Engineering calculators
The geometry and mechanics behind a design — worked from first principles, with every formula written out beside the number it produced.
What these calculators work out
A section property is a fact about a shape, not about a material. The same 200 mm box has the same second moment of area whether it is rolled from steel, extruded in aluminium or glued up from timber — which is why these numbers are the first thing a design check needs and the last thing that changes when the specification does.
The area A tells you what carries axial load. The second moment of area, Ix and Iy, tells you what resists bending about each axis, and it is where depth earns its keep: doubling the depth of a rectangle multiplies Ix by eight. The radius of gyration K = √(I/A) recasts that as a length, which is what a buckling check actually consumes.
The elastic section modulus S is the one worth reading carefully. It is I divided by the distance to the furthest fibre, not by half the depth. On a symmetric section those are the same number and the distinction never surfaces; on a tee, an angle or a half circle the centroid sits off-centre and the further face reaches yield first. The plastic modulus Z, taken about the equal-area axis rather than the centroid, is what a section can carry once that first fibre has yielded and the rest catches up. Their ratio Z/S is the shape factor — roughly 1.5 for a solid rectangle, nearer 1.12 for a rolled I-beam.
Choosing a section shape
All 36 outlines sort into four families. Straight-edged shapes — rectangles solid and hollow, squares on the flat and on the diagonal, triangles, trapezoids, hexagons, octagons and any regular polygon you name. Rolled and folded profiles — I-beams equal and unequal, tapered flanges, channels, zeds, tees and angles with equal or unequal legs. Curved shapes — circles solid, hollow and thin-walled, half and quarter circles, sectors, segments, ellipses and elliptical quadrants. And parabolic areas, full and half.
Every one of them reads only the dimensions it needs. A circle ignores the height and width boxes, a solid section ignores the hole, so whatever the other fields happen to hold cannot stop the one you picked from answering.
Frequently asked questions
What is the difference between elastic and plastic section modulus?
The elastic modulus S assumes stress varies linearly across the section and the outermost fibre governs, so S = I divided by the distance from the centroid to the furthest face. The plastic modulus Z assumes the whole section has yielded, so it is the first moment of area about the axis that splits the section into two equal areas — which on an asymmetric shape is a different line from the centroid. Z is always the larger of the two, and their ratio is the shape factor.
Which axis does Ix refer to?
Ix is the second moment of area about the horizontal centroidal axis, so it governs bending that deflects the section up and down — the usual case for a beam. Iy is about the vertical centroidal axis. A subscript 1, as in Ix₁, marks an axis along the outer edge rather than through the centroid; design almost always wants the centroidal value, and the edge value is what you carry into a further parallel-axis shift.
How do you find the centroid of a compound section?
Take the area-weighted average of the parts: multiply each part area by the distance from a common reference line to its own centroid, sum those, and divide by the total area. Subtract rather than add for a hole. Once the centroid is located, the parallel-axis theorem shifts each part inertia to it — I about the new axis equals I about the part own centroid plus that part area times the square of the distance moved.
Do these calculators apply a design code?
No, and deliberately. Everything here is classical geometry — the same closed forms appear in an 1870 textbook and in a modern steel handbook, because the geometry has not changed. No allowable stress, load factor or material grade from any code is applied, so the numbers are valid whatever standard you are working to and cannot go stale when one is revised. The design check itself remains yours.