Centroids, Second Moments and Composite Cross Sections

Geometric section properties condense cross-sectional geometry into quantities useful for stress and stiffness calculations. At the statics stage they mainly provide distributed-load centroids and axes for follow-on mechanics-of-materials work. A centroid is an area-weighted location: for a homogeneous plate of uniform thickness it matches its center of mass, but for layered materials it does not necessarily locate the mass center. The second moment of area is a fourth-power geometric property, not a physical mass moment of inertia.

Engineering boundary: These exercises analyze idealized statics models for education. They do not certify structural strength, earthquake/wind resistance, geotechnical bearing, connection design, construction safety, or compliance with any jurisdiction’s building code. Real works require local standards, authoritative loading data and qualified professional checking.

Learning outcomes

• Formulate an explicit free-body model for centroids, second moments and composite cross sections.

• Use signed forces, consistent units and geometric measurements to construct equilibrium relations.

• Check results with an independently chosen equilibrium statement or limiting case.

• Distinguish what idealized statics establishes from what structural design and field verification must establish.

Lesson 5.1: Centroids by integration and decomposition

An area centroid is found from first moments of area. Decompose complicated sections into non-overlapping rectangles, triangles and circles; holes enter with negative area. Choose a reference origin and preserve signed coordinates.

Governing relationships

For components Ai at centroids (xi,yi), xbar=Σ(Aixi)/ΣAi and ybar=Σ(Aiyi)/ΣAi. For a rectangle width b and height h, A=bh and its centroid lies at its geometric center.

Worked engineering example

For two adjacent rectangles: 200×20 mm flange centered at y=90 mm and 20×80 mm web centered y=40 mm, ybar=(4000×90+1600×40)/5600=75.714 mm above the bottom.

Model audit and common error

Overlapping area cannot be double-counted. Use consistent millimetre or metre units.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving centroids by integration and decomposition. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For centroids, second moments and composite cross sections, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 5.2: Area second moments and the parallel-axis theorem

Ixx measures the distribution of area about the x axis. Material farther from the reference axis contributes disproportionately, which is why a deep beam section often has greater bending stiffness for the same area.

Governing relationships

For a rectangle about its centroidal horizontal axis, Ixx=bh³/12. About a parallel axis at distance d, I=Icentroid+Ad². A composite section is the sum of each component transformed to the same axis.

Worked engineering example

For a 100 mm × 200 mm rectangle, Ixx=100×200³/12=66.667 million mm⁴. Rotating the section changes this property; swapping b and h gives a different result.

Model audit and common error

The symbol I here denotes area second moment, not mass rotational inertia.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving area second moments and the parallel-axis theorem. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For centroids, second moments and composite cross sections, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 5.3: Products of inertia and coordinate rotation

Unsymmetrical sections may have a nonzero product of area Ixy, coupling orientation and bending response. Principal axes rotate the coordinate system to remove the product term. Reflection symmetry about an axis passing through the centroid makes Ixy zero when evaluated in centroidal coordinates.

Governing relationships

Ixy=∫xy dA. Under coordinate rotation, second-moment and product terms transform together; principal-axis direction can be found from tan(2θ)=−2Ixy/(Ixx−Iyy) under one common sign convention.

Worked engineering example

An L-shaped equal-leg angle does not generally have the same principal axes as the global horizontal and vertical directions, despite having simple orthogonal plate elements.

Model audit and common error

State both your Ixy sign convention and the location of origin; conventions differ across references.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving products of inertia and coordinate rotation. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For centroids, second moments and composite cross sections, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 5.4: Structural section bookkeeping and unit conversions

A change of geometric scale affects area as length squared and area second moment as length to the fourth power. Keeping explicit units is crucial when transferring calculations between architectural millimetres and analysis metres.

Governing relationships

1 m⁴ =10¹² mm⁴; 1 cm⁴=10⁴ mm⁴. For a composite section, Ixx=Σ(Ixx,i,c+Ai(yi−ybar)²).

Worked engineering example

A steel plate 10 mm thick and 100 mm wide has Ixx=100×10³/12=8333.3 mm⁴ about its own horizontal centroidal axis. Placing it 100 mm away from a composite centroid adds 1000×100²=10 million mm⁴.

Model audit and common error

Never add an area in mm² directly to a second moment in mm⁴.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving structural section bookkeeping and unit conversions. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For centroids, second moments and composite cross sections, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Integrated civil case study

T-shaped bridge-girder geometry exercise: model a flange and web, compute total area, centroid and both principal-aligned section second moments. Compare the geometry with an equal-area solid rectangle; do not infer strength without material and limit-state checks.

