions to specific strips and locations.

Unit project and experiment

Analyze a 6×6 m flat slab panel supported at four columns, with an opening near one column and a specified imposed load. Prepare a plan with column strip and middle strip candidates. Discuss alternative two-way analysis methods and why unsupported bending-strip allocations are not trustworthy. Calculate gravity load totals and a first-pass concentric punching stress for an unperforated interior-column case, then sketch the truncated perimeter caused by the opening and discuss altered demand. Indicate top steel areas near column regions and bottom steel zones only after selecting an appropriate code model. Add serviceability, fire, progressive-collapse integrity and construction-stage considerations. Present a drawing with noted uncertainties.

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

• [One-way slab span and deflection laboratory](labs/lab-slab.html) — adjust inputs, inspect curves, export data and explain limits of the model.