Composite sections are not recalculated from scratch but assembled from known basic shapes: the overall centroid is the average of the individual centroids weighted with the part areas. Cut-outs count as a negative area. The same method applies unchanged to masses, lengths and volumes.
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Composite sections are not recalculated from scratch but assembled from known basic shapes: the overall centroid is the average of the individual centroids weighted with the part areas. Cut-outs count as a negative area. The same method applies unchanged to masses, lengths and volumes.
An angle section is assembled from two rectangles: A₁ = 2,000 mm² with its centroid at x₁ = 10 mm and A₂ = 1,000 mm² with its centroid at x₂ = 50 mm, both measured from the same edge. The overall centroid lies at xs = (2,000·10 + 1,000·50)/(2,000 + 1,000) = (20,000 + 50,000)/3,000 = 23.33 mm. It therefore lies between 10 and 50 mm and closer to the twice-as-large area A₁ – exactly as expected. Second example with a cut-out: a plate with A₁ = 2,000 mm² and x₁ = 10 mm receives a hole of A₂ = −500 mm² centred at x₂ = 30 mm. Then xs = (20,000 − 15,000)/1,500 = 3.33 mm. The hole therefore shifts the centroid from 10 mm to 3.33 mm, because it removes material on the side with the larger distance.
The composite method is exact and not an approximation, provided the area can be decomposed completely and without overlap into the parts entered. All distances must be counted from the same reference axis with the same sign convention, and all areas must use the same unit. For the centroid in the second direction, repeat the calculation with the y-distances. The calculator handles exactly two parts; for more parts you combine them step by step, first merging two parts into a substitute area and then combining that with the next part. If masses are entered instead of areas the result is the centre of mass – with areas the result holds only when thickness and density are the same across all parts. Second moments of area, which additionally require the parallel-axis theorem, and the centroid of lines or curved surfaces are not covered.
Hardly any cross-section is a plain rectangle. Practically all of them consist of several basic shapes, often with holes or slots. The composite method assembles the overall centroid from the known centroids of those parts – and treats cut-outs simply as a negative area.
The centroid of an area is the point at which it would have to be supported to stay in balance. It is the area-weighted mean of all its points. Decomposing an area into parts with known individual centroids turns the integral into a sum: xs = (A₁x₁ + A₂x₂)/(A₁+A₂). Each part area pulls the overall centroid towards itself in proportion to its size. The key trick for holes and slots: rather than working with the remaining area, subtract the missing material as a negative part area – the formula stays unchanged.
A seesaw with two children of different weight: the pivot must sit closer to the heavier one. Exactly the same ratio fixes the position of the area centroid – the larger part area pulls it towards itself. And a hole acts like a child pulling upwards instead of down.
xs = (A₁·x₁ + A₂·x₂) / (A₁ + A₂)
Two part areas: xs = (A₁x₁ + A₂x₂)/(A₁ + A₂)General form for several parts: xs = Σ(Aᵢxᵢ)/ΣAᵢCut-out: enter A₂ as negative, formula unchangedSecond direction: ys = (A₁y₁ + A₂y₂)/(A₁ + A₂)For masses instead of areas: xs = (m₁x₁ + m₂x₂)/(m₁ + m₂)Rectangle b×h: A = b·h, centroid at mid-heightTriangle: centroid at h/3 above the baseSemicircle: centroid at 4r/(3π) from the diameter| Symbol / input | Meaning |
|---|---|
| Centroid distance xs | The distance of the overall centroid from the chosen reference axis. With two positive part areas the result always lies between x₁ and x₂ and closer to the larger area. With a cut-out, however, it can also fall outside that range, because the negative area pushes the centroid away from itself. The reference axis may be chosen freely – its position only has to be the same for all three distances. |
| Part area A₁ | The area of the first part, i.e. the basic shape the section is assembled from – for example a rectangle with A = b·h or a circle with A = π·d²/4. If masses are entered instead of areas, the same formula gives the centre of mass; using areas presupposes the same thickness and density across all parts. |
| Centroid distance x₁ of part area A₁ | The distance of the first part's centroid from the reference axis – not the distance of an edge. For a rectangle the centroid is at mid-height, for a triangle at one third of the height above the base, for a semicircle at 4r/(3π) from the diameter. All distances must be measured from the same reference axis and counted positive in the same direction. |
| Part area A₂ | The area of the second part, in the same unit as A₁. For a cut-out, hole or slot, enter a negative value here – rather than working with the remaining area, the missing material is subtracted as a negative part area. The magnitude of the negative area must be smaller than A₁, otherwise no area would be left. |
| Centroid distance x₂ of part area A₂ | The distance of the second part's centroid from the same reference axis, likewise with a sign. For a cut-out this is the centroid of the missing material, for instance the centre of the hole – not the centroid of the remaining material. |
A₁ and A₂ are the areas of the two parts in the same unit; for a cut-out enter A₂ as negative. x₁ and x₂ are the distances of the respective part centroids from a freely chosen reference axis – not the distances of edges. For a rectangle the centroid is at mid-height, for a triangle at one third of the height above the base, for a semicircle at 4r/(3π) from the diameter.