Worked problem studio

Each problem below is a separate numerical case. Work it first without looking at the answer, then compare the force and moment checks. Changing dimensions can reveal near-singular geometry or support reactions that switch direction.

Worked problem 5.1

Given: A T section has flange 160 mm wide × 16 mm thick, web 18 mm wide and total height 180 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=5512.000 mm². ȳ=(Af yf+Aw yw)/A=123.800 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=17776376.213 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.2

Given: A T section has flange 170 mm wide × 17 mm thick, web 19 mm wide and total height 188 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=6139.000 mm². ȳ=(Af yf+Aw yw)/A=129.752 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=21501277.257 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.3

Given: A T section has flange 180 mm wide × 18 mm thick, web 20 mm wide and total height 196 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=6800.000 mm². ȳ=(Af yf+Aw yw)/A=135.694 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=25777710.431 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.4

Given: A T section has flange 190 mm wide × 19 mm thick, web 21 mm wide and total height 204 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=7495.000 mm². ȳ=(Af yf+Aw yw)/A=141.629 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=30657195.337 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.5

Given: A T section has flange 200 mm wide × 20 mm thick, web 22 mm wide and total height 212 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=8224.000 mm². ȳ=(Af yf+Aw yw)/A=147.556 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=36193539.154 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.6

Given: A T section has flange 210 mm wide × 21 mm thick, web 23 mm wide and total height 220 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=8987.000 mm². ȳ=(Af yf+Aw yw)/A=153.478 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=42442836.554 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.7

Given: A T section has flange 220 mm wide × 22 mm thick, web 24 mm wide and total height 228 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=9784.000 mm². ȳ=(Af yf+Aw yw)/A=159.394 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=49463469.634 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.8

Given: A T section has flange 230 mm wide × 23 mm thick, web 25 mm wide and total height 236 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=10615.000 mm². ȳ=(Af yf+Aw yw)/A=165.305 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=57316107.866 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.9

Given: A T section has flange 240 mm wide × 24 mm thick, web 26 mm wide and total height 244 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=11480.000 mm². ȳ=(Af yf+Aw yw)/A=171.213 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=66063708.060 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.10

Given: A T section has flange 250 mm wide × 25 mm thick, web 27 mm wide and total height 252 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=12379.000 mm². ȳ=(Af yf+Aw yw)/A=177.116 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=75771514.333 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.11

Given: A T section has flange 260 mm wide × 26 mm thick, web 28 mm wide and total height 260 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=13312.000 mm². ȳ=(Af yf+Aw yw)/A=183.016 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=86507058.083 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.12

Given: A T section has flange 270 mm wide × 27 mm thick, web 29 mm wide and total height 268 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=14279.000 mm². ȳ=(Af yf+Aw yw)/A=188.912 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=98340157.977 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Unit assessment (with guidance)

Q1. Find the centroidal height of an inverted T composed of two non-overlapping rectangles. Specify datum.

Guidance: Use the governing equation and a labeled diagram from Unit 5; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q2. A 120 mm × 220 mm rectangle is rotated by 90°. How do its centroidal second moments change?

Guidance: Use the governing equation and a labeled diagram from Unit 5; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q3. Why is a centroidal area second moment measured in mm⁴ rather than mm²?

Guidance: Use the governing equation and a labeled diagram from Unit 5; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q4. Explain when a geometric centroid is not the center of mass of a composite member.

Guidance: Use the governing equation and a labeled diagram from Unit 5; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Interactive experiment

Open [simulation 5](../simulations/simulation-05.html) and vary at least three parameters. Create an observation table from the CSV. Describe one change for which the response is nonlinear, ill-conditioned or physically infeasible.

End-of-unit checklist

• External forces and reactions identified on isolated body.

• Dimensions, datums, load units and sign convention documented.

• Vertical/horizontal resultant and moment balance checked separately.

• Real-world limits and uncertainty recorded before any design interpretation.

Unit 5: Centroids, second moments and composite cross sections

Geometric section properties condense cross-sectional geometry into quantities useful for stress and stiffness calculations. At the statics stage they mainly provide distributed-load centroids and axes for follow-on mechanics-of-materials work. A centroid is an area-weighted location: for a homogeneous plate of uniform thickness it matches its center of mass, but for layered materials it does not necessarily locate the mass center. The second moment of area is a fourth-power geometric property, not a physical mass moment of inertia.

Engineering boundary: These exercises analyze idealized statics models for education. They do not certify structural strength, earthquake/wind resistance, geotechnical bearing, connection design, construction safety, or compliance with any jurisdiction’s building code. Real works require local standards, authoritative loading data and qualified professional checking.

