What the calculation gives you
A beam carries a load across a span by bending. Two things decide whether it is good enough: the bending stress, which must stay below what the material allows, and the deflection, which must stay below what the application tolerates. For a given span, load, section and material, the calculation gives both. It is the first check on a shelf, a frame rail, a machine base, a shaft between bearings or a lintel, and it tells you which of the two limits governs, which is more often the deflection than people expect.
Section properties
The section enters through its second moment of area I (often called the moment of inertia) and the distance c from the neutral axis to the outermost fibre. Their ratio is the section modulus Z = I / c.
Rectangle b × h: I = b h3 / 12, c = h / 2
b width · h depth in the direction of bending. Depth counts with the cube: doubling it gives eight times the stiffness
Round tube D outside, d inside: I = π (D4 − d4) / 64, c = D / 2
For a solid round bar put d = 0. For standard rolled sections (I-beams, channels, angles) take I from the manufacturer's table rather than computing it.
Bending stress
σ = M c / I = M / Z
M bending moment at the section considered · σ stress at the outer fibre, tension on one face and compression on the other
The moment depends on the support and the load. For the two cases that cover most of practice:
- Simply supported, uniform total load W over span L: Mmax = W L / 8 at mid-span. With a point load P at the centre, Mmax = P L / 4.
- Cantilever, point load P at the free end of length a: Mmax = P a at the fixed end. With a uniform total load W, Mmax = W a / 2.
Deflection
Simply supported, uniform load: ymax = 5 W L3 / (384 E I)
E modulus of elasticity of the material · W the total load, not the load per unit length; with w per unit length, W = w L
Simply supported, centre point load: ymax = P L3 / (48 E I)
Cantilever, end point load: ymax = P a3 / (3 E I)
The deflection goes with the cube of the span and inversely with E I. The material enters only through E: an aluminium beam of the same section deflects about three times as much as a steel one, whatever the alloy, because E hardly changes with heat treatment. Along the span, the deflected shape of a simply supported beam under uniform load is y(x) = W x (L − x) [L2 + x (L − x)] / (24 E I L), which is what you plot when you want the shape and not only the maximum.
Worked example
The worksheet below is a simply supported span of 43.5 in carrying a total of 400 lbf, spread over the length. It compares two aluminium sections, a 1 in tube with a 0.64 in bore and a 0.25 × 1.25 in flat bar, computing I and Z for each, the stress as a function of position, the stress at mid-span and the maximum deflection of each section. Change the tube wall or the bar depth and see how the two deflections move apart; the bar is barely half the weight but, on edge, it is the stiffer of the two.
Common mistakes
- Total load and load per unit length mixed up. The 5/384 formula takes the total W; with a distributed w it is 5 w L4 / (384 E I). Check the dimensions: the result has to come out as a length.
- Bending about the wrong axis. A flat bar on edge is 25 times stiffer than the same bar laid flat (the ratio of the depths squared). Compute I about the axis the load actually bends it around.
- Mixed units. E in GPa with lengths in inches and loads in lbf cannot be combined by hand; a worksheet that carries units through catches it, a bare number does not.
- Deflection ignored. A beam can be well under the allowable stress and still sag enough to crack a finish, bind a slide or feel unsafe. Common limits are span/360 for floors and span/240 or less for machine frames; precision slides need far less.
- Point load formulas for a distributed load. For the same total load the centre point load deflects 1.6 times more and gives twice the moment.
- Shear and local effects forgotten. Short, deep beams, thin-walled tubes under point loads, and beams with holes or welds near the support need checks the bending formulas do not give.
Questions
- Which support condition do I have?
- A beam resting on two supports it can rotate on is simply supported; a beam built into a wall or bolted down with a stiff flange over a length is close to fixed. Real connections are in between. Simply supported is the conservative choice for deflection and mid-span moment; fixed ends raise the moment at the support instead.
- How do I handle a beam's own weight?
- Add it as a uniform load: the section's weight per unit length times the span. For long, light sections it can be the largest load.
- What safety factor on the stress?
- For static loads on ductile metals a factor of 1.5 to 2 on yield is usual in machine design; structural codes have their own load and resistance factors. Fatigue, impact and buckling of thin flanges need separate checks and bigger margins.
