Pick a cross-section, enter its dimensions and get every geometric property used in mechanical and structural design — area, centroid, moments and product of inertia, polar moment, principal axes, elastic and plastic section moduli, radii of gyration and torsion constant — with the section, its centroid and its axes drawn to scale.
Solid round bar or shaft, from the diameter d.
Inputs: d Open calculator →Round tube or pipe, from the outer diameter D and wall thickness t.
Inputs: D, t Open calculator →Solid square bar, from the side a.
Inputs: a Open calculator →Solid rectangular bar or plate, from width b and height h.
Inputs: b, h Open calculator →Square hollow section (SHS / box), from outer side a and wall thickness t.
Inputs: a, t Open calculator →Rectangular hollow section (RHS / box beam), from outer sides b, h and wall thickness t.
Inputs: b, h, t Open calculator →Solid ellipse, from the semi-axes a (along x) and b (along y).
Inputs: a, b Open calculator →Half-round bar (flat side down), from the radius r.
Inputs: r Open calculator →Any triangle — isosceles, right or scalene — from base b, height h and apex offset c.
Inputs: b, h, c Open calculator →Isosceles trapezoid, from top width a, bottom width b and height h.
Inputs: a, b, h Open calculator →Hexagonal bar stock, from the across-flats size s.
Inputs: s Open calculator →Doubly-symmetric I or H beam, from depth h, flange width b, web t_w and flange t_f.
Inputs: h, b, tw, tf Open calculator →T (tee) section with the flange on top, from flange width b, depth h, stem t_w and flange t_f.
Inputs: b, h, tw, tf Open calculator →Equal or unequal angle (L), from vertical leg a, horizontal leg b and thickness t.
Inputs: a, b, t Open calculator →Channel (C / U / UPN-type) with parallel flanges, from depth h, flange width b, web t_w and flange t_f.
Inputs: h, b, tw, tf Open calculator →Cross-sectional area A and the centroid position x̄, ȳ measured from the left and bottom edges — the neutral axis of the section in bending.
Ix, Iy about the centroidal axes, the product of inertia Ixy and the polar moment Ip = Ix + Iy.
Maximum and minimum second moments I1, I2 and the angle θp of the principal axes, drawn on the section when they are rotated.
Elastic S = I/c for each extreme fibre (top, bottom, left, right when they differ) and the plastic modulus Z about both axes.
rx, ry and, for sections with rotated principal axes, r1 and the minimum r2 that governs column buckling.
The torsion constant J where a reliable formula exists (exact, Roark, Bredt or thin-walled) and the shear centre S for T, angle and channel sections.
Inputs in mm, cm, m or inches; results in any of the four, converted with the correct power of the length unit (area ², moduli ³, second moments ⁴).
The second moment of area I = ∫y² dA is a purely geometric property of a cross-section (units of length⁴) that measures how far the material is spread from an axis. Engineers call it the "moment of inertia" of the section by habit, but it has nothing to do with mass: the mass moment of inertia (kg·m²) governs rotational dynamics, the second moment of area governs bending stiffness E·I and bending stress σ = M·y/I.
Because a beam bends about its neutral axis, which for a homogeneous elastic section passes through the centroid. I about any other parallel axis is larger by A·d² (parallel-axis theorem), so the centroidal value is the one that appears in σ = M·y/I and in the deflection formulas. The drawings on every page use centroidal coordinates: the bold gridlines are the centroidal x and y axes.
The principal axes are the pair of perpendicular centroidal axes about which the product of inertia I_xy is zero; they carry the maximum and minimum second moments I_1 and I_2. For sections with an axis of symmetry (rectangle, I, T, channel) they coincide with the x and y axes. For an angle or a scalene triangle they are rotated by the angle θp reported in the results — bending such a section about a leg-parallel axis produces deflection in two directions, and its buckling is governed by the minimum principal radius of gyration.
The elastic section modulus S = I/c gives the bending moment at which the extreme fibre reaches yield, M_y = S·f_y. The plastic section modulus Z is the first moment of the area about the equal-area axis and gives the fully plastic moment M_pl = Z·f_y, when the whole section has yielded. Z is always larger than S; their ratio, the shape factor, is 1.5 for a rectangle, about 1.7 for a circle and 1.10–1.15 for an I beam.
Only for circular sections (solid or hollow). For every other shape the polar moment I_p = I_x + I_y over-estimates the torsional stiffness, because a non-circular section warps when twisted. The true torsion constant J comes from Saint-Venant theory: exact for the ellipse, Roark's fit for the rectangle, Bredt's formula for closed thin-walled boxes, Σb·t³/3 for open thin-walled shapes. The calculators report J separately where a reliable formula exists, and flag its approximation.
Millimetres, centimetres, metres or inches for the inputs, and any of the four for the results, independently. Derived quantities are converted with the right power of the length factor: area with the square, section moduli with the cube, second moments and torsion constant with the fourth power — so a beam entered in millimetres can be read out in cm⁴, the unit of most steel tables.
MechaHandbook has thread charts, tolerances, standard components, tightening torque and unit conversions — all offline, no internet required.
Download the app