Learning outcomes

• Formulate an explicit free-body model for centroids, second moments and composite cross sections.

• Use signed forces, consistent units and geometric measurements to construct equilibrium relations.

• Check results with an independently chosen equilibrium statement or limiting case.

• Distinguish what idealized statics establishes from what structural design and field verification must establish.

Lesson 5.1: Centroids by integration and decomposition

An area centroid is found from first moments of area. Decompose complicated sections into non-overlapping rectangles, triangles and circles; holes enter with negative area. Choose a reference origin and preserve signed coordinates.

Governing relationships

For components Ai at centroids (xi,yi), xbar=Σ(Aixi)/ΣAi and ybar=Σ(Aiyi)/ΣAi. For a rectangle width b and height h, A=bh and its centroid lies at its geometric center.

Worked engineering example

For two adjacent rectangles: 200×20 mm flange centered at y=90 mm and 20×80 mm web centered y=40 mm, ybar=(4000×90+1600×40)/5600=75.714 mm above the bottom.

Model audit and common error

Overlapping area cannot be double-counted. Use consistent millimetre or metre units.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving centroids by integration and decomposition. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For centroids, second moments and composite cross sections, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 5.2: Area second moments and the parallel-axis theorem

Ixx measures the distribution of area about the x axis. Material farther from the reference axis contributes disproportionately, which is why a deep beam section often has greater bending stiffness for the same area.

Governing relationships

For a rectangle about its centroidal horizontal axis, Ixx=bh³/12. About a parallel axis at distance d, I=Icentroid+Ad². A composite section is the sum of each component transformed to the same axis.

Worked engineering example

For a 100 mm × 200 mm rectangle, Ixx=100×200³/12=66.667 million mm⁴. Rotating the section changes this property; swapping b and h gives a different result.

Model audit and common error

The symbol I here denotes area second moment, not mass rotational inertia.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving area second moments and the parallel-axis theorem. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For centroids, second moments and composite cross sections, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 5.3: Products of inertia and coordinate rotation

Unsymmetrical sections may have a nonzero product of area Ixy, coupling orientation and bending response. Principal axes rotate the coordinate system to remove the product term. Reflection symmetry about an axis passing through the centroid makes Ixy zero when evaluated in centroidal coordinates.

Governing relationships

Ixy=∫xy dA. Under coordinate rotation, second-moment and product terms transform together; principal-axis direction can be found from tan(2θ)=−2Ixy/(Ixx−Iyy) under one common sign convention.

Worked engineering example

An L-shaped equal-leg angle does not generally have the same principal axes as the global horizontal and vertical directions, despite having simple orthogonal plate elements.

Model audit and common error

State both your Ixy sign convention and the location of origin; conventions differ across references.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving products of inertia and coordinate rotation. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For centroids, second moments and composite cross sections, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Lesson 5.4: Structural section bookkeeping and unit conversions

A change of geometric scale affects area as length squared and area second moment as length to the fourth power. Keeping explicit units is crucial when transferring calculations between architectural millimetres and analysis metres.

Governing relationships

1 m⁴ =10¹² mm⁴; 1 cm⁴=10⁴ mm⁴. For a composite section, Ixx=Σ(Ixx,i,c+Ai(yi−ybar)²).

Worked engineering example

A steel plate 10 mm thick and 100 mm wide has Ixx=100×10³/12=8333.3 mm⁴ about its own horizontal centroidal axis. Placing it 100 mm away from a composite centroid adds 1000×100²=10 million mm⁴.

Model audit and common error

Never add an area in mm² directly to a second moment in mm⁴.

Interpretation task

Prepare a free-body sketch of a civil infrastructure element involving structural section bookkeeping and unit conversions. Mark coordinate axes, every restraint action, all known magnitudes and signed reference distances. Explain which connections or soil assumptions would invalidate the model. Report results in both a compact table and an independent equilibrium check.

Engineering discussion

This topic should be evaluated with the full load path rather than by its equation alone. For centroids, second moments and composite cross sections, the quantities that matter are the action itself, where it enters the idealized body, and which physical connection or foundation provides the balancing action. A model that omits one of these produces a formally neat answer without identifying the actual mechanism. Separating gross structural equilibrium from local member strength prevents that confusion. The analyst should first decide what the isolated body is, then measure or state geometry, then apply equilibrium. If an intermediate step changes the sign convention or datum, it should be flagged in the calculation record. Numerical solutions should preserve enough precision for checks but not imply more site accuracy than the input geometry supports.

Integrated civil case study

T-shaped bridge-girder geometry exercise: model a flange and web, compute total area, centroid and both principal-aligned section second moments. Compare the geometry with an equal-area solid rectangle; do not infer strength without material and limit-state checks.

