Stress in a rod:
how much load each square millimetre carries

Stress as load spread over a cross-section

Pull on both ends of a rod and every slice across it carries the whole load. How hard the material works depends on how that load is shared across the slice. A thick rod shares it among many square millimetres; a thin one concentrates it. Stress is the measure of that concentration.

The word 'axial' means the force acts along the rod's length through its centre. In that case the stress is spread evenly across the section and is called normal stress, because it acts perpendicular to the cut surface.

Stress, not force, is what you compare with a material's strength. Two bars of the same steel will fail at very different loads if their cross-sections differ, but at about the same stress.

The equation and a convenient unit trick

The unit shortcut works because a megapascal is one million pascals and a square millimetre is one millionth of a square metre; the factors of a million cancel. It saves converting everything to metres and pascals in routine strength checks.

For a round rod the area comes from its diameter.

Because area scales with diameter squared, a 20 percent increase in diameter lowers stress by about 31 percent for the same load.

Worked example: 12 kN on a 300 mm² section

A tie rod carries a steady tensile load of 12,000 N. Its measured cross-section is 300 mm². Find the axial stress.

Suppose the same load goes through a round rod of diameter 20 mm instead. Its area is π × 20² ÷ 4 = 314.16 mm², giving σ = 12,000 ÷ 314.16 = 38.2 MPa. If the diameter were reduced to 15 mm, the area would drop to 176.7 mm² and the stress would rise to about 67.9 MPa.

If the load is described by a hanging mass instead of a force, convert first: 12,000 N corresponds to a mass of about 1,223 kg under gravity, from 12,000 ÷ 9.81.

How diameter changes stress at the same load

Going from 10 mm to 20 mm quarters the stress, not halves it. That squared relationship is why a modest increase in diameter buys a large safety improvement, and why drilling a hole through a rod, which removes area locally, drives up stress where it matters.

Compare each row against the yield strength of the material. Mild structural steel yields at roughly 250 MPa, but the exact figure depends on grade and specification, so use the value from the material certificate.

What the average stress does not show

• Stress concentrations. Holes, notches, threads and sharp corners raise local stress to a multiple of the average, so the 40 MPa figure may be exceeded in a small zone.
• Bending and misalignment. If the load is not truly along the axis, additional stress appears on one side of the section.
• Buckling. A slender rod in compression can fail by bending long before the stress reaches the material's strength.
• Fatigue. Repeated loading can cause cracks at stresses well below yield.
• Temperature. Strength falls as a metal gets hot; check data at the working temperature.

Working backwards: required area and stretch

Designers usually start from the load and an allowable stress, and then find the size. Rearranging σ = F ÷ A gives A = F ÷ σ_allow. For the 12,000 N tie rod with an allowable stress of 120 MPa, the area needs to be at least 12,000 ÷ 120 = 100 mm².

For a round bar, solve the circle-area formula for diameter: d = √(4 × A ÷ π). With 100 mm² that is √(127.3) = 11.28 mm, so the next standard size up, such as 12 mm, would be chosen.

The elongation follows from Hooke's law, ΔL = σ × L ÷ E. Under the 40 MPa of the main example, a 2 m rod of steel with E = 200,000 MPa stretches by 40 × 2,000 ÷ 200,000 = 0.4 mm. That is small enough that, in most structures, strength rather than stretch governs the size.

• Choose a diameter above the calculated minimum, never below.
• Keep threaded ends in mind, as the root area is smaller than the shank.
• Re-check stress after rounding up, since a bigger rod is also heavier.

Tension, compression and sign

By convention a pulling load gives positive (tensile) stress and a pushing load gives negative (compressive) stress. The magnitude is calculated in the same way, but the failure behaviour is not symmetrical.

Steel is about equally strong in both directions, while concrete and cast iron are much stronger in compression than in tension. That is why concrete is reinforced with steel bars where tension appears, and why a cast-iron column performs better than a cast-iron tie.

A 40 MPa stress with a minus sign means 40 MPa of compression, and then buckling needs to be checked on top of the strength check.

A checklist for your own rod

1. 1State the force in newtons, using weight (mass × 9.81) if it comes from a hanging load.
2. 2Measure or compute the cross-sectional area in mm² at the weakest section, such as the root of a thread.
3. 3Divide force by area to get the stress in MPa.
4. 4Compare with the allowable stress for the material, which should already include a safety margin.
5. 5If the rod is critical to people's safety, have a qualified engineer verify the calculation and the standard that applies.

A circle-area tool takes care of step 2 for round sections, and a force calculator handles the conversion in step 1, leaving only the division to do by hand.