Statics · Centroids
Condition: The theoretical background develops the centre of parallel forces, distributed loads, the centroid of an area and the tabular method on the base plate of the crane post.
Open theoretical foundations →First solve each task the classic way on paper: divide the area into partial areas, set up a table, form the sums. The sub-steps on this page are your check — row by row.
Exercise 1 · Concept check
What holds for the first moments of area with respect to a centroidal axis?
If the origin is placed at the centroid, and zero — and with them the static moments and . That is the most concise characterisation of a centroidal axis.
A square plate has a circular hole. How does the hole enter the centroid calculation?
Cut-outs are regarded as "negative" areas — with area and centroid coordinates, only with a minus sign in both columns of the table. Without the coordinates the calculation would not know where the material is missing.
Match each area to the position of its centroid.
At the intersection of the two axes of symmetry
At one third of the height above the base
Outside the material, in the notch
Not on the line, between the arc and the chord
For the rectangle two axes of symmetry intersect — S lies there. For the triangle it lies at one third of the height. For the L-section it falls into the notch and thus into open space; for the semicircular arc between arc and chord. The last two show: the centroid is a calculation point, not a material point.
Conclusion: one formula, three applications
Forces, areas, lines — always the same structure: sum of the products divided by the sum of the weights. The next two exercises apply it to sections and to an area with a cut-out.
Exercise 2 · Z-section
A section is composed of three rectangles (all dimensions in mm): bottom flange , centred on it a web , on top a flange . The origin lies at the bottom left corner.
No calculation is needed here: the section has a vertical axis of symmetry for , and axes of symmetry are centroidal axes. Hence .
This too was foreseeable: top and bottom flange are equal and lie symmetrically about the height 70 mm — so the section has two axes of symmetry, and the centroid lies at their intersection.
Static moment of the section with respect to the x axis, symbolically from area and centroid height.
Allowed symbols: A, ys
Notation help: * for products.
From follows directly . The dimension length cubed is a good check.
First look for axes of symmetry — each one found saves you a whole coordinate.
For you need, for each rectangle, its area and the height of its centre: 10, 70 and 130 mm.
Conclusion: symmetry first
For this section both could be seen without a table. Looking for axes of symmetry before calculating often saves half the work with composite cross-sections — and avoids half the possible mistakes.
Exercise 3 · Plate with a cut-out
From a square plate of side length a right-angled triangle with legs is cut out at the top right corner. The origin lies at the bottom left corner.
The centroid of a triangle lies at one third of the height above each base. Measured from the right edge that is , so .
The full square would have ; the cut-out pulls the centroid almost 10 mm to the left — away from itself.
What holds for of this remaining area?
The remaining area is symmetric about the diagonal through the origin: square and cut-out both lie symmetrically about it. The diagonal is therefore a centroidal axis, and it follows that — without a second calculation.
Use two rows: the full square positive, the triangle negative.
The centroid of a right-angled triangle lies at one third of each leg — measured from the right angle.
Check the direction at the end: the cut-out must pull the centroid away from itself pull.
Conclusion: calculate negative, think positive
Cut-outs require no new formula, only a sign. And as in exercise 2 it pays to look at the symmetry: the diagonal saved half the calculation here.
Report
The completion status of all three tasks is exported as a PDF report with a verification token — checkable via the platform's report verification.
| Exercise | Status |
|---|---|
| Exercise 1: Concept check | Open |
| Exercise 2: Z-section | Open |
| Exercise 3: Plate with a cut-out | Open |