Worked problem studio

Each problem below is a separate numerical case. Work it first without looking at the answer, then compare the force and moment checks. Changing dimensions can reveal near-singular geometry or support reactions that switch direction.

Worked problem 5.1

Given: A T section has flange 160 mm wide × 16 mm thick, web 18 mm wide and total height 180 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=5512.000 mm². ȳ=(Af yf+Aw yw)/A=123.800 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=17776376.213 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.2

Given: A T section has flange 170 mm wide × 17 mm thick, web 19 mm wide and total height 188 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=6139.000 mm². ȳ=(Af yf+Aw yw)/A=129.752 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=21501277.257 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.3

Given: A T section has flange 180 mm wide × 18 mm thick, web 20 mm wide and total height 196 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=6800.000 mm². ȳ=(Af yf+Aw yw)/A=135.694 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=25777710.431 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.4

Given: A T section has flange 190 mm wide × 19 mm thick, web 21 mm wide and total height 204 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=7495.000 mm². ȳ=(Af yf+Aw yw)/A=141.629 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=30657195.337 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.5

Given: A T section has flange 200 mm wide × 20 mm thick, web 22 mm wide and total height 212 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=8224.000 mm². ȳ=(Af yf+Aw yw)/A=147.556 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=36193539.154 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.6

Given: A T section has flange 210 mm wide × 21 mm thick, web 23 mm wide and total height 220 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=8987.000 mm². ȳ=(Af yf+Aw yw)/A=153.478 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=42442836.554 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.7

Given: A T section has flange 220 mm wide × 22 mm thick, web 24 mm wide and total height 228 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=9784.000 mm². ȳ=(Af yf+Aw yw)/A=159.394 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=49463469.634 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.8

Given: A T section has flange 230 mm wide × 23 mm thick, web 25 mm wide and total height 236 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=10615.000 mm². ȳ=(Af yf+Aw yw)/A=165.305 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=57316107.866 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.9

Given: A T section has flange 240 mm wide × 24 mm thick, web 26 mm wide and total height 244 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=11480.000 mm². ȳ=(Af yf+Aw yw)/A=171.213 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=66063708.060 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.10

Given: A T section has flange 250 mm wide × 25 mm thick, web 27 mm wide and total height 252 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=12379.000 mm². ȳ=(Af yf+Aw yw)/A=177.116 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=75771514.333 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.11

Given: A T section has flange 260 mm wide × 26 mm thick, web 28 mm wide and total height 260 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=13312.000 mm². ȳ=(Af yf+Aw yw)/A=183.016 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=86507058.083 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Worked problem 5.12

Given: A T section has flange 270 mm wide × 27 mm thick, web 29 mm wide and total height 268 mm. Find area, bottom-referenced centroid and centroidal Ixx.

Solution: Use non-overlapping flange and web rectangles. A=Af+Aw=14279.000 mm². ȳ=(Af yf+Aw yw)/A=188.912 mm. Add each rectangle’s centroidal Ixx and Ad² terms about the composite centroid to obtain Ixx=98340157.977 mm⁴. Units of Ixx are fourth power of length.

Review check: State the units and verify whether the result reflects the applied load path. Identify one assumption whose failure would require a new analysis.

Unit assessment (with guidance)

Q1. Find the centroidal height of an inverted T composed of two non-overlapping rectangles. Specify datum.

Guidance: Use the governing equation and a labeled diagram from Unit 5; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q2. A 120 mm × 220 mm rectangle is rotated by 90°. How do its centroidal second moments change?

Guidance: Use the governing equation and a labeled diagram from Unit 5; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q3. Why is a centroidal area second moment measured in mm⁴ rather than mm²?

Guidance: Use the governing equation and a labeled diagram from Unit 5; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Q4. Explain when a geometric centroid is not the center of mass of a composite member.

Guidance: Use the governing equation and a labeled diagram from Unit 5; report direction, magnitude, units and model assumptions. Numerical assessment answers should be obtained from the worked methods, not from inspection of figures alone.

Interactive experiment

Open [simulation 5](../simulations/simulation-05.html) and vary at least three parameters. Create an observation table from the CSV. Describe one change for which the response is nonlinear, ill-conditioned or physically infeasible.

End-of-unit checklist

• External forces and reactions identified on isolated body.

• Dimensions, datums, load units and sign convention documented.

• Vertical/horizontal resultant and moment balance checked separately.

• Real-world limits and uncertainty recorded before any design interpretation.

1. [Friction, sliding, overturning and stability](units/unit-06.md)