First choose a reference axis – usually an outer edge, so that all distances come out positive – and keep it for every entry. Then decompose the area into basic shapes whose centroids are known, making sure the parts complement each other completely and do not overlap. Enter holes and slots as a negative area with the centroid of the missing material. After the calculation in the first direction, repeat it with the distances in the second direction to locate the centroid fully. For more than two parts, combine step by step: merge two parts into a substitute area and combine that with the next part.
An angle section consists of two rectangles: A₁ = 2,000 mm² with its centroid at x₁ = 10 mm and A₂ = 1,000 mm² with its centroid at x₂ = 50 mm, both measured from the same edge. Then xs = (2,000·10 + 1,000·50)/3,000 = 70,000/3,000 = 23.33 mm. The result lies between the two individual values and closer to the twice-as-large area – a good plausibility check. With a cut-out: a plate with A₁ = 2,000 mm² at x₁ = 10 mm receives a hole of A₂ = −500 mm² at x₂ = 30 mm. Then xs = (20,000 − 15,000)/1,500 = 3.33 mm. The centroid moves from 10 mm to 3.33 mm, because the hole removes material on the more distant side – and it therefore falls outside the range between x₁ and x₂, which is normal with negative areas.
With two positive part areas the result always lies between x₁ and x₂, in the inverse ratio of the areas: if A₁ is twice as large as A₂, the centroid sits twice as close to x₁. As soon as a cut-out is involved this bracketing no longer holds – the negative area pushes the centroid away from itself, even beyond the individual values. The overall centroid is the basis of every further section calculation: bending stresses refer to the centroidal axis, and second moments of area are transferred to it with the parallel-axis theorem. An error here therefore propagates into every subsequent calculation.
The calculation runs in square metres and metres internally. Areas may be entered in mm², cm², m², in² or ft² and distances in mm, cm, m or inches – but both areas must use the same unit, and so must both distances. Since only the ratio of the areas enters, the result always carries the unit of the distances. If masses are entered instead of areas, the same formula applies unchanged.
Centroidal axis of composite profiles made of plates, angles and tubes; sections with holes, slots, cut-outs and notches; welded girders from a web plate and flanges; position of the neutral axis for bending-stress calculations; overturning stability and footprint of machines and vessels; centre of mass of assemblies when masses are entered instead of areas.
The method is exact and not an approximation, provided the area can be decomposed completely and without overlap into the parts entered. All distances must be counted from the same reference axis with the same sign convention. The calculator handles exactly two parts; more parts are combined step by step. With areas, the result is the centre of mass only if thickness and density are the same across all parts – for a composite section of different materials, work with the masses or with modulus-weighted areas instead. Second moments of area, which additionally require the parallel-axis theorem, and centroids of lines and curved surfaces are not covered.
Common mistake: The most common error is entering edge distances instead of the part centroid distances – for a rectangle that is half its height off. Second, the distances are measured from different reference axes; the axis may be chosen freely but must be the same for every entry. Third, for a hole people work with the remaining area although its centroid is precisely what is being sought – the correct approach is the negative part area with the centroid of the missing material. And fourth, repeating the calculation for the second direction is forgotten: a centroid has two coordinates.
Centroidal axis of composite profiles made of plates, angles and tubes; sections with holes, slots, cut-outs and notches; welded girders from a web plate and flanges; position of the neutral axis for bending-stress calculations; overturning stability and footprint of machines and vessels; centre of mass of assemblies when masses are entered instead of areas.
A₁ and A₂ are the areas of the two parts in the same unit; for a cut-out enter A₂ as negative. x₁ and x₂ are the distances of the respective part centroids from a freely chosen reference axis – not the distances of edges. For a rectangle the centroid is at mid-height, for a triangle at one third of the height above the base, for a semicircle at 4r/(3π) from the diameter.
The method is exact and not an approximation, provided the area can be decomposed completely and without overlap into the parts entered. All distances must be counted from the same reference axis with the same sign convention. The calculator handles exactly two parts; more parts are combined step by step. With areas, the result is the centre of mass only if thickness and density are the same across all parts – for a composite section of different materials, work with the masses or with modulus-weighted areas instead. Second moments of area, which additionally require the parallel-axis theorem, and centroids of lines and curved surfaces are not covered.